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Higher genus capillary surfaces in the unit ball of R3

Filippo Morabito

Author Affiliations

Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, 291 Daehak-ro, Yuseong-gu, Daejeon, South Korea

School of Mathematics, Korea Institute for Advanced Study, 87 Hoegi-ro, Cheongryangry 2-dong, Seoul, South Korea

Boundary Value Problems 2014, 2014:130  doi:10.1186/1687-2770-2014-130


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/130


Received:16 December 2013
Accepted:30 April 2014
Published:22 May 2014

© 2014 Morabito; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Abstract

We construct the first examples of capillary surfaces of positive genus, embedded in the unit ball of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M2">View MathML</a> with vanishing mean curvature and locally constant contact angles along their three boundary curves. These surfaces come in families depending on one parameter and they converge to the triple equatorial disk. Such surfaces are obtained by deforming the Costa-Hoffman-Meeks minimal surfaces.

MSC: 53A10, 35R35, 53C21.

Keywords:
minimal surface; perturbation method; nonlinear pde’s

1 Introduction

The study of capillarity started in the beginning of the 19th century by the work of PS de Laplace and T Young. They considered a liquid contained in a vertical tube of small radius dipped in a reservoir and studied the shape of the free surface interface between the liquid and the air. Such a surface is called capillary surface. More generally a capillary surface is the surface interface between a liquid situated adjacent to another immiscible liquid or gas.

PS de Laplace proved that the height u of a capillary surface over a domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M3">View MathML</a> satisfies the differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M4">View MathML</a>

(1)

where H is the mean curvature, λ is a constant to be determined by physical condition (volume of the fluid and boundary conditions) and k is positive (resp. negative) when denser fluid lies below (resp. above) the interface.

T Young, who considered the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M5">View MathML</a>, understood that the capillary surface meets the tube (or more generally the container) making an angle, called contact angle, which depends on the liquid and on the material which composes the container and not on the gravity. For liquids in tubes (i.e. cylindrical containers) we see that the following additional boundary condition (Young condition) is satisfied:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M6">View MathML</a>

Here ν is the unit normal vector to the tube along the boundary of the surface. It says that the capillary surface meets the tube in a constant contact angle (equal to α). See Finn [1], for a survey on more recent discoveries about capillarity.

Existence and uniqueness for the solution of capillarity problem for graphs over domains of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M7">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M8">View MathML</a> (also in the more general form where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M9">View MathML</a>, for an assigned function f), has been extensively studied in the past, see e.g. Gerhardt [2], Lieberman [3], Simon and Spruck [4], Spruck [5], Uraltseva [6].

A more recent series of works (see e.g.[7-9]) deals with the existence and regularity of capillary graphs with constant mean curvature in vertical cylinders containing corners or cusps. Huff and McCuan [10] showed the existence of Scherk-type capillary minimal graphs.

Very recently, Calle and Shahriyari in [11] have solved the prescribed mean curvature equation with a boundary contact angle condition. They show the existence of graphs over domains in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M10">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M11">View MathML</a> is a n-dimensional Riemannian submanifold of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M12">View MathML</a>. In [12] Lira and Wanderley show the existence of Killing graphs with prescribed mean curvature and prescribed contact angle along their boundary in a wide class of Riemannian manifolds endowed with a Killing vector field.

Fall and Mercuri in [13] constructed by a perturbation method disk-type minimal surfaces embedded in an infinite cylinder in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M2">View MathML</a> and which intersect its boundary orthogonally. In [14] they extended this result to Riemannian manifolds.

In [15] Fall showed that, given a bounded domain of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M2">View MathML</a> there exist embedded constant mean curvature (cmc) surfaces contained in Ω and whose boundary intersects Ω orthogonally. Also he showed that, given a stable stationary point p for the mean curvature of Ω, there exists near p a family of embedded surfaces with cmc equal to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M15">View MathML</a>, which, after scaling and translation, converges to a hemisphere of radius 1 as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M16">View MathML</a>.

In [16] Fall and Mahmoudi showed that if Ω is a domain of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M17">View MathML</a> and K a k-dimensional non-degenerate minimal submanifold, then there exists a family of embedded constant mean curvature hypersurfaces which, as their mean curvature tends to infinity, concentrate along K and intersect Ω orthogonally.

In this work we show the existence of higher genus minimal capillary surfaces by a perturbation method. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M18">View MathML</a> be the unit ball centered at the origin of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M2">View MathML</a>. For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M21">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M22">View MathML</a> small enough, there exists a surface <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M23">View MathML</a> of genus k, embedded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M18">View MathML</a> with non-empty boundary which consists in three simple closed curves <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M27">View MathML</a> which lie in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M28">View MathML</a> and such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M29">View MathML</a>

(2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M30">View MathML</a> denotes the mean curvature at the point p; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M31">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M32">View MathML</a> denote, respectively, the unit normal vector to the surface <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M23">View MathML</a> and to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M28">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M35">View MathML</a>. The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M36">View MathML</a> are decreasing smooth and non-zero for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M21">View MathML</a>. We will describe them below.

The solution of the previous system is based on the deformation of a compact piece of a scaled Costa-Hoffman-Meeks minimal surface contained in the unit ball. More precisely we consider the image by a homothety of ratio τ. Such a surface is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38">View MathML</a>. As we will explain in Section 2.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38">View MathML</a> is asymptotic to a top half catenoid, to a bottom half catenoid and to a horizontal plane. The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M40">View MathML</a> are defined to be the values of the scalar product <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M41">View MathML</a> we obtain if we replace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M23">View MathML</a> by the two halves catenoid and the plane. In particular <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M43">View MathML</a>.

We provide the first examples of capillary type surfaces with non-trivial topology, having vanishing mean curvature and locally constant contact angles with the sphere. They are equal to the contact angles made by the asymptotic catenoids and the plane described above with the sphere. Such surfaces are obtained by deformation of minimal surfaces by a function in the space described by Definition 2.1.

Here is the statement of the result we get. The cartesian coordinates in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M2">View MathML</a> are denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M45">View MathML</a>.

Theorem 1.1For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M20">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M47">View MathML</a>positive and small enough, such that for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M21">View MathML</a>there exists a surface<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M23">View MathML</a>embedded in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M18">View MathML</a>, of genusk, whose boundary<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M51">View MathML</a>is composed by three simple Jordan curves<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M27">View MathML</a>and satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M55">View MathML</a>

(3)

Such surfaces are invariant under the action of the rotation of angle<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M56">View MathML</a>about the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M57">View MathML</a>-axis, under the action of the reflection in the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M58">View MathML</a>plane and under the action of the composition of a rotation of angle<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M59">View MathML</a>about the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M57">View MathML</a>-axis and the reflection in the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M61">View MathML</a>plane.

We observe that for values of τ in the range of validity of our theorem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M62">View MathML</a>. In other terms the surface cannot make a constant angle equal to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M63">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M28">View MathML</a> along <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M25">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M27">View MathML</a>. We point out that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M67">View MathML</a>. As τ is the homothety ratio, this says that, as τ tends to 0 the limit of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M23">View MathML</a> consists in the triple equatorial disk.

The proof can easily be modified in order to handle the case of capillary surfaces with boundary on a vertical cylinder.

Among the works dealing with capillary surfaces in a ball we cite [17] by Ros and Souam. They showed that a stable minimal capillary surface (that is, stationary surfaces with non-negative second variation of the area) in a ball of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M2">View MathML</a> is a totally geodesic disk or a surfaces of genus 1 with boundary having at most 3 connected components. Consequently, at least for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M70">View MathML</a>, the surfaces described by Theorem 1.1 are unstable.

The interest in capillary surfaces in the unit ball has been rekindled by the recent works of Fraser and Schoen [18,19]. They considered free boundary minimal surfaces embedded in the unit ball of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M7">View MathML</a>, i.e. surfaces which meet orthogonally the boundary of the ball.

Free boundary minimal submanifolds are critical for the problem of extremizing the volume among deformations which preserve the ball. Such solutions arise from variational min/max constructions, and examples include equatorial disks, the (critical) catenoid, as well as the cone over any minimal submanifold of the sphere. If Σ is a compact Riemannian surface with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M72">View MathML</a> then the Dirichlet-to-Neumann operator maps a function u on Σ to the normal derivative of the harmonic extension of u to the interior. A submanifold properly immersed in the unit ball is a free boundary submanifold if and only if its coordinate functions are Steklov eigenfunctions with eigenvalue 1. Using this characterization they prove the existence of free boundary minimal surfaces in the unit ball of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M2">View MathML</a> of genus 0 with boundary having k connected components, for any finite <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M74">View MathML</a>. The authors conjecture the existence of higher genus examples of free boundary embedded minimal surfaces which have three boundary components and converge to the union of the critical vertical catenoid and the equatorial disk.

The minimal surfaces described in Theorem 1.1 come in 1-parameter families, they have finite genus ≥1, they meet orthogonally the boundary of the ball only along the middle boundary curve. Furthermore, for any value of the genus, the limit for values of the parameter close to zero consists in the triple equatorial disk.

2 Preliminaries

The proof of the existence of solutions of the capillarity type problem is based on the deformation of a compact piece of the minimal surfaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38">View MathML</a>. We describe this family of surfaces in Section 2.1.

We will show that it is possible to deform a surface Σ in this family in order to get a surface satisfying (3). More precisely we will prove the existence of a function u defined on Σ and of small norm such that its normal graph <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M76">View MathML</a> over Σ has vanishing mean curvature and the scalar product of the unit normal vectors, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M77">View MathML</a>, equals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M78">View MathML</a> at each point of the ith component of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M79">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M80">View MathML</a>.

We will adapt to our setting some arguments used in [20,21].

2.1 The scaled Costa-Hoffman-Meeks surface

The Costa-Hoffman-Meeks surface of genus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M20">View MathML</a> embedded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M2">View MathML</a> (see [22]) is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M83">View MathML</a>.

After suitable rotation and translation, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M83">View MathML</a> enjoys the following properties.

1. It has one planar end <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M85">View MathML</a> asymptotic to the horizontal plane <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M86">View MathML</a>, one top end <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M87">View MathML</a> and one bottom end <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M88">View MathML</a> that are, respectively, asymptotic to the upper end and to the lower end of a catenoid having the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M57">View MathML</a>-axis as axis of rotation. The planar end <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M85">View MathML</a> is located between the two catenoidal ends.

2. It is invariant under the action of the rotation of angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M56">View MathML</a> about the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M57">View MathML</a>-axis, under the action of the reflection in the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M58">View MathML</a> plane and under the action of the composition of a rotation of angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M59">View MathML</a> about the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M57">View MathML</a>-axis and the reflection in the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M61">View MathML</a> plane.

3. It intersects the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M61">View MathML</a> plane in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M98">View MathML</a> straight lines, which intersect themselves at the origin with angles equal to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M59">View MathML</a>. The intersection of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M83">View MathML</a> with the plane <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M101">View MathML</a> (≠0) is a single Jordan curve. The intersection of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M83">View MathML</a> with the upper half space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M103">View MathML</a> (resp. with the lower half space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M104">View MathML</a>) is topologically an open annulus.

The parameterization of the end <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M105">View MathML</a> is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M106">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M107">View MathML</a>, and the parameterization of the corresponding end <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M108">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38">View MathML</a> is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M110">View MathML</a>. We recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38">View MathML</a> is the image of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M83">View MathML</a> by the homothety of ratio τ.

Now we provide a local description of the surface <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38">View MathML</a> near its ends and we introduce the coordinates that we will use.

2.2 The planar end

The planar end <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M114">View MathML</a> of the surface <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38">View MathML</a> can be parametrized by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M116">View MathML</a>

(4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M117">View MathML</a>. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M118">View MathML</a> is fixed small enough. In the sequel, where necessary, we will consider on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M119">View MathML</a> also the polar coordinates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M120">View MathML</a>. The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M121">View MathML</a> satisfies the minimal surface equation, which has the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M122">View MathML</a>

(5)

It can be shown (see [20]) that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M121">View MathML</a> can be extended at the origin continuously by using Weierstrass representation. In particular we can prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M124">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M125">View MathML</a>, where the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M126">View MathML</a> denotes a function that, together with its partial derivatives of order less than or equal to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M127">View MathML</a> is bounded by a constant times g. Furthermore, by taking into account the symmetries of the surface, it is possible to show the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M121">View MathML</a>, in polar coordinates, has to be collinear to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M129">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M130">View MathML</a> and odd.

2.3 The catenoidal ends

The parametrization of the standard catenoid C, whose axis of revolution is the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M57">View MathML</a>-axis, is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M132">View MathML</a>. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M133">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M134">View MathML</a>. The unit normal vector field to C is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M135">View MathML</a>

(6)

The catenoid C may be divided in two pieces, denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M136">View MathML</a>, which are defined as the image by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M132">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M138">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M139">View MathML</a>, we define the catenoid <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M140">View MathML</a> as the image of C by a homothety of ratio τ. Its parametrization is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M141">View MathML</a>. Of course, by this transformation, the two surfaces correspond to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M136">View MathML</a>. They are denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M143">View MathML</a>.

Up to some dilation, we can assume that the two ends <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M144">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M145">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38">View MathML</a> are asymptotic to some translated copy of the two halves of the catenoid parametrized by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M147">View MathML</a> in the vertical direction. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M144">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M145">View MathML</a> can be parametrized, respectively, by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M150">View MathML</a>

(7)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M151">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M152">View MathML</a>

(8)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M153">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M154">View MathML</a>, functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M155">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M156">View MathML</a> tend exponentially fast to 0 as s goes to ±∞ reflecting the fact that the ends are asymptotic to a catenoidal end. More precisely it is known that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M157">View MathML</a>. Furthermore, taking into account the symmetries of the surface, it is easy to show the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M155">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M156">View MathML</a>, in terms of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M160">View MathML</a> coordinates, have to be collinear to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M161">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M162">View MathML</a> and must satisfy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M163">View MathML</a>. Furthermore we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M164">View MathML</a>. In the sequel we will omit the indices t, b and we will use the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M165">View MathML</a>. We assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M166">View MathML</a>, κ being a constant.

For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M167">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M168">View MathML</a>, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M169">View MathML</a>

(9)

The parametrizations of the three ends of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38">View MathML</a> induce a decomposition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38">View MathML</a> into slightly overlapping components: a compact piece <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M172">View MathML</a> and three noncompact pieces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M173">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M174">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M175">View MathML</a>.

We define a weighted space of functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38">View MathML</a>.

Definition 2.1 Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M177">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M178">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M179">View MathML</a>, the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M180">View MathML</a> is defined to be the space of functions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M181">View MathML</a> for which the following norm is finite:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M182">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M183">View MathML</a>

and which are invariant with respect to the reflection in the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M184">View MathML</a> plane, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M185">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M186">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M187">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M188">View MathML</a>, invariant with respect to a rotation of angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M56">View MathML</a> about the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M57">View MathML</a> axis and to the composition of a rotation of angle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M59">View MathML</a> about the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M57">View MathML</a> axis and the reflection in the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M86">View MathML</a> plane.

We remark that there is no weight on the middle end. In fact we compactify this end and we consider a weighted space of functions defined on a two ended surface.

The proof of Theorem 1.1 consists of two steps. Firstly we will show that for each choice of the genus k there exists, for τ sufficiently small, a family of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M194">View MathML</a> such that their normal graph over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38">View MathML</a> satisfies the first equation in (3). To do that we need to find the expression of the mean curvature operator for normal graphs of functions defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38">View MathML</a>. This is the aim of following section. Secondly we prove that in the family of solutions described above there is a function satisfying also the capillarity condition in (3).

3 The mean curvature of a graph over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38">View MathML</a>

It is well known that the mean curvature <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M198">View MathML</a> of the normal graph of a function u over a minimal surface Σ can be decomposed as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M199">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M200">View MathML</a> denotes a linear second order elliptic operator and Q is a nonlinear differential operator of higher order. The operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M200">View MathML</a> is known under the name of Jacobi operator and it is defined as the linearized of the mean curvature operator. For a minimal surface Σ in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M2">View MathML</a> its expression is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M203">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M204">View MathML</a> denotes the Laplace-Beltrami operator and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M205">View MathML</a> is the norm of the second fundamental form on the surface.

As for the majority of minimal surfaces, unfortunately the explicit expression of the mean curvature operator of the Costa-Hoffman-Meeks surfaces is not known. The knowledge of the geometric behavior of such surfaces (we recall that their ends are asymptotic to the two halves of a catenoid and to a plane) allows us to get information about the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M206">View MathML</a> and more generally of the mean curvature operator at the ends of the surfaces.

3.1 Mean curvature operator at the catenoidal ends

The surface parametrized by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M207">View MathML</a> is minimal if and only if the function w satisfies the minimal surface equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M208">View MathML</a>

(10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M209">View MathML</a> being the Jacobi operator of the catenoid, i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M210">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M211">View MathML</a>

(11)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M212">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M213">View MathML</a> are nonlinear second order differential operators which are bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M214">View MathML</a>, for every l, and satisfy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M215">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M216">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M217">View MathML</a> together with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M218">View MathML</a>

(12)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M219">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M220">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M221">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M222">View MathML</a>. The positive constant c does not depend on s.

Finally we observe that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M223">View MathML</a> maps the functional space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M224">View MathML</a>

3.2 Mean curvature operator at the planar end

If we linearize the nonlinear equation (5) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M225">View MathML</a>

(13)

If we consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M226">View MathML</a> we get an operator which equals, up to a multiplication by τ, the Jacobi operator of the plane, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M227">View MathML</a>. The graph surface of the function u is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M228">View MathML</a> and its mean curvature by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M198">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M230">View MathML</a>, the mean curvature of the graph of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M231">View MathML</a>, in terms of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M198">View MathML</a>, is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M233">View MathML</a>

(14)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M234">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M235">View MathML</a>

Since we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M228">View MathML</a> is a minimal surface, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M237">View MathML</a>. So we get the following equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M238">View MathML</a>

(15)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M239">View MathML</a> is a second order linear operator with operator with coefficients in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M240">View MathML</a>.

We recall that if the function v satisfies the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M241">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M242">View MathML</a> then the graph of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M243">View MathML</a> is minimal. Now we are interested in finding the equation which a function w must satisfy in such a way the surface parametrized by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M244">View MathML</a>, that is the graph of w over the middle end <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M114">View MathML</a>, is minimal. That is equivalent to require that the graph of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M246">View MathML</a> is minimal. Then we can obtain the wanted equation by replacing v by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M247">View MathML</a> in (15). So we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M248">View MathML</a>

(16)

If we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M249">View MathML</a> to simplify the notation, we can write this equation in the following way:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M250">View MathML</a>

(17)

We observe that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M251">View MathML</a> clearly maps the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M252">View MathML</a> into the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M253">View MathML</a>.

3.3 Properties of the Jacobi operator of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38">View MathML</a>

The Jacobi operator of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38">View MathML</a>, up to a multiplicative factor, is asymptotic, respectively, to the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M256">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M209">View MathML</a> at the planar end and the catenoidal end.

In this subsection we will describe the mapping properties of an elliptic operator related to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M206">View MathML</a>. It will be used to solve the first equation of (3).

The volume form on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38">View MathML</a> is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M260">View MathML</a>. In the parameterization of the ends introduced above, such form can be written as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M261">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M262">View MathML</a> near the catenoidal type ends and as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M263">View MathML</a> near the middle end. Now we can define globally on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38">View MathML</a> a smooth function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M265">View MathML</a>

(18)

that is identically equal to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M266">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M267">View MathML</a> and equal to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M268">View MathML</a> (resp. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M269">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M270">View MathML</a>) on the end <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M144">View MathML</a> (resp. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M145">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M85">View MathML</a>). They are defined in such a way that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M274">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M275">View MathML</a> we have, respectively,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M276">View MathML</a>

Finally on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M277">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M278">View MathML</a>

It is possible to check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M279">View MathML</a>

is a bounded linear operator.

As in [21] (see also [20] for the same result in a less symmetric setting), using the non-degeneracy of the Costa-Hoffman-Meeks surfaces shown in [23,24], it is possible to show the following result.

Proposition 3.1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M280">View MathML</a>, then the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M281">View MathML</a>is surjective and has a kernel of dimension one. Moreover, there exists a right inverse<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M282">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M281">View MathML</a>whose norm is bounded.

4 Construction of a family of solutions to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M284">View MathML</a>

In this section we will prove the existence of a family of embedded minimal surfaces and which are close to the piece of surface <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38">View MathML</a> contained in the unit ball <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M18">View MathML</a>.

We set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M287">View MathML</a>

and we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M288">View MathML</a> to be the value of s such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M289">View MathML</a>

(19)

We get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M290">View MathML</a>

We define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M291">View MathML</a> so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M292">View MathML</a>

The value of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M293">View MathML</a> has been chosen so that the image of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M294">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M295">View MathML</a>, by the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M296">View MathML</a> (compare (4)) is the circumference <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M297">View MathML</a> of radius 1 in the horizontal plane <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M86">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M288">View MathML</a> is the value of s for which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M300">View MathML</a> is the height of the curves <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M301">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M302">View MathML</a> which are the intersection of the unit sphere with the top and bottom halves of the catenoid parametrized by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M140">View MathML</a> and translated vertically by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M304">View MathML</a>, respectively.

We define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M305">View MathML</a> to be equal to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38">View MathML</a> from which we have removed the image of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M307">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M308">View MathML</a>, the image of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M309">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M310">View MathML</a> and the image of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M311">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M312">View MathML</a>. The boundary curves of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M305">View MathML</a> do not lie in the unit sphere but they are in a tubular neighborhood of the curves <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M301">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M302">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M297">View MathML</a>. In the sequel we will use also the cylindrical coordinates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M317">View MathML</a> (of course <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M318">View MathML</a>). The circumferences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M301">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M302">View MathML</a> are contained, respectively, in the horizontal planes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M321">View MathML</a> and their vertical projection on the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M322">View MathML</a> plane is the circumference of radius <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M323">View MathML</a>. The middle boundary curve of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M324">View MathML</a> is located in a small neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M297">View MathML</a>. Points in the middle boundary curve have a height which can be estimated by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M326">View MathML</a>.

By using (4), (7), and (8) we get easily the following lemma. It describes the region of the surface <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38">View MathML</a> which is a graph over the annular domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M328">View MathML</a> of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M86">View MathML</a> plane.

Lemma 4.1There exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M330">View MathML</a>such that, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M331">View MathML</a>an annular part of the ends<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M144">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M145">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M114">View MathML</a>of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38">View MathML</a>can be written as vertical graphs over the annulusAof the functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M336">View MathML</a>

(20)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M337">View MathML</a>

(21)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M338">View MathML</a>

(22)

Here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M339">View MathML</a>are the polar coordinates in the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M86">View MathML</a>plane. The functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M341">View MathML</a>are defined in the annulusAand are bounded in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M342">View MathML</a>topology by a constant (independent byf) multiplied byf, where the partial derivatives are computed with respect to the vector fields<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M343">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M344">View MathML</a>.

We will make a slight modification to the parametrization of the ends <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M345">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M346">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M114">View MathML</a>, for s and ρ in a small neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M348">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M293">View MathML</a>, respectively.

The unit normal vector field to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38">View MathML</a> is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M351">View MathML</a>. Firstly we modify the vector field <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M351">View MathML</a> into a transverse unit vector field <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M353">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M354">View MathML</a> is a smooth interpolation of the following vector fields defined on different pieces of the surface:

• at the top (resp. bottom) catenoidal end, the unit normal vector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M355">View MathML</a> (resp. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M356">View MathML</a>) for s in a small neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M357">View MathML</a> (resp. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M358">View MathML</a>); we recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M359">View MathML</a> are the unit normal vectors to the translated copy of the halves catenoid parametrized by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M360">View MathML</a> along the curves <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M301">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M302">View MathML</a>;

• at the middle planar end, the vertical vector field <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M363">View MathML</a> for ρ in a small neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M364">View MathML</a>;

• the normal vector field <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M365">View MathML</a> on the remaining part of the surface.

We observe that at the top end <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M144">View MathML</a>, we can give the following estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M367">View MathML</a>

(23)

This follows easily from (10) together with the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M155">View MathML</a> decays at least like <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M369">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M144">View MathML</a>. Similar considerations hold at the bottom end <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M145">View MathML</a>. Near the middle planar end <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M114">View MathML</a>, we observe that the following estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M373">View MathML</a>

(24)

This follows easily from (13) together with the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M121">View MathML</a> decays at least like <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M375">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M114">View MathML</a>.

The mean curvature of the graph <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M228">View MathML</a> of a function u in the direction of the vector field <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M353">View MathML</a> is the image of u by a second order nonlinear elliptic operator:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M379">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M380">View MathML</a> is the Jacobi operator of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M381">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M382">View MathML</a> is a nonlinear second order differential operator and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M383">View MathML</a> is a linear operator which takes into account the change of the normal vector field <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M351">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M353">View MathML</a>.

The operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M383">View MathML</a> is supported in a neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M387">View MathML</a> and of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M388">View MathML</a>. It is possible to show that the coefficients of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M383">View MathML</a> are uniformly bounded by a constant times <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M266">View MathML</a>. First we observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M391">View MathML</a> in a neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M387">View MathML</a> and of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M393">View MathML</a> and the result of [20] Appendix B show that the change of vector field induces a linear operator whose coefficients are bounded by a constant times <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M266">View MathML</a>.

As we will see in the sequel, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M395">View MathML</a> which solves <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M396">View MathML</a>, depends nonlinearly by a triple of functions defined on the boundary curves of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M305">View MathML</a>. Here is the definition of the functional space we will consider.

Definition 4.2 Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M398">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M178">View MathML</a>, the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M400">View MathML</a> is defined to be the space of triples of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M401">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M402">View MathML</a> and even, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M403">View MathML</a> is collinear to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M161">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M405">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M406">View MathML</a> is collinear to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M407">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M408">View MathML</a> and odd, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M409">View MathML</a>, and whose norm, defined below, is finite.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M410">View MathML</a>

(25)

Now we consider the triple of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M411">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M412">View MathML</a>

(26)

We define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M413">View MathML</a> to be the function equal to

1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M414">View MathML</a> on the image of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M308">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M416">View MathML</a> is a cut-off function equal to 0 for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M417">View MathML</a> and identically equal to 1 for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M418">View MathML</a>;

2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M419">View MathML</a> on the image of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M310">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M421">View MathML</a> is a cut-off function equal to 0 for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M422">View MathML</a> and identically equal to 1 for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M423">View MathML</a>;

3. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M424">View MathML</a> on the image of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M312">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M426">View MathML</a> is a cut-off function equal to 0 for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M427">View MathML</a> and identically equal to 1 for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M428">View MathML</a>;

4. zero on the remaining part of the surface <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M305">View MathML</a>.

The cut-off functions just introduced must enjoy the same symmetry properties as the functions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M430">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M431">View MathML</a> and H are harmonic extension operators introduced, respectively, in Propositions A.1 and A.2.

We will prove that, under appropriates hypotheses, the graph <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M432">View MathML</a> over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M305">View MathML</a> of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M434">View MathML</a>, is a surface whose mean curvature vanishes.

The equation to solve is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M435">View MathML</a>

Since we are looking for solutions having the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M436">View MathML</a>, we can write it as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M437">View MathML</a>

The resolution of the previous equation is obtained by the one of the following fixed point problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M438">View MathML</a>

(27)

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M439">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M440">View MathML</a>, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M441">View MathML</a> is defined in Proposition 3.1 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M442">View MathML</a> is a linear extension operator such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M443">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M444">View MathML</a> denotes the space of functions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M445">View MathML</a> restricted to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M305">View MathML</a>. It is defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M447">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M305">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M449">View MathML</a> in the image of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M450">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M308">View MathML</a>, in the image of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M452">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M310">View MathML</a> and in the image of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M454">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M312">View MathML</a>. Finally <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M456">View MathML</a> is an interpolation of these values in the remaining part of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M457">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M458">View MathML</a>

Remark 4.3 From the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M442">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M460">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M461">View MathML</a>

This phenomenon of explosion of the norm does not occur near the catenoidal type ends:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M462">View MathML</a>

A similar equation holds for the bottom end. In the following we will assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M463">View MathML</a> and close to zero.

The existence of a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M464">View MathML</a> for (27) is a consequence of the following result, which proves that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M465">View MathML</a> is a contraction mapping.

Proposition 4.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M440">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M467">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M468">View MathML</a>satisfying (26) and enjoying the properties given above. There exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M469">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M470">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M471">View MathML</a>

(28)

and, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M472">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M473">View MathML</a>

wherecis a positive constant, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M474">View MathML</a>and satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M475">View MathML</a>and for all boundary data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M476">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M477">View MathML</a>, enjoying the same properties as Φ.

Proof We recall that the Jacobi operator associated to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M38">View MathML</a>, is asymptotic (up to a multiplication by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M479">View MathML</a>) to the Jacobi operator of the catenoid (respectively, of the plane) plane at the catenoidal ends (respectively, at the planar end). The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M413">View MathML</a> is identically zero far from the ends where the explicit expression of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M206">View MathML</a> is not known: this is the reason for our particular choice in the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M413">View MathML</a>. Then from the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M413">View MathML</a> and thanks to Proposition 3.1 we obtain the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M484">View MathML</a>

To obtain this estimate we used the following ones:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M485">View MathML</a>

(a similar estimate holds for the bottom end) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M486">View MathML</a>

together with the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M487">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M488">View MathML</a>, from which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M489">View MathML</a>.

Using the estimates of the coefficients of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M383">View MathML</a> and the definition of γ (see (18)), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M491">View MathML</a>

As for the last term, we recall that the expression of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M382">View MathML</a> depends on the type of end we are considering (see (17) and (11)). We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M493">View MathML</a>

In fact

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M494">View MathML</a>

As for the second estimate, we recall that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M495">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M496">View MathML</a>

We observe that from the considerations above it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M497">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M498">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M499">View MathML</a>

To get the last estimate it suffices to observe that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M500">View MathML</a>

 □

Theorem 4.5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M501">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M502">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M503">View MathML</a>. Then the nonlinear mapping<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M465">View MathML</a>defined above has a unique fixed pointvinB.

Proof The previous lemma shows that, if τ is chosen small enough, the nonlinear mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M465">View MathML</a> is a contraction mapping from the ball B of radius <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M506">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M430">View MathML</a> into itself. This value follows from the estimate of the norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M508">View MathML</a>. Consequently thanks to Schäuder fixed point theorem, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M465">View MathML</a> has a unique fixed point w in this ball. □

This argument provides a new surface <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M510">View MathML</a> whose mean curvature equals zero, which is close to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M305">View MathML</a> and has three boundary curves.

The surface <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M510">View MathML</a> is, close to its upper and lower boundary curve, the graph over the catenoidal ends in the direction given by the vector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M353">View MathML</a> of the functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M514">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M515">View MathML</a>. Nearby the middle boundary the surface is the vertical graph of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M516">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M488">View MathML</a>. All the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M518">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M107">View MathML</a>, depend nonlinearly on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M520','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M520">View MathML</a>.

Lemma 4.6The function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M521">View MathML</a>, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M522">View MathML</a>, satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M523">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M524">View MathML</a>

(29)

The function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M525">View MathML</a>satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M526">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M527">View MathML</a>

(30)

Proof We recall that the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M528">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M529">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M530','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M530">View MathML</a> are the restrictions to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M144">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M145">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M114">View MathML</a> of a fixed point v for the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M465">View MathML</a>. The estimates of their norm are a consequence of Proposition 4.4. Observe that to derive the estimate of the norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M528">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M529">View MathML</a> we use the better estimate for the norm of the fixed point v which holds at the catenoidal type ends. Precisely stated: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M537','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M537">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M522">View MathML</a>. Then (29) follows from

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M539">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M522">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M541','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M541">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M542','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M542">View MathML</a>. To get the estimate (30) we observe that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M543">View MathML</a>

 □

Remark 4.7 In next section we will use previous result to prove Theorem 1.1 under the additional assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M544','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M544">View MathML</a>. Consequently in (29) it appears a positive power of τ. The previous result can be reformulated as follows: all of the mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M545','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M545">View MathML</a> are contracting. Furthermore the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M546','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M546">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M547','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M547">View MathML</a>.

5 Proof of Theorem 1.1

The surface <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M548','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M548">View MathML</a> we constructed in previous section, has three boundary curves. Such curves do not lie in the sphere <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M28">View MathML</a>. So we introduce a new surface <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M550','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M550">View MathML</a>.

To prove the main theorem we need to show that there exists Φ such that also the second equation of (3) is satisfied.

We recall that we modified the immersion of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M305">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M2">View MathML</a> in order to have the normal vector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M353">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M305">View MathML</a> equal to the normal vectors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M359">View MathML</a> in a neighborhood of its top and bottom boundary curves and equal to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M363">View MathML</a> in a neighborhood of the middle boundary curve. Precisely, at the catenoidal type ends, from (6), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M353">View MathML</a> in a neighborhood of the boundary curves equals the vector fields (here we use the basis <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M558','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M558">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M559','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M559">View MathML</a>

Near the boundary curves, the surface <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M560','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M560">View MathML</a> is the graph in the direction of the vectors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M561','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M561">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M562','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M562">View MathML</a>, over the ends of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M305">View MathML</a> of the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M564','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M564">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M562','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M562">View MathML</a>.

As a consequence the top and bottom ends of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M510">View MathML</a>, near the boundary curves, can be parametrized as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M567">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M568','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M568">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M569','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M569">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M570','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M570">View MathML</a> be the function of θ defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M571','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M571">View MathML</a>

(31)

In other terms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M572','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M572">View MathML</a> is the value of the r-variable for which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M573','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M573">View MathML</a> is the parametrization of a curve on the sphere <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M28">View MathML</a>. More precisely it is one of the boundary curves of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M560','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M560">View MathML</a>.

Using the expression of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M576','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M576">View MathML</a> we get the following estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M577','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M577">View MathML</a>

(32)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M578','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M578">View MathML</a> denotes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M291">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M522">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M581','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M581">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M582','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M582">View MathML</a>. They are the values taken by r for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M357">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M364">View MathML</a>.

In order to compute a unit normal vector to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M28">View MathML</a> along the boundary curves of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M560','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M560">View MathML</a> we will consider cylindrical coordinates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M317">View MathML</a>. It is clear that the vector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M588','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M588">View MathML</a> is orthogonal to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M28">View MathML</a> at the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M317">View MathML</a>. So three unit normal vectors to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M28">View MathML</a> along the top (resp. bottom, middle) boundary curve of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M560','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M560">View MathML</a> are obtained replacing r by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M572','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M572">View MathML</a> and z by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M594">View MathML</a> in the formula giving <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M595','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M595">View MathML</a>. We get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M596','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M596">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M597','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M597">View MathML</a>.

A non-unit normal vector to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M560','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M560">View MathML</a> along its boundary curves is given, in the frame <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M558','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M558">View MathML</a>, by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M600','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M600">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M601','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M601">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M522">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M603','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M603">View MathML</a>

We get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M604','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M604">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M605','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M605">View MathML</a>

In Section 4 we proved the existence of a family of solutions to the first equation of (3). It remains to show the existence of one solution in such a family which satisfies the other equations in (3). It is clear that the last equations in (3) are equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M606','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M606">View MathML</a>

(33)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M607','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M607">View MathML</a> denotes the length of the vector v. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M301">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M302">View MathML</a> are the intersection curves of the asymptotic halves catenoid and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M28">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M611','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M611">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M612','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M612">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M613','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M613">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M614','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M614">View MathML</a> are, respectively, the normal vectors to the asymptotic halves catenoid and to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M28">View MathML</a> along <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M301">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M302">View MathML</a>. We observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M613','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M613">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M614','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M614">View MathML</a> have unit length. Such normal vectors can be computed as done for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M620','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M620">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M621','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M621">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M622','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M622">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M623','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M623">View MathML</a>.

The computation of the scalar product yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M624','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M624">View MathML</a>

We can compute <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M625','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M625">View MathML</a> by using previous formula: indeed it suffices to assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M626','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M626">View MathML</a> and to replace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M627','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M627">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M578','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M578">View MathML</a>. We get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M629','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M629">View MathML</a>

The square of the length of the normal vectors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M620','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M620">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M621','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M621">View MathML</a> are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M632','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M632">View MathML</a>

By construction of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M633','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M633">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M634','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M634">View MathML</a> and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M635','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M635">View MathML</a>, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M636','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M636">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M637','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M637">View MathML</a> can be estimated as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M638','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M638">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M639','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M639">View MathML</a>, respectively. If we replace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M640','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M640">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M641','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M641">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M642','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M642">View MathML</a>, and we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M643','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M643">View MathML</a>, we get the values of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M644','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M644">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M645','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M645">View MathML</a>. In conclusion <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M636','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M636">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M637','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M637">View MathML</a> are small perturbations of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M644','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M644">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M645','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M645">View MathML</a>.

The equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M650','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M650">View MathML</a> is equivalent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M651','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M651">View MathML</a>. In view of previous observations this last equation can be seen as a small perturbation of the simpler equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M652','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M652">View MathML</a>.

The advantage of solving <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M653','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M653">View MathML</a> is that it reduces to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M654','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M654">View MathML</a>

Similarly, instead of solving <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M655','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M655">View MathML</a>, we consider the simpler equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M656','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M656">View MathML</a>, which reduces to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M657','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M657">View MathML</a>

The equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M658','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M658">View MathML</a> is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M659','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M659">View MathML</a>

To establish the proof we need to find a more explicit expression of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M660','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M660">View MathML</a>. We get easily

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M661','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M661">View MathML</a>

Observe that if we evaluate first two functions at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M662','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M662">View MathML</a> (the value taken by r if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M357">View MathML</a>) and third one at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M664','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M664">View MathML</a> (the value taken by r if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M364">View MathML</a>) then we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M666','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M666">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M667','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M667">View MathML</a> is the operator defined as follows. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M668','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M668">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M669','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M669">View MathML</a>

Let us consider the operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M670','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M670">View MathML</a>

see (33).

The definition of the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M671','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M671">View MathML</a> is similar to definition 4.2, with the unique difference of the lower regularity.

We want to show the existence of a solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M672','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M672">View MathML</a>.

We define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M673','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M673">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M674','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M674">View MathML</a>

Proposition 5.1There exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M675','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M675">View MathML</a>such that if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M676','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M676">View MathML</a>then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M677','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M677">View MathML</a>for which, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M21">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M672','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M672">View MathML</a>has a solution in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M680','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M680">View MathML</a>, the ball centered at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M681','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M681">View MathML</a>and of radius<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M682','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M682">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M683','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M683">View MathML</a>.

Proof Let us consider the operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M684','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M684">View MathML</a>

The operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M685','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M685">View MathML</a> can be seen as an approximation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M686','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M686">View MathML</a>: indeed we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M685','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M685">View MathML</a> from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M686','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M686">View MathML</a> omitting some nonlinear terms and evaluating the remaining ones at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M578','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M578">View MathML</a> instead of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M627','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M627">View MathML</a>.

Equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M691','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M691">View MathML</a> has a unique solution, because the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M692','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M692">View MathML</a> is easily seen to be invertible. By elliptic regularity theory this result extends to the operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M693','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M693">View MathML</a>

From (32) and Lemma 4.6 we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M694','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M694">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M695','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M695">View MathML</a>. We would like to show existence of a solution to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M672','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M672">View MathML</a> by the Leray-Schauder degree theory but the nonlinear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M697','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M697">View MathML</a> is not compact. We apply the same technique as in Proposition 15 of [25].

Let us introduce a family of smoothing operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M698','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M698">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M699','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M699">View MathML</a>, defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M700','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M700">View MathML</a>

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M701','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M701">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M702','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M702">View MathML</a>. The operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M698','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M698">View MathML</a> satisfies for fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M704','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M704">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M705','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M705">View MathML</a>

(34)

where c does not depend on q.

We approximate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M706','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M706">View MathML</a> by the family of compact operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M707','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M707">View MathML</a> defined as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M708','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M708">View MathML</a>

Now we can apply Leray-Schauder degree theory to prove the existence of a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M709','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M709">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M710','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M710">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M680','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M680">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M21">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M22">View MathML</a> small enough and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M714','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M714">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M715','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M715">View MathML</a> chosen large enough.

Since the norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M709','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M709">View MathML</a> is bounded uniformly in q, we can extract a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M717','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M717">View MathML</a> converging to 0 such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M718','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M718">View MathML</a> converges in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M719','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M719">View MathML</a> for any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M720','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M720">View MathML</a>. Thanks to the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M707','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M707">View MathML</a> and to (34), the limit of this sequence converges to a solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M672','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M672">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M21">View MathML</a>. □

The zero of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M706','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M706">View MathML</a> provides the boundary data Φ for which the surface <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M560','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M560">View MathML</a> meets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M28">View MathML</a> in order to make (33) satisfied. That finishes the proof of Theorem 1.1.

Appendix

Results in this section are about the existence of some harmonic extension operators.

The following result gives a harmonic extension of a function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M727','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M727">View MathML</a>.

Proposition A.1There exists an operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M728','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M728">View MathML</a>

such that for each even function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M729','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M729">View MathML</a>, which is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M730','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M730">View MathML</a>-orthogonal to the constant function then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M731','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M731">View MathML</a>solves

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M732','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M732">View MathML</a>

Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M733','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M733">View MathML</a>

(35)

for some constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M734','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M734">View MathML</a>.

Proof We consider the decomposition of the function φ with respect to the basis <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M735','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M735">View MathML</a>, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M736','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M736">View MathML</a>

Then the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M737','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M737">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M738','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M738">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M739','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M739">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M740','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M740">View MathML</a>, we can conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M741','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M741">View MathML</a> and then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M742','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M742">View MathML</a>. □

Proposition A.2There exists an operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M743','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M743">View MathML</a>

such that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M744','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M744">View MathML</a>, even function and orthogonal to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M745','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M745">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M746','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M746">View MathML</a>in the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M730','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M730">View MathML</a>-sense, the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M748','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M748">View MathML</a>solves

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M749','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M749">View MathML</a>

Moreover

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M750','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M750">View MathML</a>

for some constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M734','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M734">View MathML</a>.

The proof is immediate once we observe that, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M752','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M752">View MathML</a>, then the solution is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M753','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/130/mathml/M753">View MathML</a>.

Competing interests

The author declares that he has no competing interests.

Acknowledgements

This research was supported by Basic Science Research Program through the National Research Foundation of South Korea (NRF) funded by the Ministry of Education, Grant NRF-2013R1A1A1013521.

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