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Asymptotics of solutions of the heat equation in cones and dihedra under minimal assumptions on the boundary

Vladimir A Kozlov1 and Jürgen Rossmann2*

Author affiliations

1 Institute of Mathematics, Linköping University, Linköping, SE-58183, Sweden

2 Institute of Mathematics, University of Rostock, Rostock, D-18051, Germany

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Citation and License

Boundary Value Problems 2012, 2012:142  doi:10.1186/1687-2770-2012-142


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/142


Received:25 June 2012
Accepted:14 November 2012
Published:3 December 2012

© 2012 Kozlov and Rossmann; 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

In the first part of the paper, the authors obtain the asymptotics of Green’s function of the first boundary value problem for the heat equation in an m-dimensional cone K. The second part deals with the first boundary value problem for the heat equation in the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M1">View MathML</a>. Here the right-hand side f of the heat equation is assumed to be an element of a weighted <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M2">View MathML</a>-space. The authors describe the behavior of the solution near the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M3">View MathML</a>-dimensional edge of the domain.

Introduction

The paper is concerned with the first boundary value problem for the heat equation

(1)

(2)

in the domain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M6">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M7">View MathML</a> is a cone in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M9">View MathML</a>, Ω denotes a subdomain of the unit sphere, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M10">View MathML</a> is the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M3">View MathML</a>-dimensional edge of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M12">View MathML</a>. We are interested in the asymptotics of solutions in the class of the weighted Sobolev spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M13">View MathML</a>. Here the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M14">View MathML</a> is defined for an arbitrary integer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M15">View MathML</a> and real <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M17">View MathML</a>, β as the set of all function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M18">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M19">View MathML</a> with the finite norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M20">View MathML</a>

(3)

In the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M21">View MathML</a>, we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M22">View MathML</a>. If, moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M23">View MathML</a>, then we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M24">View MathML</a>.

For the case of smooth boundary Ω (of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M25">View MathML</a>), the asymptotics of solutions was obtained in our previous paper [1]. For the particular case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M27">View MathML</a>, we refer also to the paper [2] by Kozlov and Maz’ya, and for the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M29">View MathML</a>, to the paper [3] by de Coster and Nicaise. The goal of the present paper is to describe the asymptotics of solutions with a remainder in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M13">View MathML</a> under minimal smoothness assumptions on the boundary. Throughout the paper, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M31">View MathML</a>.

The paper consists of two parts. The first part (Section 1) deals with the asymptotics of the Green function for the heat equation in the cone K. We obtain the same decomposition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M32">View MathML</a>

as in [4,5] (for the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M36">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M37">View MathML</a>, see Section 1.1). However, the proof in [4,5] does not work if Ω is only of the class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M38">View MathML</a>. We give a new proof, which is completely different from that in [4,5]. Our tools are estimates for solutions of the Dirichlet problem for the Laplace equation in a cone in weighted <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M39">View MathML</a> Sobolev spaces and asymptotic formulas for solutions of this problem which were obtained in the papers [6,7] by Maz’ya and Plamenevskiĭ. Moreover, we use the estimates of the Green function in the recent paper [8] by Kozlov and Nazarov. In contrast to the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M40">View MathML</a>, the estimates for the second order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M41">View MathML</a>- and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M42">View MathML</a>-derivatives of the remainder <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M43">View MathML</a> contain an additional factor <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M44">View MathML</a> with a negative exponent −ε. Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M45">View MathML</a> is the distance from the boundary of ∂K.

In the second part of the paper (Section 2), we apply the results of Section 2 in order to obtain the asymptotics of solutions of the problem (1), (2) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M46">View MathML</a>. We show that, under a certain condition on β, there exists a solution of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M47">View MathML</a>

with a remainder <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M48">View MathML</a>. Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M49">View MathML</a> is an extension of the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M50">View MathML</a>

Φ denotes the fundamental solution of the heat equation in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M51">View MathML</a>. The proof of this result (Theorem 2.2) is essentially the same as in [1]. However, the proofs of some lemmas in [1] have to be modified under our weaker assumptions on Ω.

At the end of the paper, we show that the extensions of the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M52">View MathML</a> can be defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M53">View MathML</a>

where T and R are certain smooth functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M54">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M51">View MathML</a>, respectively (see the beginning of Section 3 for their definition). This extends the result of [[1], Corollary 4.5] to the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M56">View MathML</a>.

1 The Green function of the heat equation in a cone

We start with the problem

(4)

(5)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M59">View MathML</a> be the Green function for the problem (4), (5). It is defined for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M60">View MathML</a> as the solution of the problem

Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M62">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M63">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M64">View MathML</a> are defined below), and ζ is a function in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M65">View MathML</a> equal to one in a neighborhood of the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M66">View MathML</a>. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M67">View MathML</a> is the space of all functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M68">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M69">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M70">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M71">View MathML</a>. The goal of this section is to describe the behavior of the Green function for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M72">View MathML</a>.

1.1 Asymptotics of Green’s function

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M73">View MathML</a> be the nondecreasing sequence of eigenvalues of the Beltrami operator −δ on Ω (with the Dirichlet boundary condition) counted with their multiplicities, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M74">View MathML</a> be an orthonormal (in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M75">View MathML</a>) sequence of eigenfunctions corresponding to the eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M76">View MathML</a>. Furthermore, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M77">View MathML</a>

This means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M78">View MathML</a> are the solutions of the quadratic equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M79">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M80">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M81">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M82">View MathML</a> .

By [[8], Theorem 3],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M83">View MathML</a>

(6)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M84">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M85">View MathML</a>. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M45">View MathML</a> denotes the distance of the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M41">View MathML</a> from the boundary ∂K. Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M88">View MathML</a> is defined as zero for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M89">View MathML</a>, while <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M88">View MathML</a> is an arbitrarily small positive real number if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M91">View MathML</a>. Actually, the estimate (6) is proved in [8] only for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M92">View MathML</a>, but for a more general class of operators, parabolic operators with discontinuous in time coefficients. If the coefficients in [8] do not depend on t, then one can use the same argument as in the proof of [[8], Theorem 3] when treating the derivatives along the edge of the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M93">View MathML</a>. This argument shows that the kth derivative with respect to t will bring only an additional factor <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M94">View MathML</a> to the right-hand side of (6).

The following lemma will be applied in the proof of Lemma 1.2. Here and in the sequel, we use the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M95">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M96">View MathML</a>.

Lemma 1.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M59">View MathML</a>be the Green function introduced above, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M98">View MathML</a>denote the Green function of the initial-boundary value problem

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M100">View MathML</a>

(7)

Proof The solution of the problem

(8)

(9)

is given by the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M103">View MathML</a>

We define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M104">View MathML</a>

Then it follows from (8) and (9) that

Furthermore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M106">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M107">View MathML</a>

Comparing this with the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M108">View MathML</a>

we get (7). □

In the sequel, σ is an arbitrary real number satisfying the conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M109">View MathML</a>

(10)

We define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M110">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M111">View MathML</a>, while

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M112">View MathML</a>

(11)

where

(12)

(13)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M115">View MathML</a>. Here, we used the notation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M116">View MathML</a>

We define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M117">View MathML</a> as the weighted Sobolev space with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M118">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M119">View MathML</a> and integer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M15">View MathML</a>.

Lemma 1.2Suppose thatσis a real number such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M121">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M122">View MathML</a>is not integer for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M123">View MathML</a>. Furthermore, let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M119">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M125">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M126">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M127">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M129">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M85">View MathML</a>.

Proof We prove the lemma by induction in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M131">View MathML</a>.

First, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M132">View MathML</a>. Then it follows from [[7], Corollary 4.1 and Theorem 4.2] (see also [[6], Theorem 3.2]) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M133">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M129">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M85">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M137">View MathML</a>. Thus, the assertion of the lemma is true for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M111">View MathML</a>.

Suppose the assertion is proved for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M139">View MathML</a>. Now let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M140">View MathML</a>. We set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M141">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M142">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M143">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M21">View MathML</a>, where ε is a sufficiently small positive number. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M145">View MathML</a>

By the induction hypothesis, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M146">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M147">View MathML</a> is given by (11) (with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M148">View MathML</a> instead of σ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M149">View MathML</a> instead of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M35">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M151">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M152">View MathML</a>. The coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M153">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M147">View MathML</a> are given by (13) and satisfy the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M155">View MathML</a>. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M156">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M157">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M129">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M159">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M160">View MathML</a>. Using the same equality for the Green function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M59">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M162">View MathML</a>

Furthermore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M163">View MathML</a>

Using the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M164">View MathML</a>

we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M165">View MathML</a>

(14)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M166">View MathML</a>

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M167">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M21">View MathML</a>). Let χ be a smooth function with compact support on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M169">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M170">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M171">View MathML</a>. Using the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M95">View MathML</a>, the function χ can be also considered as a function in K. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M173">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M174">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M175">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M176">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M129">View MathML</a>. Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M179">View MathML</a> and

Applying [[7], Theorem 4.2], we obtain

(15)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M182">View MathML</a>. The coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M183">View MathML</a> are given by the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M184">View MathML</a>

(16)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M185">View MathML</a>. The integral in (16) is well defined, since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M186">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M187">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M188">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M189">View MathML</a>. The remainder <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M190">View MathML</a> and the coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M183">View MathML</a> in (15) satisfy the estimate

(17)

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M193">View MathML</a>. This means that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M194">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M195">View MathML</a>. Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M196">View MathML</a>

(18)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M197">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M198">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M199">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M85">View MathML</a>. Using (18) and the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M201">View MathML</a>

we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M202">View MathML</a>

It remains to show that the coefficients

(19)

in (15) have the form (13) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M204">View MathML</a>. First, note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M205">View MathML</a>

since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M206">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M207">View MathML</a>.

Obviously, the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M208">View MathML</a> and

contain only functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M210">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M211">View MathML</a>. Thus, the orthogonality of the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M34">View MathML</a> implies

(20)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M214">View MathML</a>. Applying Lemma 1.1, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M215">View MathML</a> has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M216">View MathML</a>

(21)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M217">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M218">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M219">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M160">View MathML</a>, it follows from (18) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M221">View MathML</a>

The function on the right-hand side belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M222">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M129">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M160">View MathML</a>, while the left-hand side belongs only to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M222">View MathML</a> if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M227">View MathML</a>

Combining the last equality with (21), we get the representation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M228">View MathML</a>

Inserting this into the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M229">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M230">View MathML</a>

The substitution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M231">View MathML</a> leads to the differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M232">View MathML</a>

which has the solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M233">View MathML</a>

with arbitrary constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M234">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M235">View MathML</a>. Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M236">View MathML</a>

(22)

Using (6) and (17), one gets the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M237">View MathML</a>

with certain functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M238">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M239">View MathML</a>. Thus, the constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M235">View MathML</a> in (22) must be zero. Integrating (19), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M241">View MathML</a>

by means of (20). Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M242">View MathML</a>

The integral on the left-hand side is equal to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M243">View MathML</a>. Thus, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M244">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M245">View MathML</a>

This means that the formula (13) is valid for the coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M36">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M247">View MathML</a>. The proof of the lemma is complete. □

1.2 Point estimates for the remainder in the asymptotics of Green’s function

We are interested in point estimates for the remainder <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M248">View MathML</a> in Lemma 1.2 in the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M72">View MathML</a>. For this, we need the following lemma.

Lemma 1.3Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M250">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M251">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M252">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M253">View MathML</a>

with a constantcindependent ofu.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M254">View MathML</a> be a point int K, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M255">View MathML</a> be a ball centered at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M254">View MathML</a> with radius <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M257">View MathML</a>. We introduce the new coordinates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M258">View MathML</a> and set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M259">View MathML</a>. Obviously, the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M260">View MathML</a> has the distance 1 from ∂K. Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M261">View MathML</a>

This implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M262">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M263">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M264">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M265">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M266">View MathML</a>

The result follows. □

Using the last two lemmas, we can prove the following theorem.

Theorem 1.1Suppose thatσis a real number satisfying (10). Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M267">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M268">View MathML</a>

(23)

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M72">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M84">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M85">View MathML</a>. Here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M272">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M89">View MathML</a>, while<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M88">View MathML</a>is an arbitrarily small positive real number if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M91">View MathML</a>.

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M276">View MathML</a> for small positive ε, we may assume, without loss of generality, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M122">View MathML</a> is not integer for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M278">View MathML</a>. We prove the theorem by induction in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M131">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M132">View MathML</a>, then the assertion of the theorem follows from [[8], Theorem 3]. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M140">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M282">View MathML</a>, and that the theorem is proved for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M139">View MathML</a>. We set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M141">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M142">View MathML</a>. In the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M21">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M148">View MathML</a> be an arbitrary real number satisfying the inequalities <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M288">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M289">View MathML</a>. By the induction hypothesis, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M290">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M147">View MathML</a> is given by (11) (with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M148">View MathML</a> instead of σ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M149">View MathML</a> instead of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M35">View MathML</a>). Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M295">View MathML</a> for sufficiently small δ, it follows from the induction hypothesis that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M296">View MathML</a>

(24)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M297">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M84">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M85">View MathML</a>. As was shown in the proof of Lemma 1.2, the remainder <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M300">View MathML</a> admits the decomposition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M301">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M302">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M303">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M129">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M85">View MathML</a>. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M137">View MathML</a>. Furthermore (cf. (14)),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M308">View MathML</a>

Let χ be a smooth cut-off function on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M169">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M310">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M311">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M312">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M313">View MathML</a>. We define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M314">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M315">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M129">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M317">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M318">View MathML</a>

Thus, by [[7], Theorem 4.1], there exists a constant c such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M319">View MathML</a>

(25)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M129">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M85">View MathML</a>. We estimate the norm of f. Using (24), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M323">View MathML</a>

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M324">View MathML</a>. Thus,

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M326">View MathML</a> vanishes outside the region <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M327">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M328">View MathML</a>, the estimate (24) also yields

Finally, it follows from the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M330">View MathML</a>

that

Consequently, by (25),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M332">View MathML</a>

(26)

with a positive constant κ. Applying the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M333">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M252">View MathML</a> (cf. [[9], Lemma 1.2.3]), we obtain (23) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M89">View MathML</a>.

It remains to prove the estimate (23) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M91">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M337">View MathML</a> be the “regularized distance” of the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M41">View MathML</a> to the boundary ∂K, i.e., ρ is a smooth function in K satisfying the inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M339">View MathML</a>

with positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M340">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M341">View MathML</a> (cf. [[10], Chapter VI, § 2.1]). Moreover, ρ satisfies the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M342">View MathML</a>

(27)

We consider the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M343">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M344">View MathML</a>. It follows from the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M345">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M346">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M347">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M348">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M349">View MathML</a>. Using (24) and (27), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M350">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M351">View MathML</a>. The inequalities <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M352">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M353">View MathML</a> yield

(see (26)). Consequently by [[7], Theorem 4.1], the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M355">View MathML</a> satisfies the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M356">View MathML</a>

Applying Lemma 1.3 to the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M357">View MathML</a> with an arbitrary multi-index α with length <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M91">View MathML</a>, we get

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M91">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M85">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M252">View MathML</a>. Since p can be chosen arbitrarily large, the estimate (23) holds in the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M91">View MathML</a>. The proof is complete. □

2 Asymptotics of solutions of the problem in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M12">View MathML</a>

Now we consider the problem (1), (2) in the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M12">View MathML</a>. Throughout this section, it is assumed that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M46">View MathML</a>, where p and β satisfy the inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M367">View MathML</a>

(28)

and q is an arbitrary real number >1. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M59">View MathML</a> be the Green function of the problem (4), (5). Furthermore, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M369">View MathML</a>

be the fundamental solution of the heat equation in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M51">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M371">View MathML</a>

is the Green function of the problem (1), (2). We consider the solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M372">View MathML</a>

(29)

of the problem (1), (2).

We again denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M373">View MathML</a> the function (11) introduced in Section 1. In the sequel, σ is an arbitrary real number such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M374">View MathML</a>

(30)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M375">View MathML</a>

(31)

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M376">View MathML</a>. Let χ be an infinitely differentiable function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M377">View MathML</a> equal to one on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M378">View MathML</a> and vanishing on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M313">View MathML</a>. We define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M380">View MathML</a>

Obviously,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M381">View MathML</a>

where

(32)

(33)

We also consider the decomposition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M384">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M385">View MathML</a>

(34)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M386">View MathML</a>

(35)

is an extension of the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M387">View MathML</a>

(36)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M36">View MathML</a> defined by (13). Our goal is to show that both remainders v and w are elements of the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M13">View MathML</a>. We start with the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M390">View MathML</a>.

2.1 Estimates in weighted <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M39">View MathML</a> Sobolev spaces

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M14">View MathML</a> be the weighted Sobolev space with the norm (3). Furthermore, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M393">View MathML</a>

In this subsection, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M394">View MathML</a>, where p and β satisfy (28). First, we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M395">View MathML</a>. This was shown in [[1], Corollary 2.3] for the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M40">View MathML</a>. In the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M397">View MathML</a>, we must keep in mind that the second-order derivatives of the eigenfunctions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M34">View MathML</a> must not be bounded. Then we have the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M399">View MathML</a>

(37)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M84">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M272">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M89">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M88">View MathML</a> is an arbitrarily small positive real number if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M89">View MathML</a>. However, this requires only a small modification of the proof in [1].

Lemma 2.1Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M405">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M406">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M407">View MathML</a>

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M84">View MathML</a>and allk.

Proof A simple calculation (see the proof of [[1], Corollary 1]) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M409">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M410">View MathML</a> denotes the commutator of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M411">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M412">View MathML</a>. Obviously, the inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M413">View MathML</a>

are satisfied on the support of the kernel

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M414">View MathML</a>

(38)

Since, moreover, the eigenfunctions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M34">View MathML</a> satisfy the inequality (37) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M84">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M417">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M84">View MathML</a>. Using Hölder’s inequality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M419">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M420">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M421">View MathML</a>

The substitution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M422">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M423">View MathML</a> yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M424">View MathML</a>

i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M425">View MathML</a>. Consequently,

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M427">View MathML</a>

Substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M428">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M429">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M430">View MathML</a>

This means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M431">View MathML</a> is a constant. This proves the lemma. □

Next, we estimate the first-order x-derivatives of the remainder v. For this, we employ the following lemma (cf. [[11], Lemma A.1]).

Lemma 2.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M432">View MathML</a>be the integral operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M433">View MathML</a>

(39)

with a kernel<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M434">View MathML</a>satisfying the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M435">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M436">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M437">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M438">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M439">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M432">View MathML</a>is bounded on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M441">View MathML</a>.

In the proof of the following assertion, we use another decomposition of the remainder v as in [[1], Lemma 2.4]. This allows us to apply directly the estimate in Theorem 1.1.

Lemma 2.3Letpandβsatisfy the condition (28). Furthermore, letvbe the function (33), where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M442">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M119">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M444">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M89">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M446">View MathML</a>

with a constantcindependent off. The same is true for the functionw.

Proof Obviously,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M447">View MathML</a>

where

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M449">View MathML</a>

We show that the integral operators with the kernels

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M450">View MathML</a>

are bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M451">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M452">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M453">View MathML</a>. Using Theorem 1.1, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M454">View MathML</a>

where ε is an arbitrarily small positive number. Applying Lemma 2.2 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M455">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M456">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M457">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M458">View MathML</a>, we conclude that the integral operator with the kernel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M459">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M451">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M89">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M462">View MathML</a> on the support of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M463">View MathML</a>, the estimate (6) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M464">View MathML</a>

with arbitrary real a. Thus, by Lemma 2.2, the integral operator with the kernel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M465">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M451">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M89">View MathML</a>.

We consider the kernel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M468">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M373">View MathML</a> has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M470">View MathML</a>

we get the representation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M471">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M472">View MathML</a>

Here we used the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M473">View MathML</a> on the support of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M474">View MathML</a>. The inequalities <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M473">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M476">View MathML</a> imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M477">View MathML</a>

It is no restriction to assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M478">View MathML</a> in addition to (30) and (31). Therefore, we can apply Lemma 2.2 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M455">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M457">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M481">View MathML</a> to the integral operator with the kernel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M482">View MathML</a>. It follows that the integral operator with the kernel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M483">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M451">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M89">View MathML</a>. Consequently, the integral operator with the kernel

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M486">View MathML</a>

is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M451">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M89">View MathML</a>. This proves the lemma. □

Furthermore, the assertions of [[1], Lemmas 2.5, 2.6, Theorem 2.7] are also valid if Ω is only of the class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M38">View MathML</a>. The proof under this weaker assumption on Ω does not require any modifications of the method in [1]. We give here only the formulation of [[1], Theorem 2.7].

Theorem 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M490">View MathML</a>, wherepandβsatisfy the condition (28). Then there exists a solution of the problem (1), (2) which has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M491">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M492">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M493">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M49">View MathML</a>are given by (12), (31) and (35), respectively. The functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M49">View MathML</a>depend only on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M497">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M498">View MathML</a>andtand satisfy the estimates

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M499">View MathML</a>

(40)

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M500">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M501">View MathML</a>

(41)

for allk, α, γ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M502">View MathML</a>.

2.2 Weighted <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M2">View MathML</a> estimates for the remainder

We assume now that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M504">View MathML</a> and consider the decomposition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M505">View MathML</a>

of the solution (29), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M506">View MathML</a> is defined by (34). Our goal is to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M507">View MathML</a> if p and β satisfy the condition (28). For the proof, we will use the next lemma which follows directly from [[12], Theorem 3.8].

Lemma 2.4Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M432">View MathML</a>is a linear operator on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M509">View MathML</a>satisfying the following conditions:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M510">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M511">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M512">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M513">View MathML</a>and for all functionshwith support in the layer<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M514">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M515">View MathML</a> .

Then the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M516">View MathML</a>

holds for arbitraryq, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M517">View MathML</a>. Here the constantcdepends only on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M340">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M341">View MathML</a>, pandq.

The condition (ii) of the last lemma can be verified in some cases by means of the following lemma (cf. [[8], Lemma 10]).

Lemma 2.5Suppose that the kernel of the integral operator (39) satisfies the estimate

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M521">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M522">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M436">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M524">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M438">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M439">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M527">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M528">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M529">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M530','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M530">View MathML</a>with support in the layer<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M531','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M531">View MathML</a>. Here, the constantcis independent of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M532">View MathML</a>andδ.

It is more easy to estimate the remainder <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M533','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M533">View MathML</a>, where Σ is defined by (32). For this reason, we estimate the difference <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M534">View MathML</a> first.

Lemma 2.6Let Σ and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M506">View MathML</a>be the functions (32) and (34), respectively. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M46">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M537','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M537">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M538','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M538">View MathML</a>

for allkandα, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M84">View MathML</a>. Here, the constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M540','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M540">View MathML</a>are independent off. In particular, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M541','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M541">View MathML</a>.

Proof We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M542','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M542">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M543">View MathML</a> is given by (38). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M544','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M544">View MathML</a> be the integral operator with the kernel

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M545','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M545">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M84">View MathML</a>. As was shown in the proof of Lemma 2.1, this operator is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M451">View MathML</a>. Now let h be a function in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M548','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M548">View MathML</a> with support in the layer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M531','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M531">View MathML</a> satisfying the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M550','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M550">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M551','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M551">View MathML</a>

Analogously to the proof of Lemma 2.1, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M552','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M552">View MathML</a>

(42)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M84">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M554','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M554">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M555','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M555">View MathML</a> on the support of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M556','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M556">View MathML</a>, we can append the factors

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M557','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M557">View MathML</a>

with arbitrary exponents a and b on the right-hand side of (42). For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M521">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M559','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M559">View MathML</a>, we obviously have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M560','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M560">View MathML</a>. Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M561','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M561">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M521">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M563','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M563">View MathML</a>, where a and b are arbitrary real numbers and ε is an arbitrarily small positive real number. Hence, by Lemmas 2.4 and 2.5, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M544','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M544">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M565','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M565">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M566','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M566">View MathML</a>.

We consider the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M567">View MathML</a> with the kernel

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M568','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M568">View MathML</a>

It follows from the boundedness of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M544','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M544">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M39">View MathML</a> that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M567">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M572','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M572">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M188">View MathML</a>. Furthermore, one can check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M574','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M574">View MathML</a>

with arbitrary a and b. Thus, as in the first part of the proof, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M567">View MathML</a> (and therefore also the adjoint operator of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M544','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M544">View MathML</a>) is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M577','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M577">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M578','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M578">View MathML</a>. This means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M544','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M544">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M565','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M565">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M581','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M581">View MathML</a>. The lemma is proved. □

By means of Lemma 2.5, it is also possible to prove the assertion of [[1], Theorem 3.7] under the weaker assumption on Ω of the present paper.

Theorem 2.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M504">View MathML</a>, wherepandβsatisfy the condition (28) andqis an arbitrary real number, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M583','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M583">View MathML</a>. Then there exists a solution of the problem (1), (2) which has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M584','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M584">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M585','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M585">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M49">View MathML</a>are given by (12) and (35), respectively, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M48">View MathML</a>. The functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M49">View MathML</a>are extensions of the functions (36) depending only on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M497">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M498">View MathML</a>andtand satisfy the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M591','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M591">View MathML</a>

(43)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M592','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M592">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M502">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M594">View MathML</a>.

Proof We have to show that the integral operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M595','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M595">View MathML</a> with the kernel

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M596','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M596">View MathML</a>

is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M565','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M565">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M71">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M390">View MathML</a> this is true by Theorem 2.1. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M600','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M600">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M601','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M601">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M602','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M602">View MathML</a> be the same functions as in the proof of Lemma 2.3 and let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M603','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M603">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M604','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M604">View MathML</a>. We show that the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M605','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M605">View MathML</a> satisfy the condition (ii) of Lemma 2.4. Let h be a function in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M548','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M548">View MathML</a> with support in the layer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M531','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M531">View MathML</a> satisfying the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M608','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M608">View MathML</a> for all x. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M609','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M609">View MathML</a>

Using Theorem 1.1, we get

Thus,

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M521">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M563','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M563">View MathML</a>. Applying Lemma 2.5 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M614','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M614">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M615','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M615">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M458">View MathML</a>, we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M617','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M617">View MathML</a>

(44)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M618','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M618">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M71">View MathML</a>. Analogously, the estimate (6) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M620','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M620">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M521">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M563','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M563">View MathML</a>, where a is an arbitrary real number. Here, we used the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M462">View MathML</a> on the support of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M624','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M624">View MathML</a>. Thus, by Lemma 2.5, the inequality (44) holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M625','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M625">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M71">View MathML</a>.

Analogously to the estimation of the kernel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M468">View MathML</a> in the proof of Lemma 2.3, we obtain the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M628','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M628">View MathML</a>

by means of (37). We may assume, without loss of generality, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M478">View MathML</a> in addition to (30) and (31). Then we conclude from Lemma 2.5 that (44) is valid for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M630','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M630">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M71">View MathML</a>. Hence, by Lemma 2.4, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M595','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M595">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M565','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M565">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M566','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M566">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M71">View MathML</a>.

In order to prove this for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M636','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M636">View MathML</a>, we consider the adjoint operator. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M637','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M637">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M638','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M638">View MathML</a> be the integral operators with the kernels

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M639','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M639">View MathML</a>

respectively. From the boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M595','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M595">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M451">View MathML</a> it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M637','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M637">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M572','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M572">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M188">View MathML</a>. We show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M645','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M645">View MathML</a>

(45)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M513">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M452">View MathML</a> and for all functions h with support in the layer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M514">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M649','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M649">View MathML</a>. Let h be such a function. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M650','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M650">View MathML</a>

By means of 1.1, we obtain

Analogously, the estimate (6) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M652','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M652">View MathML</a>

where a is an arbitrary real number, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M653','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M653">View MathML</a> on the support of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M654','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M654">View MathML</a>. Applying Lemma 2.5, we obtain (45) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M71">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M656','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M656">View MathML</a>. Using the representation for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M657','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M657">View MathML</a>, the estimate (37), and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M658','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M658">View MathML</a> on the support of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M659','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M659">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M660','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M660">View MathML</a>

We may assume again that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M478">View MathML</a> in addition to (30) and (31). Then it follows from Lemma 2.5 that (45) is valid for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M630','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M630">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M71">View MathML</a>. Therefore, by Lemma 2.4, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M664','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M664">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M577','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M577">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M578','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M578">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M71">View MathML</a>. This means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M595','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M595">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M565','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M565">View MathML</a> for all q if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M71">View MathML</a>. The proof of the theorem is complete. □

3 Another representation for the coefficients

As was proved [[1], Lemma 4.1], the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M49">View MathML</a> in Theorem 2.1 can be replaced by other extensions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M672','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M672">View MathML</a> of the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M673','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M673">View MathML</a> provided these extensions also satisfy the conditions (40) and (41). Note that the proof of this assertion in [1] is also correct under our assumptions on the boundary of Ω. Moreover, it was proved in [[1], Lemma 4.4], for the particular case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M390">View MathML</a>, that the extension

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M675','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M675">View MathML</a>

satisfies the conditions (40) and (41). Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M676','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M676">View MathML</a> is a smooth function with support in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M169">View MathML</a> satisfying the conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M678','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M678">View MathML</a>

with certain positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M679','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M679">View MathML</a>, κ and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M680','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M680">View MathML</a>

Furthermore, R is a smooth function with support on the cube <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M681','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M681">View MathML</a> having the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M682','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M682">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M683','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M683">View MathML</a>

(46)

with a sufficiently large integer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M684','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M684">View MathML</a>.

We extend the result of [[1], Lemma 4.4] to the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M685','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M685">View MathML</a>. First, note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M686','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M686">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M687','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M687">View MathML</a> is the integral operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M688','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M688">View MathML</a>

with the kernel

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M689','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M689">View MathML</a>

Our goal is to show that the operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M690','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M690">View MathML</a>

is bounded if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M502">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M692','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M692">View MathML</a>. Since the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M693','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M693">View MathML</a> depends only on the variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M95">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M498">View MathML</a>, and t, it suffices to prove that the operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M696','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M696">View MathML</a>

is bounded if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M697','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M697">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M500">View MathML</a>.

We define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M699','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M699">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M700','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M700">View MathML</a>

This means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M699','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M699">View MathML</a> is the integral operator with the kernel

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M702','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M702">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M95">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M217">View MathML</a>. As was shown in [1], the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M699','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M699">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M451">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M697','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M697">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M500">View MathML</a>. In order to prove the boundedness in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M565','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M565">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M685','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M685">View MathML</a>, we verify the condition (ii) of Lemma 2.4. For this, we apply the following lemma.

Lemma 3.1Suppose that the kernel of the integral operator (39) satisfies the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M711','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M711">View MathML</a>

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M521">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M522">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M95">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M715','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M715">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M436">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M717','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M717">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M718','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M718">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M719','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M719">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M530','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M530">View MathML</a>with support in the layer<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M531','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M531">View MathML</a>. Here, the constantcis independent of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M532">View MathML</a>andδ.

Proof Obviously,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M723','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M723">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M724','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M724">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M725','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M725">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M726','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M726">View MathML</a>. Consequently, it follows from our assumption on K that

Thus, we can apply Lemma 2.5. □

We will show that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M699','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M699">View MathML</a> satisfies the condition of the last lemma. This leads to the following assertion.

Lemma 3.2Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M729','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M729">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M730','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M730">View MathML</a>and that at least one of the conditions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M697','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M697">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M500">View MathML</a>is satisfied. Furthermore, we assume that the number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M684','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M684">View MathML</a>in (46) is greater than<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M734','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M734">View MathML</a>. Then the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M699','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M699">View MathML</a>is bounded in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M565','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M565">View MathML</a>.

Proof For the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M737','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M737">View MathML</a>, we refer to [[1], Lemma 4.4].

We consider the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M517">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M530','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M530">View MathML</a> be an arbitrary function with support in the layer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M514">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M741','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M741">View MathML</a> for all x. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M742','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M742">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M743','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M743">View MathML</a>, while

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M744','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M744">View MathML</a>

(47)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M745','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M745">View MathML</a>. We verify the condition of Lemma 3.1 for the kernel of the last integral operator. To this end, we use the same decomposition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M746','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M746">View MathML</a>

for the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M747','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M747">View MathML</a>-derivatives of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M748','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M748">View MathML</a> as in the proof of [[1], Lemma 4.4], where

and

Here we used the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M751','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M751">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M752','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M752">View MathML</a>. Applying the estimates

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M754','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M754">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M755','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M755">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M756','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M756">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M757','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M757">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M758','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M758">View MathML</a>

with arbitrary positive M and certain positive <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M759','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M759">View MathML</a>. Furthermore, the estimates

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M760','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M760">View MathML</a>

and

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M762','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M762">View MathML</a> with certain positive κ and arbitrary positive M yield

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M763','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M763">View MathML</a>

Finally, (cf. formulas (4.7) and (4.8) in [1]), we get the estimates

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M764','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M764">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M765','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M765">View MathML</a>

if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M697','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M697">View MathML</a>. Thus,

(48)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M768','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M768">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M521">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M563','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M563">View MathML</a>, and s lies between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M532">View MathML</a> and τ, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M772','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M772">View MathML</a>. Consequently, it follows from (48) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M773','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M773">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M521">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M563','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M563">View MathML</a>. This means that the kernel of the integral operator (47) satisfies the condition of Lemma 3.1 if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M776','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M776">View MathML</a>. Hence, by Lemmas 2.4 and 3.1, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M699','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M699">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M565','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M565">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M697','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M697">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M500">View MathML</a>.

In order to prove this for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M636','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M636">View MathML</a>, we consider the adjoint operator. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M782','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M782">View MathML</a> be the integral operator with the kernel

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M783','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M783">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M699','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M699">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M451">View MathML</a> under the assumptions of the lemma, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M782','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M782">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M572','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M572">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M188">View MathML</a>. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M789','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M789">View MathML</a> is a function with support in the layer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M514">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M741','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M741">View MathML</a> for all x. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M792','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M792">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M745','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M745">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M794','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M794">View MathML</a>

As was shown above, the derivatives of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M543">View MathML</a> satisfy the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M796','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M796">View MathML</a>

with the same M as before. This implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M797','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M797">View MathML</a>

Therefore, it follows from Lemma 3.1 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M798','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M798">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M789','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M789">View MathML</a> with support in the layer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M531','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M531">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M697','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M697">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M500">View MathML</a>. Applying Lemma 2.4, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M782','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M782">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M804','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M804">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M578','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M578">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M697','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M697">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M500">View MathML</a>. Consequently, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M699','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M699">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M565','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M565">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M810','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M810">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M697','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M697">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M500">View MathML</a>. The proof is complete. □

Using the last lemma, we obtain the following result which generalizes [[1], Corollary 4.5].

Theorem 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M504">View MathML</a>, wherepandβsatisfy the condition (28) andqis an arbitrary real number, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M583','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M583">View MathML</a>. Then there exists a solution of the problem (1), (2) which has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M815','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M815">View MathML</a>

(49)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M585','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M585">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M52">View MathML</a>are given by (12) and (36), respectively, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M48">View MathML</a>.

Proof By Lemma 3.2, the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M819','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M819">View MathML</a> satisfy the same condition (43) as the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M49">View MathML</a> in Theorem 2.2. Thus, it follows from [[1], Lemma 4.1] that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M821','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M821">View MathML</a>

This together with Theorem 2.2 implies (49) with a remainder <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/142/mathml/M48">View MathML</a>. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors achieved the key results of the paper during a research stay of JR in Linköping in October 2012. Both authors read and approved the final manuscript.

Acknowledgements

The paper partially arose during the stay of J. Rossmann in Linköping in October 2011. The second author thanks the Department of Mathematics at the University of Linköping for the hospitality.

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