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Lp-estimates for a transmission problem of mixed elliptic-parabolic type

Robert Denk* and Tim Seger

Author Affiliations

Department of Mathematics and Statistics, University of Konstanz, Konstanz, 78457, Germany

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Boundary Value Problems 2014, 2014:22  doi:10.1186/1687-2770-2014-22


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/22


Received:25 July 2013
Accepted:18 December 2013
Published:22 January 2014

© 2014 Denk and Seger; 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 cited.

Abstract

We consider the situation when an elliptic problem in a subdomain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M2">View MathML</a> of an n-dimensional bounded domain Ω is coupled via inhomogeneous canonical transmission conditions to a parabolic problem in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M3">View MathML</a>. In particular, we can treat elliptic-parabolic equations in bounded domains with discontinuous coefficients. Using Fourier multiplier techniques, we prove an a priori estimate for strong solutions to the equations in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M1">View MathML</a>-Sobolev spaces.

MSC: 35B45, 35M12.

Keywords:
transmission problem; elliptic-parabolic equation; a priori estimates

1 Introduction

In the present paper we prove a priori estimates in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M1">View MathML</a>-Sobolev spaces for the solution of a transmission problem of elliptic-parabolic type with discontinuous coefficients. More precisely, we consider a bounded domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M6">View MathML</a> which is divided into two subdomains <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M8">View MathML</a> separated by a closed contour <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M9">View MathML</a> and a boundary value problem of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M10">View MathML</a>

(1.1)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M11">View MathML</a> is a differential operator of order 2m, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M12">View MathML</a> is a column of boundary operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M13">View MathML</a>, and λ is a complex parameter. We assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M14">View MathML</a> and are looking for a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M15">View MathML</a>. The top-order coefficients of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M11">View MathML</a> are assumed to be continuous up to the boundary in each subdomain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M17">View MathML</a> but may have jumps across the interface Γ. The condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M15">View MathML</a> leads to the canonical transmission conditions along Γ, given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M19">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M20">View MathML</a> stands for the jump of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M21">View MathML</a>th normal derivative of u along the interface Γ. Generalizing (1.2), we will consider inhomogeneous transmission conditions of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M22">View MathML</a>

(1.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M23">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M24">View MathML</a>

Here we have set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M25">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M26">View MathML</a>.

The aim of the paper is to prove uniform a priori estimates for the solutions of (1.1) and (1.3) under suitable ellipticity and smoothness assumptions on A and C, see Section 2 below for the precise formulations. To give an idea of our results, let us for the moment assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M27">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M28">View MathML</a> in (1.1) and (1.3). In classical elliptic theory, in the case of an uncoupled system we would expect a uniform a priori estimate of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M29">View MathML</a>

On the other hand, the classical parabolic (in the sense of parameter-elliptic) a priori estimate would read

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M30">View MathML</a>

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M31">View MathML</a> is the typical parameter-dependent norm appearing in parabolic theory. Concerning the coupled system (1.1), (1.3), the question arises if we still have similar estimates for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M32">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M33">View MathML</a>. We will see below that this is true in some sense. More precisely, we will obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M34">View MathML</a>

This can be seen as a mixture of elliptic and parabolic a priori estimates. Note that we do not reach the full order 2m with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M33">View MathML</a> in the first inequality and not the full power <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M36">View MathML</a> with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M32">View MathML</a> in the second inequality. The general result for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M38">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M39">View MathML</a> and the precise formulation are stated in Section 2 below.

Applications of problem (1.1), (1.3) (in its parabolic form, i.e., the parameter λ being replaced by the time derivative) can be found, e.g., in [1], including the heat equation in a domain with vanishing thermal capacity in some subdomain and a model of an electric field generated by a current in a partially non-conducting domain. On the other hand, the problem under consideration is closely related to spectral problems with indefinite weight functions of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M40">View MathML</a>

Here ω is a weight function which may change sign and may vanish on a set of positive measure. Such spectral problems have been investigated, e.g., in a series of papers by Faierman (see [2-4]) and by Pyatkov [5,6], see also [7] and the references therein. In particular, in the paper [4] a Calderón method of reduction to the boundary was applied to deal with the case where ω vanishes on a set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M2">View MathML</a> of positive measure. For this, unique solvability of the Dirichlet boundary value problem in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M2">View MathML</a> had to be assumed. Transmission problems of purely parabolic type (where the parameter λ is present in each subdomain) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M1">View MathML</a>-a priori estimates for their solution were considered in [8] and [23]. Transmission problems in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M1">View MathML</a> were also studied, with the same methods as in the present note, by Shibata and Shimizu in [9].

A standard approach to treat transmission problems is to use (locally) a reflection technique in one subdomain resulting in a system of differential operators which are coupled by the transmission conditions. A general theory of parameter-dependent systems can be found in a series of papers by Volevich and his co-authors (see [10] and the references therein). Here the so-called Newton polygon method leads to uniform a priori estimates for the solution. However, in the present case the Newton polygon is of trapezoidal form and thus not regular. Therefore, the Newton polygon approach cannot be applied to the transmission problem (1.1). On the other side, the resulting system is not parameter-elliptic in the classical sense ([11]) and is not covered by the standard parameter-elliptic theory. We also note the connection to singularly perturbed problems where a similar Newton polygon structure appears, cf.[12]. The analysis of the elliptic-parabolic system below also serves as a starting point for more general (and nonlinear) elliptic-parabolic systems as, for instance, appearing in lithium battery models (see [13]). A detailed investigation of the nonlinear elliptic-parabolic lithium battery model and solvability in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M1">View MathML</a>-Sobolev spaces can be found in the second author’s thesis [14]. In [15] and [16] mathematical models for lithium battery systems can be found which lead to inhomogeneous transmission conditions.

In Section 2 we will state the precise assumptions and the main result of the present paper. The boundary value problem is analyzed by a localization method and the investigation of the model problem in the half-space. An explicit description of the solution of the model problem (in terms of Fourier multipliers) and resulting estimates can be found in Section 3. Finally, the proof of the main a priori estimate is given in Section 4.

2 Statement of the problem and main result

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M47">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M48">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M49">View MathML</a> be open. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M50">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M51">View MathML</a> we denote the Lebesgue and Sobolev spaces on Ω with their standard norms. We will further make use of the seminorms

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M52">View MathML</a>

where we used the standard notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M53">View MathML</a>. For real non-integer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M54">View MathML</a> let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M55">View MathML</a> denote the Besov space on Ω with its standard norm. Besides the standard norms, for the treatment of parameter-elliptic problems the following parameter-dependent norms will be convenient: Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M56">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M57">View MathML</a> be a complex parameter, varying in a closed sector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M58">View MathML</a> with vertex at 0 where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M59">View MathML</a>. Then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M60">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M61">View MathML</a>, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M62">View MathML</a>

(2.1)

On the boundary, we will consider parameter-dependent trace norms given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M63">View MathML</a>

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M64">View MathML</a> we denote the Fourier transform of u, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M65">View MathML</a> stands for the partial Fourier transform with respect to the first <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M66">View MathML</a> variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M67">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M49">View MathML</a> be a bounded domain with boundary Ω of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M69">View MathML</a>, and let Γ be a closed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M69">View MathML</a> Jordan contour in Ω, having no points with Ω in common. Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M71">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M72">View MathML</a> the resulting subdomains such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M74">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M75">View MathML</a>. Note that, due to our assumptions, there is no contact point of Γ and Ω. We define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M76">View MathML</a> and will consider the differential operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M77">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M78">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M79">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M80">View MathML</a>. Slightly generalizing the form of equation (1.1), we consider differential operators of even order 2m of the following structure:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M81">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M60">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M83">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M84">View MathML</a>. Furthermore, let the boundary operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M85">View MathML</a> of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M86">View MathML</a> be of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M87">View MathML</a>

being defined on Ω. We will write for short <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M88">View MathML</a> when we refer to the boundary value problem (1.1).

Assumption 2.1

(1) Smoothness assumptions on the coefficients. We assume

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M89">View MathML</a>

for the coefficients of the differential operators and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M90">View MathML</a> for the coefficients of the boundary operators.

(2a) Ellipticity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M91">View MathML</a>. For the principal symbol <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M92">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M93">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M94">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M95">View MathML</a>).

(2b) Ellipticity with parameter of the boundary value problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M96">View MathML</a>. The principal symbol of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M97">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M98">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M99">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M100">View MathML</a>, and the Shapiro-Lopatinskii condition is satisfied for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M101">View MathML</a> at each point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M102">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M103">View MathML</a> denotes the principal symbol of the boundary operator, this condition reads as follows: For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M102">View MathML</a> let the boundary value problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M105">View MathML</a> be rewritten in local coordinates associated with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M106">View MathML</a>, i.e. in coordinates resulting from the original ones by rotation and translation such that the positive <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M107">View MathML</a>-axis coincides with the direction of the inner normal vector. Then for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M108">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M109">View MathML</a>, the ODE problem on the halfline

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M110">View MathML</a>

admits a unique solution. Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M111">View MathML</a>.

(3) Assumptions on the data. We assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M112">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M113">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M114">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M115">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M116">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M117">View MathML</a>.

(4) In addition, we assume proper ellipticity, i.e. the polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M118">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M119">View MathML</a> of order 2m from conditions (2a) and (2b) have exactly m roots in each half-plane <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M120">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M121">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M122">View MathML</a>, respectively, and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M123">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M124">View MathML</a>. Proper ellipticity allows a decomposition of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M125">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M126">View MathML</a>

(2.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M127">View MathML</a> denote the roots in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M128">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M129">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M130">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M131">View MathML</a>), respectively. A similar decomposition with an additional dependence on λ also holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M132">View MathML</a>. We remark that proper ellipticity holds automatically if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M133">View MathML</a>.

Under these assumptions, we consider the inhomogeneous transmission boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M134">View MathML</a>

(2.3)

Here we have set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M135">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M136">View MathML</a> denotes the derivative in direction of the outer normal with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M8">View MathML</a>. Our main result is the following a priori estimate for solutions to (2.3). Here, a solution of (2.3) is defined as a pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M138">View MathML</a> belonging to the Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M139">View MathML</a> for which the system (2.3) is satisfied as equality of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M1">View MathML</a>-functions.

Theorem 2.2 (A priori estimate for the transmission boundary value problem)

Let Assumption 2.1 be satisfied and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M141">View MathML</a>be a solution to the transmission problem (2.3). Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M142">View MathML</a>such that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M143">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M144">View MathML</a>the following estimates hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M145">View MathML</a>

(2.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M146">View MathML</a>

(2.5)

Note that with respect to g, inequality (2.4) is of elliptic type and (2.5) is of parameter-elliptic type. Due to the fact that the boundary operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M85">View MathML</a> act on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M33">View MathML</a>, we have parameter-elliptic norms with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M149">View MathML</a> in both inequalities.

Remark 2.3 Our main task will be to study the problem for constant coefficient operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M150">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M151">View MathML</a> in the half-spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M152">View MathML</a> without lower order terms. This simplification can be justified by performing a localization procedure, using a finite covering <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M153">View MathML</a> with appropriate open sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M154">View MathML</a>, a corresponding partition of unity and perturbation results. For a detailed explanation of the localization procedure, we refer to [8], pp.151-153, but here we briefly list the types of local problems one has to deal with. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M155">View MathML</a>, one faces a local elliptic (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M156">View MathML</a>) or parameter-elliptic <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M157">View MathML</a> operator in the whole space. For these situations, the estimates for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M158">View MathML</a> are well-known results, see [17], Theorem 5.3.2, for the elliptic and [8], Proposition 2.5, for the parameter-elliptic case. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M159">View MathML</a>, the local problem is a standard boundary value problem in the half-space and the desired estimate is contained in [8], Proposition 2.6. It remains to consider the case where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M154">View MathML</a> intersects both <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M71">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M72">View MathML</a>, and in the sequel we restrict our considerations to the corresponding local model problem. This reads

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M163">View MathML</a>

(2.6)

The reflection <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M164">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M165">View MathML</a> will be useful to treat problem (2.6). Therefore, we will use the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M166">View MathML</a> for the symbol of the reflected operator, which is parameter-elliptic in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M167">View MathML</a>. We set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M168">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M169">View MathML</a>.

By this substitution, we may rewrite (2.6) as a system in the half-space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M167">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M171">View MathML</a>

(2.7)

Here we have set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M172">View MathML</a>

Remark 2.4 We see that the determinant of the principal symbol <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M173">View MathML</a> vanishes at the points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M174">View MathML</a>. Hence the standard theory for parameter-elliptic systems is not applicable in this case. Due to continuity and homogeneity of the principal symbols we have the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M175">View MathML</a>

(2.8)

with a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M176">View MathML</a>. Operators whose principal symbols allow an estimate of the form (2.8) are also called N-elliptic with parameter. Here the ‘N’ stands for the Newton polygon which is related to the principal symbol. In the case of (2.8), the Newton polygon is not regular, and therefore this equation is not covered by the results on N-ellipticity as in [10].

Remark 2.5 The boundary conditions in (2.6) are called canonical transmission conditions. In the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M177">View MathML</a>, they are equivalent to the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M178">View MathML</a> for

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M179">View MathML</a>

Note that in (2.6) the number of conditions equals the order of the operator, in contrast to boundary value problems. We will show in Lemma 3.1 below that the ODE system corresponding to the transmission problem (2.6) is uniquely solvable. This is an analogue of the Dirichlet boundary conditions which are absolutely elliptic, i.e., for every properly elliptic operator the Dirichlet boundary value problem satisfies the Shapiro-Lopatinskii condition.

3 Fundamental solutions and solution operators

To represent the solution in terms of fundamental solutions, we start with the observation that the ODE system obtained from (2.6) by partial Fourier transform is uniquely solvable. This is the analogue of the Shapiro-Lopatinskii condition for transmission problems. For detailed discussions of this condition for boundary value problems, we refer to [18], Section 6.2 and [19], Chapter 11. The assertion of the following lemma is formulated for our situation of one elliptic and one parameter-elliptic operator but of course it also holds in the cases when both operators are of the same type.

To simplify our notation, we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M180">View MathML</a> and consider the differential operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M181">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M182">View MathML</a>.

Lemma 3.1Suppose the operators<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M183">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M184">View MathML</a>are elliptic and parameter-elliptic in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M185">View MathML</a>, respectively. Fix<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M186">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M123">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M188">View MathML</a>and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M189">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M190">View MathML</a>). Then the ODE problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M191">View MathML</a>

(3.1)

admits a unique solution.

Proof In the sequel, we do not write down the dependence of the polynomials and their roots on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M106">View MathML</a> explicitly and fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M123">View MathML</a> as well as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M194">View MathML</a>. We decompose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M195">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M196">View MathML</a> as indicated in (2.2) into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M197">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M198">View MathML</a> denote the m-dimensional space of stable solutions to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M199">View MathML</a>

and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M200">View MathML</a> denote the m-dimensional space of stable solutions to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M201">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M202">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M203">View MathML</a> be a basis of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M198">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M200">View MathML</a>, respectively. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M206">View MathML</a> is obviously a subset of the 2m-dimensional space of solutions to the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M207">View MathML</a>

(3.2)

and B is linearly independent: Suppose there are nontrivial <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M208">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M117">View MathML</a>) with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M210">View MathML</a>

Then (3.2) would possess a solution which is bounded on the entire real line, which contradicts the fact that the polynomial <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M211">View MathML</a> has only roots with nonzero imaginary part. Hence B is a fundamental system to (3.2) and the determinant of the Wronskian matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M212">View MathML</a> is nonzero:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M213">View MathML</a>

(3.3)

Now suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M214">View MathML</a> is a solution to (3.1). Then there exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M215">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M216">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M217">View MathML</a>

If we plug in this approach into the transmission conditions, we obtain the system of linear equations to determine <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M218">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M219">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M220">View MathML</a>

From (3.3) it now follows that the coefficients exist and are uniquely determined, which proves the assertion. □

From now on, we restrict ourselves to the model problem (2.7) which is the only non-standard step in the proof of the main theorem, see Remark 2.3. We first consider the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M221">View MathML</a> in (2.7), i.e. we study

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M222">View MathML</a>

(3.4)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M223">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M224">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M225">View MathML</a>

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M226">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M227">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M228">View MathML</a> are also called generalized Dirichlet and Neumann conditions, respectively.

Due to Lemma 3.1, the ODE system corresponding to (3.4) is uniquely solvable. The main step in the proof of Theorem 2.2 will be to find a priori estimates for the fundamental solutions of this ODE system. In the following, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M229">View MathML</a> stands for the (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M230">View MathML</a>)-dimensional unit matrix.

Definition 3.2 The fundamental solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M231">View MathML</a>

is defined as the unique solution of the ODE system (in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M107">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M233">View MathML</a>

Following an idea of Leonid Volevich [20], we represent the solutions in a specific way. For this, we consider the elliptic boundary value problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M234">View MathML</a> and the parameter-elliptic boundary value problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M235">View MathML</a> separately. It is well known that the (generalized) Dirichlet and Neumann boundary conditions are absolutely elliptic, hence the Shapiro-Lopatinskii condition holds for both subproblems. We will call the canonical basis for these boundary value problems the basic solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M236">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M237">View MathML</a>. More precisely, we define the following.

Definition 3.3 We define the basic solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M238">View MathML</a>

as the unique solution of the ODE system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M239">View MathML</a>

(3.5)

Analogously, the basic solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M240">View MathML</a>

is defined as the unique solution of the ODE system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M241">View MathML</a>

(3.6)

We set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M242">View MathML</a>

The advantage of the basic solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M243">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M244">View MathML</a> lies in the fact that classical (parameter-)elliptic estimates are easily available for them. We have to compare these solutions with the fundamental solution ω. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M245">View MathML</a>. As the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M246">View MathML</a> is a solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M247">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M248">View MathML</a>), it can be written as a linear combination of the basic solutions. Therefore, we can write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M249">View MathML</a>

with unknown coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M250">View MathML</a>. The analogous representation holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M251">View MathML</a>. In matrix notation, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M252">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M253">View MathML</a>. By the definition of the fundamental solution, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M254">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M255">View MathML</a>

(3.7)

Remark 3.4 Due to the unique solvability of equations (3.5) and (3.6), we have for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M256">View MathML</a> the following scaling properties for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M257">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M258">View MathML</a>

where we used the abbreviations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M259">View MathML</a>

(See also (3.10) below for an explicit representation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M236">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M237">View MathML</a>.) We will apply this with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M262">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M243">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M264">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M244">View MathML</a>. Note that these scaling properties also yield the identities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M266">View MathML</a>

(3.8)

We summarize the representation of the solution in form of solution operators:

Lemma 3.5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M267">View MathML</a>, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M268">View MathML</a>be a solution of (3.4). Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M269">View MathML</a>be an extension ofgto the half-space. Thenuhas the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M270">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M271">View MathML</a>and where the solution operators<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M272">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M273">View MathML</a>are given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M274">View MathML</a>

Here the basic solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M275">View MathML</a>is defined in Definition 3.3, and the coefficient matrix<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M276">View MathML</a>is defined in (3.7).

Proof By definition of the fundamental solution, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M277">View MathML</a>. Writing this in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M278">View MathML</a>

(‘Volevich trick’) and noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M279">View MathML</a>, we obtain the above representation. □

Our proofs are based on the Fourier multiplier concept, see, e.g., [18]. Here a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M280">View MathML</a> is called an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M1">View MathML</a>-Fourier multiplier if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M282">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M283">View MathML</a> (being defined on the Schwartz space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M284">View MathML</a>) extends to a continuous mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M285">View MathML</a>. We will apply Michlin’s theorem to prove the Fourier multiplier property. For this, we introduce the notion of a Michlin function.

Definition 3.6 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M286">View MathML</a> be a matrix-valued function. Then we call M a Michlin function if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M287">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M288">View MathML</a> and if there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M176">View MathML</a>, independent of q, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M290">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M291">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M292">View MathML</a>

Remark 3.7 (a) Michlin’s theorem (see [17], Section 2.2.4) states that every Michlin function is an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M1">View MathML</a>-Fourier multiplier for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M294">View MathML</a>.

(b) By the product rule one immediately sees that the product of Michlin functions is a Michlin function, too.

(c) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M295">View MathML</a> be a Michlin function, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M296">View MathML</a> be invertible for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M291">View MathML</a> and q. If the norm of the inverse matrix is bounded by a constant independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M291">View MathML</a> and q, then also <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M299">View MathML</a> is a Michlin function. This follows iteratively noting that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M300">View MathML</a>

Now we will show that the basic solution Y as well as the coefficient matrix Ψ satisfy uniform estimates. Here and in the following, C stands for a generic constant which may vary from inequality to inequality but is independent of the variables appearing in the inequality. We will scale the functions with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M301">View MathML</a> and with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M302">View MathML</a>

(3.9)

Lemma 3.8 (a) For all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M303">View MathML</a>and all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M248">View MathML</a>, the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M305">View MathML</a>

is a Michlin function with constant independent of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M306">View MathML</a>.

(b) The functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M307">View MathML</a>

are Michlin functions.

Proof We use an explicit description of the basic solutions. According to [21], Section 1, there exist polynomials (with respect to τ) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M308">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M309">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M310">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M311">View MathML</a> being the Kronecker delta symbol. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M312">View MathML</a> is a smooth closed contour in the upper half-plane <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M128">View MathML</a>, depending on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M291">View MathML</a> and enclosing the m roots of the polynomial <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M315">View MathML</a> with positive imaginary part, while <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M316">View MathML</a> is a smooth closed contour in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M128">View MathML</a> depending on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M318">View MathML</a> and enclosing the m roots of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M319">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M128">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M321">View MathML</a> is positively homogeneous in its arguments of degree <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M322">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M323">View MathML</a> while <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M324">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M325">View MathML</a> are positively homogeneous in their arguments of degree m.

This leads to the following representation for the basic solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M326">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M327">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M328">View MathML</a>

(3.10)

To prove part (a), we will show that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M329">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M330">View MathML</a>

(3.11)

are Michlin functions. Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M331">View MathML</a> and noting the definitions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M332">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M333">View MathML</a>, this immediately implies (a). Similarly, to show (b) we have to prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M334">View MathML</a>

(3.12)

are Michlin functions. We will restrict ourselves to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M335">View MathML</a>, the result for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M336">View MathML</a> follows in the same way.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M329">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M338">View MathML</a>, we substitute <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M339">View MathML</a> in the integral representation (3.10) and obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M340">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M341">View MathML</a>. Note for the first equality that it is not necessary to differentiate the contour <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M342">View MathML</a> because it may be chosen locally independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M291">View MathML</a>. In the last equality, we replaced the contour <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M344">View MathML</a> by a fixed contour <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M345">View MathML</a> which is possible by a compactness argument.

Due to the properties of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M321">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M325">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M348">View MathML</a> is homogeneous of degree −j in its arguments. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M349">View MathML</a> is homogeneous of degree <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M350">View MathML</a> in its arguments, and we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M351">View MathML</a>

From the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M345">View MathML</a> may be chosen in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M128">View MathML</a> and the elementary inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M354">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M355">View MathML</a>) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M356">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M357">View MathML</a>. Inserting this and the homogeneity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M348">View MathML</a> into the above representation, we see

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M359">View MathML</a>

which shows (3.11). In the same way, for the proof of (3.12) we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M360">View MathML</a> in the above integral representation and obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M361">View MathML</a>

This finishes the proof of (3.11) and (3.12) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M362">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M363">View MathML</a>, we use the substitution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M364">View MathML</a> in the integral representation. As indicated above, (a) and (b) are immediate consequences of (3.11) and (3.12), respectively. □

The last lemma in connection with the following result is the essential step for the proof of the a priori estimates from the main theorem.

Lemma 3.9The functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M365">View MathML</a>

are Michlin functions.

Proof By Lemma 3.8(b), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M366">View MathML</a>

with Michlin functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M367">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M368">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M369">View MathML</a> we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M370">View MathML</a>

(3.13)

By a homogeneity argument we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M371">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M372">View MathML</a> are Michlin functions, and therefore the matrix on the right-hand side of (3.13) is a Michlin function. In order to apply Remark 3.7(c), we have to show that the norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M373">View MathML</a> is uniformly bounded.

For this, we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M373">View MathML</a> in the form of a Schur complement: For an invertible block matrix, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M375">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M376">View MathML</a>. Applied to the matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M369">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M378">View MathML</a>

(3.14)

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M379">View MathML</a>

(3.15)

By (3.8), the matrices <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M367">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M368">View MathML</a> and, consequently, the matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M369">View MathML</a> are homogeneous of degree 0 in their arguments. Thus we can write S in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M383">View MathML</a>

We set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M384">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M385">View MathML</a>

and write S as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M386">View MathML</a>

The matrices <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M367">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M368">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M389">View MathML</a> are bounded for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M390">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M288">View MathML</a>. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M392">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M393">View MathML</a>, we see that there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M394">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M395">View MathML</a>

holds for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M390">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M288">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M398">View MathML</a>. For these <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M291">View MathML</a> and q, a Neumann series argument shows that the norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M400">View MathML</a> is bounded by 2.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M390">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M288">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M403">View MathML</a>, the tuple <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M404">View MathML</a> belongs to the compact set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M405">View MathML</a>. Now we use the fact that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M390">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M288">View MathML</a>, the matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M408">View MathML</a> is invertible, and therefore the matrix on the right-hand side of (3.13) is invertible, too. This yields the invertibility of S, and by continuity the inverse matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M400">View MathML</a> is bounded for these <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M291">View MathML</a> and q.

Therefore, we have seen that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M411">View MathML</a> holds for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M412">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M288">View MathML</a>. From the explicit description of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M373">View MathML</a> in (3.14) and the uniform boundedness of the other coefficients in (3.14), we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M415">View MathML</a> holds for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M291">View MathML</a> and q. By Remark 3.7(c), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M369">View MathML</a> is a Michlin function.

The above proof also shows that the modification <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M418">View MathML</a> is a Michlin function. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M419">View MathML</a> remains unchanged and that we obtain an additional factor <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M420">View MathML</a> in the right upper corner which does not affect the boundedness. □

4 Proof of the a priori estimate

In this section, we will investigate the mapping properties of the solution operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M272">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M273">View MathML</a> introduced in Lemma 3.5. As above, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M423">View MathML</a>. In the following, we will use the abbreviations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M424">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M425">View MathML</a>. Based on Lemma 3.8 and 3.9 and on the continuity of the Hilbert transform, it is not difficult to obtain the following result.

Lemma 4.1 (a) Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M426">View MathML</a>

Then for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M427">View MathML</a>and all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M303">View MathML</a>we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M429">View MathML</a>

The same holds when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M430">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M431">View MathML</a>are replaced by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M432">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M433">View MathML</a>, respectively.

(b) Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M434">View MathML</a>

Then for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M427">View MathML</a>and all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M303">View MathML</a>we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M437">View MathML</a>

Proof (a) For fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M303">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M439">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M440">View MathML</a>

We have to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M441">View MathML</a>. For this, we write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M442">View MathML</a>

Inserting this into the definition of the solution operator, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M443">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M444">View MathML</a>

Here we used the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M445">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M369">View MathML</a> are Michlin functions and therefore Fourier multipliers and that the (one-sided) Hilbert transform

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M447">View MathML</a>

induces a bounded operator in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M448">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M294">View MathML</a>.

This shows the first statement in (a). Obviously, the uniform estimate also holds in the case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M430">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M431">View MathML</a> are multiplied with the same factor, as this factor cancels out.

The proof of (b) follows exactly in the same way with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M369">View MathML</a> being replaced by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M418">View MathML</a> from Lemma 3.5. □

The next result shows the key estimate for the solution of (3.4).

Theorem 4.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M454">View MathML</a>be a solution of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M455">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M456">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M267">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M269">View MathML</a>be an extension ofgto the half-space. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M459">View MathML</a>. Then for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M288">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M461">View MathML</a>, the inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M462">View MathML</a>

hold.

Proof In this proof, we will write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M463">View MathML</a>. We use the equivalences

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M464">View MathML</a>

which can easily be seen by a Michlin type argument. With this, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M465">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M466">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M454">View MathML</a> be a solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M468">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M469">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M470">View MathML</a> be an extension of g. By a density argument, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M471">View MathML</a>. By Lemma 3.5, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M472">View MathML</a>. Applying Lemma 4.1, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M473">View MathML</a>

(4.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M474">View MathML</a>

(4.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M475">View MathML</a>

(4.3)

Inserting the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M476">View MathML</a>

into the right-hand side, we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M477">View MathML</a>

(4.4)

On the other hand, inserting the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M478">View MathML</a>

(which holds for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M461">View MathML</a> with a constant C depending on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M480">View MathML</a>), we get in particular

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M481">View MathML</a>

Dividing by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M482">View MathML</a>, we see that this implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M483">View MathML</a>

(4.5)

In the same way as above, we can apply Lemma 4.1 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M484">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M485">View MathML</a> instead of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M430">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M431">View MathML</a>, respectively. We see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M488">View MathML</a>

With the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M489">View MathML</a>

this gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M490">View MathML</a>

This and equations (4.4) and (4.5) yield the statements of the theorem. □

Now we can consider the problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M491">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M492">View MathML</a> in the half-space. As mentioned in Remark 2.3, this finishes the proof of the main theorem.

Theorem 4.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M454">View MathML</a>be a solution of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M494">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M456">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M496">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M497">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M459">View MathML</a>. Then for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M288">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M461">View MathML</a>the following a priori estimates hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M501">View MathML</a>

(4.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M502">View MathML</a>

(4.7)

Proof (i) We start the proof with some preliminary remarks. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M503">View MathML</a> be the restriction operator from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M504">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M167">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M506">View MathML</a> is a retraction from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M507">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M508">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M509">View MathML</a>, and there exists a co-retraction (independent of k), i.e. a total extension operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M510">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M511">View MathML</a> for all k (see [22], Theorem 5.21).

For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M512">View MathML</a>, the trace operator to the boundary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M513">View MathML</a> is a bounded operator from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M508">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M515">View MathML</a>. This holds both with respect to the parameter-independent norms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M516">View MathML</a> and the parameter-dependent norms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M517">View MathML</a>. For the latter, we refer to [8], Proposition 2.2. There exists a parameter-dependent extension operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M518">View MathML</a> which satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M519">View MathML</a> and whose operator norm with respect to the parameter-dependent norms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M520','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M520">View MathML</a> is bounded by a constant independent of q for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M288">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M461">View MathML</a> (see, e.g., [8], Proposition 2.3). In particular, we will consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M523">View MathML</a> which is a parameter-independent continuous extension operator.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M524">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M525">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M526">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M527">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M528">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M529">View MathML</a>. Then a simple application of Michlin’s theorem shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M530','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M530">View MathML</a> induces a bounded linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M531','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M531">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M509">View MathML</a>. Due to the compact support of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M533','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M533">View MathML</a>, the related operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M534">View MathML</a> belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M535','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M535">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M509">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M150">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M538','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M538">View MathML</a> commute due to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M539">View MathML</a>.

(ii) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M454">View MathML</a> be a solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M541','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M541">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M542','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M542">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M288">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M461">View MathML</a>. We define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M545','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M545">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M546','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M546">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M547','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M547">View MathML</a>. For

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M548','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M548">View MathML</a>

we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M549','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M549">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M550','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M550">View MathML</a>

By the continuity of the involved operators, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M551','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M551">View MathML</a>

(4.8)

(iii) Similarly, we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M552','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M552">View MathML</a>. It is well known (or easily seen by Michlin’s theorem) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M553','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M553">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M554','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M554">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M555','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M555">View MathML</a>

(4.9)

(iv) We define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M556','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M556">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M557','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M557">View MathML</a>. Then w is a solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M558','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M558">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M559','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M559">View MathML</a>. Applying the parameter-independent extension operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M523">View MathML</a> to every component of g, we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M561','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M561">View MathML</a>. An extension <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M562','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M562">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M563','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M563">View MathML</a> is given by omitting the trace to the boundary. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M564','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M564">View MathML</a>.

For the left-hand side of (4.6), we remark that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M565','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M565">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M566','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M566">View MathML</a>

By Theorem 4.3, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M567">View MathML</a>

(4.10)

From (4.8) we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M568','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M568">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M569','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M569">View MathML</a> we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M570','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M570">View MathML</a> in the same way from (4.9). Inserting this into (4.10), we obtain the first inequality (4.6) of the theorem.

(v) The proof of (4.7) follows the same lines. However, here we start with the refined estimate (4.3). For the left-hand side of (4.7), we note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M571','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M571">View MathML</a>

Now we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M572','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M572">View MathML</a> with the parameter-dependent extension operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M573','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M573">View MathML</a> from part (i). Then the term on the right-hand side of (4.3) equals

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M574','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M574">View MathML</a>

(4.11)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M117">View MathML</a>, we can estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M576','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M576">View MathML</a>

Concerning the terms involving <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M577','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M577">View MathML</a>, we use

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M578','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M578">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M117">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M580','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M580">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M581','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M581">View MathML</a>. Finally, the terms involving <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M569','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M569">View MathML</a> can be estimated by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M583','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M583">View MathML</a>

So we see that all terms in (4.11) can be estimated by the right-hand side of (4.7), and the proof of (4.7) is finished. □

Remark 4.4 (a) The estimate (2.5) does not imply uniqueness of a solution to (2.3) because the elliptic part <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M32">View MathML</a> of the solution appears in a norm of lower order on the right-hand side of the estimate. Nevertheless, in bounded domains such estimates give rise to the Fredholm property of a corresponding solution operator.

(b) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M585','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M585">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M586','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M586">View MathML</a>, we obtain in particular

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M587','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M587">View MathML</a>

from (4.7). This is the basis for resolvent estimates and spectral properties of the corresponding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M1">View MathML</a>-realization in the case where the Dirichlet problem for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M183">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M590','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/22/mathml/M590">View MathML</a> is invertible. Here we have a connection to eigenvalue problems with weights and the Calderón method as studied in, e.g., [4].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally to this work.

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