This article is part of the series Proceedings of the International Congress in Honour of Professor Hari M. Srivastava.

Open Access Research

Diffraction problems for quasilinear parabolic systems with boundary intersecting interfaces

Qi-Jian Tan* and Chao-Yi Pan

Author Affiliations

Department of Mathematics, Chengdu Normal University, Chengdu, 611130, P.R. China

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Boundary Value Problems 2013, 2013:99  doi:10.1186/1687-2770-2013-99


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/99


Received:19 January 2013
Accepted:9 April 2013
Published:22 April 2013

© 2013 Tan and Pan; 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 this paper, we discuss the n-dimensional diffraction problem for weakly coupled quasilinear parabolic system on a bounded domain Ω, where the interfaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M1">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M2">View MathML</a>) are allowed to intersect with the outer boundary Ω and the coefficients of the equations are allowed to be discontinuous on the interfaces. The aim is to show the existence of solutions by approximation method. The approximation problem is a diffraction problem with interfaces, which do not intersect with Ω.

MSC: 35R05, 35K57, 35K65.

Keywords:
diffraction problem; quasilinear parabolic system; interface; approximation method

1 Introduction

Let Ω be a bounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M3">View MathML</a> with boundary Ω (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M4">View MathML</a>), and let Ω be partitioned into a finite number of subdomains <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M5">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M6">View MathML</a>) separated by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M7">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M2">View MathML</a>, are interfaces, which do not intersect with each other. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M10">View MathML</a>, set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M11">View MathML</a>

In this paper, we consider the diffraction problem for quasilinear parabolic reaction-diffusion system in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M12">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M17">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M18">View MathML</a>

(1.2)

repeated indices i or j indicate summation from 1 to n, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M19">View MathML</a> is the unit normal vector to Γ (the positive direction of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M20">View MathML</a> is fixed in advance), the symbol <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M21">View MathML</a> denotes the jump of a quantity across <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M22">View MathML</a>, and the coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M25">View MathML</a> are allowed to be discontinuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M22">View MathML</a>. In the following, we refer to the conditions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M22">View MathML</a> in (1.1) as diffraction conditions.

The diffraction problems often appear in different fields of physics, ecology, and technics. In some of them, the interfaces are allowed to intersect with the outer boundary Ω (see [1-5]). The linear diffraction problems have been treated by many researchers (see [1-10]). For the quasilinear parabolic and elliptic diffraction problems, when all of the interfaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M7">View MathML</a> do not intersect with Ω, the existence and uniqueness of the solutions have been investigated in [11-14] by Leray-Schauder principle and the method of upper and lower solutions. In this paper, we investigate the existence of solutions of (1.1) when the interfaces are allowed to intersect with Ω. In this case, because of the existence of the intersection of Γ and Ω, the methods in [11-14] can not be extended. We shall show the existence of solutions by approximation method. The approximation problem is a diffraction problem with interfaces which do not intersect with Ω.

The plan of the paper is as follows. In Sect. 2, we give the notations, hypotheses and an example, and state the existence theorem of the solutions. Section 3 is devoted to the proof of the existence theorem.

2 The hypotheses, main result and example

2.1 The notations, hypotheses and main result

First, let us introduce more notations and function spaces.

For any set S, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M29">View MathML</a> denotes its closure. The symbol <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M30">View MathML</a> means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M31">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M32">View MathML</a>.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M33">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M35">View MathML</a>, intersect with the outer boundary Ω, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M37">View MathML</a> do not intersect with Ω. Assume that the domain Ω is partitioned into subdomains <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M39">View MathML</a>, separated by interfaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M34">View MathML</a>, and partitioned into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M42">View MathML</a>, separated by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M36">View MathML</a>. The interface of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M38">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M45">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M34">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M47">View MathML</a>. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M48">View MathML</a>

We see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M49">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M50">View MathML</a> is the spaces of Hölder continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M51">View MathML</a> with exponent <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M52">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M53">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M54">View MathML</a> are the Hilbert spaces with scalar products <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M55">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M56">View MathML</a>, respectively. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M57">View MathML</a>

For the vector functions with N-components we denote the above function spaces by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M59">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M61">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M62">View MathML</a>, respectively.

Moreover, we recall the following.

Definition 2.1 (see [13,15])

Write u in the split form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M63">View MathML</a>

The vector function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M64">View MathML</a> is said to be mixed quasimonotone in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M65">View MathML</a> with index vector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M66">View MathML</a> if for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M67">View MathML</a>, there exist nonnegative integers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M69">View MathML</a>, satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M70">View MathML</a>

such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M71">View MathML</a> is nondecreasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M72">View MathML</a>, and is nonincreasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M73">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M74">View MathML</a>.

The following hypotheses will be used in this paper:

(H) (i) Ω and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M2">View MathML</a>, are of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M77">View MathML</a> for some exponent <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M78">View MathML</a> and there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M79">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M80">View MathML</a> such that for every open ball <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M81">View MathML</a> centered at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M82">View MathML</a> and radius <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M83">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M84">View MathML</a>

Assume that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M35">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M86">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M87">View MathML</a>

(2.1)

(ii) Assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M88">View MathML</a>

(2.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M89">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M90">View MathML</a> are defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M91">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M92">View MathML</a> are defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M93">View MathML</a>, and all of them are allowed to be discontinuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M94">View MathML</a>.

(iii) There exist constant vectors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M95">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M96">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M97">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M98">View MathML</a>

(2.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M69">View MathML</a> are all independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M101">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M102">View MathML</a>

The vector functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M103">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M39">View MathML</a>, are mixed quasimonotone in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M105">View MathML</a> with the same index vector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M66">View MathML</a>.

(iv) For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M67">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M89">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M111">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M112">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M113">View MathML</a>. There exist a positive nonincreasing function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M114">View MathML</a> and a positive nondecreasing function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M115">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M116">View MathML</a> such that

(2.4)

(2.5)

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M67">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M120">View MathML</a> for some domain Ξ with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M122">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M6">View MathML</a>), and the following compatibility condition on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M124">View MathML</a> holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M125">View MathML</a>

(2.6)

Definition 2.2 A function u is said to be a solution of (1.1) if u possesses the following properties: (i) For some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M126">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M6">View MathML</a>. For any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M30">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M130">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M131">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M132">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M133">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M134">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M136">View MathML</a>; (ii) u satisfies the equations in (1.1) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M137">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M6">View MathML</a>, the diffraction conditions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M139">View MathML</a> and the parabolic boundary conditions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M140">View MathML</a>.

The main result in this paper is the following existence theorem.

Theorem 2.1Let Hypothesis (H) hold. Then problem (1.1) has a solutionuin<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M105">View MathML</a>.

2.2 An example

We next give an example satisfying the conditions in Hypothesis (H).

Example 2.1 In problem (1.1), let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M142">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M143">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M144">View MathML</a>, and let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M145">View MathML</a>

The outer boundary of domain is a circle of radius 10 with the center at the origin, whereas the interface curves are two parabolas and a smaller circle of radius 1 (see Figure 1). We see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M146">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M147">View MathML</a> intersect with Ω, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M148">View MathML</a> does not.

thumbnailFigure 1. The example of the domain and the interfaces for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M149">View MathML</a>.

For the coefficients of the equations and the boundary values in (1.1) we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M150">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M151">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M152">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M153">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M154">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M155">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M156">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M157">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M158">View MathML</a> are all positive constants for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M159">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M160">View MathML</a>.

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M161">View MathML</a>

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M67">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M163">View MathML</a>

We find that these functions satisfy (2.2) and the hypothesis (iv) of (H). Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M164">View MathML</a>. Then the requirements on M in (2.3) become

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M165">View MathML</a>

It follows from these inequalities that there exist positive constant vector M, such that m and M satisfy (2.3). Furthermore, the vector functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M103">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M167">View MathML</a>, are mixed quasimonotone in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M105">View MathML</a> with the same index vector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M169">View MathML</a>. The above arguments show that the conditions in Hypothesis (H) can be satisfied.

3 The proof of the existence theorem

3.1 Preliminaries

Lemma 3.1The following statements hold true:

(i) For any given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M170">View MathML</a>, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M171">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M172">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M173">View MathML</a>

(ii) There exists a positive number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M174">View MathML</a>such that for any given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M175">View MathML</a>, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M176">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M177">View MathML</a>

Proof By (2.1), if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M170">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M171">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M180">View MathML</a>. Thus for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M181">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M182">View MathML</a>. Again by (2.1) we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M183">View MathML</a>. This proves the result in (i).

For any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M184">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M170">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M186">View MathML</a>, then it follows from (i) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M187">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M188">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M189">View MathML</a>, there exist positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M190">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M191">View MathML</a>

Hence, the conclusion in (ii) follows from the above relation by taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M192">View MathML</a>. □

For an arbitrary ε, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M193">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M194">View MathML</a> be smooth function with values between 0 and 1 such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M195">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M196">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M197">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M198">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M199">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M200">View MathML</a>. Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M201">View MathML</a>

(3.1)

Lemma 3.2<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M202">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M39">View MathML</a>, are smooth functions with values between 0 and 1, and possess the property

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M204">View MathML</a>

(3.2)

Let functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M205">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M39">View MathML</a>, be defined on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M207">View MathML</a>, and let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M208">View MathML</a>

(3.3)

Then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M170">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M210">View MathML</a>

(3.4)

Proof Since (3.2) is a special case of (3.4) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M211">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M212">View MathML</a>, we only prove (3.4).

Case 1. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M213">View MathML</a>, then the conclusion of (i) in Lemma 3.1 implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M214">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M215">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M216">View MathML</a>. (3.1) yields that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M217">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M218">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M219">View MathML</a>. These, together with (3.3), imply that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M220">View MathML</a>.

Case 2. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M221">View MathML</a>, then the conclusion of (ii) in Lemma 3.1 shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M222">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M223">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M216">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M225">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M218">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M227">View MathML</a>. Again by (3.3) we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M228">View MathML</a>.

Case 3. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M214">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M230">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M184">View MathML</a>, then Lemma 3.1 yields that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M232">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M233">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M234">View MathML</a>, and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M235">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M236">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M237">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M238">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M239">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M240">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M241">View MathML</a>.

Case 4. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M242">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M184">View MathML</a>, then it follows from Lemma 3.1 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M244">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M245">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M246">View MathML</a>, and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M247">View MathML</a>. Again by the conclusion of (i) in Lemma 3.1 we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M248">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M249">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M250">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M251">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M252">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M253">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M254">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M255">View MathML</a>. □

3.2 The approximation problem of (1.1)

In this subsection, we construct a problem to approximate (1.1).

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M67">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M257">View MathML</a>

(3.5)

It follows from hypothesis (iv) of (H), (3.2) and (3.5) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M258">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M259">View MathML</a> are in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M260">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M112">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M262">View MathML</a> is in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M263">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M42">View MathML</a>), the vector function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M265">View MathML</a> is mixed quasimonotone in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M105">View MathML</a> with index vector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M66">View MathML</a>, and

(3.6)

(3.7)

We note that the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M258">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M259">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M262">View MathML</a> are continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M273">View MathML</a>, and are allowed to be discontinuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M94">View MathML</a>.

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M37">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M276">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M277">View MathML</a>. Take two subdomains <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M278">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M279">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M280">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M281">View MathML</a> be an arbitrary smooth function taking values in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M282">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M283">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M284">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M285">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M286">View MathML</a>. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M287">View MathML</a>

(3.8)

with a sufficiently smooth nonnegative averaging kernel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M288">View MathML</a> that is equal to zero for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M289">View MathML</a> and is such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M290">View MathML</a>. Then from the hypothesis (iv) of (H) and [[1], Chapter II] we know that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M67">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M292">View MathML</a> is in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M293">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M294">View MathML</a> is in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M295">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M42">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M297">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M298">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M297">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M300">View MathML</a>. Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M301">View MathML</a>

(3.9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M302">View MathML</a> is a positive constant, independent of ε. Furthermore, (3.4), (3.5) and (3.8) show that for small enough ε,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M303">View MathML</a>

These, together with (2.6), imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M304">View MathML</a>

(3.10)

For any given ε, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M193">View MathML</a>, consider the approximation diffraction problem of (1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M306">View MathML</a>

(3.11)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M307">View MathML</a>

We note that the interfaces in (3.11) are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M308">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M37">View MathML</a>) which do not intersect with Ω. In view of (3.10), the compatibility condition on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M124">View MathML</a> holds.

Proposition 3.1Problem (3.11) has a unique piecewise classical solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M311">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M105">View MathML</a>possessing the following properties:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M313">View MathML</a>

(3.12)

Proof Problem (3.11) is a special case of [[13], problem (1.1)] without time delays. Formulas (2.3) and (3.5) show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M314">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M315">View MathML</a> are a pair of bounded and coupled weak upper and lower solutions of (3.11) in the sense of [[13], Definition 2.2]. We find that the conditions of [[13], Theorem 4.1] are all fulfilled. Then from [[13], Theorem 4.1], we obtain that problem (3.11) has a unique piecewise classical solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M311">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M105">View MathML</a> possessing the properties in (3.12). □

3.3 The uniform estimates of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M318">View MathML</a>

In the following discussion, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M81">View MathML</a> be an arbitrary open ball of radius ρ with center at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M320">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M321">View MathML</a> be an arbitrary cylinder of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M322">View MathML</a>.

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M67">View MathML</a>, consider the equality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M324">View MathML</a> for any function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M325">View MathML</a> from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M326">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M327">View MathML</a> and for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M328">View MathML</a>, t from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M329">View MathML</a>. In view of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M330">View MathML</a>, it follows from (3.6), (3.7), (3.9) and the formula of integration by parts that

(3.13)

(3.14)

Similarly, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M333">View MathML</a> and for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M334">View MathML</a> we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M335">View MathML</a>

(3.15)

Lemma 3.3There exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M336">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M337">View MathML</a>) andCdepending only on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M338">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M339">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M340">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M341">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M342">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M343">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M344">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M302">View MathML</a>, independent ofε, such that

(3.16)

(3.17)

Proof (3.16) follows from (3.14), (3.6), (3.7), (3.9) and [[1], Chapter V, Theorem 1.1 and Remark 1.2]. Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M348">View MathML</a> in (3.14) and using Cauchy’s inequality, we can obtain (3.17). □

Lemma 3.4For any given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M349">View MathML</a>, let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M350">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M130">View MathML</a>. Then there exist positive constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M352">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M353">View MathML</a>) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M354">View MathML</a>depending only on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M355">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M356">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M357">View MathML</a>and the parameters<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M338">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M340">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M341">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M342">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M343">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M344">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M302">View MathML</a>, independent ofε, such that for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M41">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M366">View MathML</a>,

(3.18)

(3.19)

For any given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M369">View MathML</a>, let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M370">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M371">View MathML</a>. Then there exist positive constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M372">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M373">View MathML</a>) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M374">View MathML</a>depending only on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M375">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M376">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M377">View MathML</a>and the parameters<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M338">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M340">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M341">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M342">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M382">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M344">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M302">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M385">View MathML</a>

(3.20)

Proof Choose a subdomain B satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M386">View MathML</a>. (3.4) and (3.5) show that for small enough ε,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M387">View MathML</a>

(3.21)

Then the same proofs as those of [[13], formulas (3.30) and (3.31)] give (3.18) and (3.19). If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M370">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M389">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M349">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M391">View MathML</a>. Hence, the conclusion in (3.20) follows from (3.18), (3.19), (3.21) and the same argument as that for [[13], formula (3.37)]. □

In the rest of this subsection, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M392">View MathML</a> be an arbitrary fixed number in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M393">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M394">View MathML</a> be an arbitrary fixed subdomain satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M395">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M396">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M397">View MathML</a>. We next investigate the uniform estimates in the neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M398">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M320">View MathML</a> be any point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M400">View MathML</a>. [[2], Chapter 3, Section 16] and [13] show that there exists a ball <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M81">View MathML</a> with center at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M320">View MathML</a> such that we can straighten <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M403">View MathML</a> out by introducing a local coordinate system <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M404">View MathML</a>. Our assumptions concerning Γ imply that we can divide <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M405">View MathML</a> into a finite number of pieces and to introduce for each of them coordinates y. Since the investigations in the rest of this subsection are local properties, we can assume without loss of generality that the interface <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M406">View MathML</a> lies in the plane <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M407">View MathML</a>. Then by (3.4), when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M408">View MathML</a> the coefficients of problem (3.11) can be represented in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M409">View MathML</a>

(3.22)

and the diffraction conditions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M410">View MathML</a> in problem (1.1) can be represented in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M411">View MathML</a>

(3.23)

Lemma 3.5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M412">View MathML</a>. Then there exist positive constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M413">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M414">View MathML</a>) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M415">View MathML</a>depending only on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M416">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M417">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M357">View MathML</a>, and the parameters<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M338">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M340">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M341">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M342">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M343">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M344">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M302">View MathML</a>, independent ofε, such that

(3.24)

(3.25)

(3.26)

Proof It follows from (3.22) and Hypothesis (H) that

(3.27)

and from the equations in (3.11) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M430">View MathML</a>

(3.28)

Then using (3.13), (3.15), (3.22), (3.27) and (3.28), we can prove (3.24)-(3.26) by a slight modification of the proofs of [[13], formulas (3.30) and (3.31)]. The detailed proofs are omitted. □

3.4 The proof of Theorem 2.1

From estimates (3.16), (3.17) and the Arzela-Ascoli theorem it follows that we can find a subsequence (we retain the same notation for it) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M431">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M432">View MathML</a> converges in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M433">View MathML</a> to u and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M434">View MathML</a> converges weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M435">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M436">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M136">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M438">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M439">View MathML</a>. Furthermore, the parabolic boundary conditions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M318">View MathML</a> in (3.11) imply that u satisfies the parabolic boundary conditions in (1.1).

For any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M369">View MathML</a>, and for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M370">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M371">View MathML</a>, (3.20) yields that there exists a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M444">View MathML</a> (denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M431">View MathML</a> still) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M431">View MathML</a> converges in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M447">View MathML</a> to u. By letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M448">View MathML</a>, from (3.21) and the equations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M449">View MathML</a> in (3.11) we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M450">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M451">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M377">View MathML</a> are arbitrary, then u satisfies the equations in (3.11) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M137">View MathML</a>.

For any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M349">View MathML</a> and for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M350">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M130">View MathML</a>, we see from (3.18), (3.19) that there exists a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M444">View MathML</a> (denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M431">View MathML</a> still) such that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M136">View MathML</a> and for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M41">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M461">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M434">View MathML</a> converges in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M463">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M436">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M465">View MathML</a> converges in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M466">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M467">View MathML</a>. Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M468">View MathML</a>

(3.29)

By letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M448">View MathML</a> we conclude from (3.21) and the diffraction conditions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M470">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M318">View MathML</a> in (3.11) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M472">View MathML</a>

(3.30)

For any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M473">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M394">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M395">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M396">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M397">View MathML</a>, the estimates (3.24)-(3.26) imply that for any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M130">View MathML</a> there exists a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M444">View MathML</a> (denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M431">View MathML</a> still) such that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M481">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M67">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M483">View MathML</a>

(3.31)

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M484">View MathML</a>

(3.32)

We next show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M485">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M486">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M487">View MathML</a>

By (3.27), (3.17), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M488">View MathML</a>

and by (2.2), (3.22),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M489">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M434">View MathML</a> converges weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M435">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M436">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M136">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M494">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M448">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M496">View MathML</a> converges weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M497">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M498">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M136">View MathML</a>. This, together with (3.31), implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M500">View MathML</a>

(3.33)

and u satisfies the diffraction conditions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M501">View MathML</a> in (3.23).

In view of (3.30) u satisfies the diffraction conditions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M502">View MathML</a> in (1.1). Furthermore, (3.29), (3.32) and (3.33) imply that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M369">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M30">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M505">View MathML</a>

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/99/mathml/M506">View MathML</a>. Therefore, u is a solution of (1.1). This completes the proof of Theorem 2.1.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The authors would like to thank the reviewers and the editors for their valuable suggestions and comments. The work was supported by the research fund of Department of Education of Sichuan Province (10ZC127) and the research fund of Chengdu Normal University (CSYXM12-06).

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