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On the maximum principle for elliptic operators in weighted spaces

Loredana Caso* and Roberta D’Ambrosio

Author Affiliations

Dipartimento di Matematica, Università di Salerno, Via Giovanni Paolo II, I - 84084, Fisciano, Italy

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Boundary Value Problems 2014, 2014:91  doi:10.1186/1687-2770-2014-91

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/91


Received:16 January 2014
Accepted:15 April 2014
Published:6 May 2014

© 2014 Caso and D¿Ambrosio; licensee Springer.

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

Abstract

We establish a maximum principle for subsolutions of second order elliptic equations. In particular, we consider some linear operators with leading coefficients locally VMO, while the other coefficients and the boundary conditions involve a suitable weight function.

MSC: 35J25, 35R05, 35B50.

Keywords:
elliptic operators; VMO-coefficients; weighted Sobolev spaces; maximum principle

1 Introduction

It is well known that a priori estimates and uniqueness results, which are necessary in the proof of the well-posedness for boundary value problems for elliptic equations in nondivergence form, are based on Aleksandrov type estimates, i.e., on estimates for the maximum of a solution in terms of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M1">View MathML</a>-norm of the right-hand side.

If Ω is a bounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M2">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M3">View MathML</a>) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M4">View MathML</a>

(1.1)

is a uniformly elliptic operator in Ω, the classical result of AD Aleksandrov states that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M5">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M6">View MathML</a> in ∂Ω, verifies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M7">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M8">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M9">View MathML</a>), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M10">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M11">View MathML</a> depends only on n, Ω, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M12">View MathML</a> and on the ellipticity constant.

There have been various directions of developments and extensions of Aleksandrov estimate. For example, maximum principles have been established in different types of boundary problems, such as in the stationary oblique derivative problem or in the stationary Venttsel’ problem. Another direction of development of the Aleksandrov ideas is the extension of maximum estimates to equations with lower order coefficients and right-hand sides in other function classes (for example, in spaces with anisotropic norms or weighted spaces). In particular, a large number of works is devoted to the weakening of requirements for the right-hand side of the equation considered (see, for example, [1] and its large bibliography).

In this framework, it is well known that additional hypotheses on the leading coefficients are necessary to obtain the estimates. Several authors have obtained estimates for the maximum of a solution through the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M13">View MathML</a>-norms of the right-hand side (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M14">View MathML</a>) under different conditions on the leading coefficients.

For instance, if Ω is an arbitrary open subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M16">View MathML</a>, a bound of type (1.2) and a consequent uniqueness result can be found in [2]. In fact, it has been proved that, if the coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M17">View MathML</a> are bounded and locally VMO, the coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M18">View MathML</a>, a satisfy suitable summability conditions, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M19">View MathML</a>, then for any solution u of the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M20">View MathML</a>

(1.3)

there exist a ball <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M21">View MathML</a> and a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M11">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M23">View MathML</a>

(1.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M24">View MathML</a> is the negative part of f,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M25">View MathML</a>

and c depends on n, p, on the ellipticity constant and on the regularity of the coefficients of L.

If the boundary of a domain has various singularities, as for example corners or edges, then, in accordance with the linear theory, it is natural to assume that the lower order coefficients and the right-hand side of the equation belong to some weighted spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M13">View MathML</a>, where the weight is usually a power of the distance function from the ‘singular set’ on the boundary of domain. In these cases, the estimates on the solutions are obtained in terms of such weight function.

For instance, if ρ is a bounded weight function related to the distance function from a non-empty subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M27">View MathML</a> of the boundary of an arbitrary domain Ω, not necessarily bounded and regular (see Section 2 for the definition of such weight function), in [3] has been studied a problem similar to the problem (1.3) with boundary conditions and data related to the weight function ρ. In particular, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M29">View MathML</a>, the coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M17">View MathML</a> are bounded and locally VMO, the coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M18">View MathML</a>, a belong to suitable weighted spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M32">View MathML</a>, in [3] the author has proved that the solution u of the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M33">View MathML</a>

(1.5)

verifies the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M34">View MathML</a>

(1.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M21">View MathML</a> is an open ball and the constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M11">View MathML</a> depends on n, p, s, ρ, on the ellipticity constant and on the regularity of the coefficients of L. As a consequence, some uniqueness results are also obtained. Results of this type are also established in [4] under the more general hypothesis <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M37">View MathML</a>, but for an operator L with coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M38">View MathML</a>.

The aim of this paper is to improve the above quoted results in [3] by obtaining a similar estimate under much weaker assumptions. In particular, the main difference lies in the hypotheses on the coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M18">View MathML</a>, a which are not supposed to belong to weighted spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M32">View MathML</a> but just to appropriate weighted Sobolev spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M41">View MathML</a> (see Section 2 for the definition of such weighted spaces), which strictly contain the weighted spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M32">View MathML</a>. Moreover, as in [4], we consider the more general hypothesis <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M43">View MathML</a>.

2 Notation

In this section we introduce some notation used throughout this paper. Moreover, we recall the definitions of a class of weight functions and of some function spaces in which the coefficients of our operator will be chosen.

Let A be a Lebesgue measurable subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M2">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M45">View MathML</a> be the collection of all Lebesgue measurable subsets of A. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M46">View MathML</a>, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M47">View MathML</a> the Lebesgue measure of F and by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M48">View MathML</a> the class of restrictions to F of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M49">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M50">View MathML</a>. Moreover, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M51">View MathML</a> is a space of functions defined on F, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M52">View MathML</a> the class of all functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M53">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M54">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M55">View MathML</a>. Furthermore, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M56">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M57">View MathML</a>), we put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M58">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M59">View MathML</a> is a decreasing function and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M60">View MathML</a>, we can refer to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M61">View MathML</a> as the modulus of continuity of g in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M62">View MathML</a>.

Let Ω be an open subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M64">View MathML</a>. We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M65">View MathML</a> the class of measurable weight functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M66">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M67">View MathML</a>

(2.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M68">View MathML</a> is independent of x and y, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M69">View MathML</a> is the open ball of radius <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M70">View MathML</a> centered at y.

We remark that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M65">View MathML</a> contains the class of all functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M72">View MathML</a> which are Lipschitz continuous in Ω with Lipschitz constant less than 1.

Typical examples of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M73">View MathML</a> are the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M74">View MathML</a>

if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M75">View MathML</a> and, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M76">View MathML</a> and S is a nonempty subset of ∂Ω, the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M77">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M73">View MathML</a> we put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M79">View MathML</a>

(2.2)

We recall that the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M27">View MathML</a> is a closed subset of ∂Ω and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M81">View MathML</a>

(see [5]).

It is well known that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M82">View MathML</a>

(2.3)

and, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M83">View MathML</a>[5,6],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M84">View MathML</a>

(2.4)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M73">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M86">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M87">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M28">View MathML</a>, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M89">View MathML</a> the space of distributions u on Ω such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M90">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M91">View MathML</a>. We observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M89">View MathML</a> is a Banach space with the norm defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M93">View MathML</a>

Moreover, it is separable if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M94">View MathML</a>, reflexive if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M95">View MathML</a>, and, in particular, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M96">View MathML</a> is an Hilbert space. We put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M97">View MathML</a>, and we observe that the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M98">View MathML</a> is dense in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M99">View MathML</a> (see [7,8]).

A more detailed account of properties of the above defined weighted Sobolev spaces can be found in [7,9] and [8].

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M100">View MathML</a>, we put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M101">View MathML</a>

(2.5)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M73">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M94">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M28">View MathML</a>, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M105">View MathML</a> the class of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M106">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M107">View MathML</a>

(2.6)

Obviously <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M105">View MathML</a> is a Banach space with the norm defined by (2.6). It is easy to prove that the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M109">View MathML</a> is a subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M105">View MathML</a> (see [10]). Thus, we can define a new space of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M111">View MathML</a> as the closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M109">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M105">View MathML</a>.

We recall the following characterization of the above defined space (see [10]):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M114">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M115">View MathML</a> denotes the characteristic function of the set E.

Therefore, we define modulus of continuity of g in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M111">View MathML</a> as a map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M117">View MathML</a> such that [11]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M118">View MathML</a>

(2.7)

Further properties of above mentioned function spaces can be found in [5,10], and [11].

If Ω has the property

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M119">View MathML</a>

(2.8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M120">View MathML</a> and A is a positive constant independent of x and r, it is possible to consider the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M121">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M122">View MathML</a>) composed by all functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M123">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M124">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M125">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M126">View MathML</a>, with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M127">View MathML</a>

we will say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M128">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M129">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M130">View MathML</a>. A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M131">View MathML</a> is called a modulus of continuity of g in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M132">View MathML</a> if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M133">View MathML</a>

We say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M134">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M135">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M136">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M137">View MathML</a> denotes the zero extension of ζg outside of Ω. A more detailed account of properties of the above defined spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M138">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M132">View MathML</a> can be found in [12].

We conclude this section introducing a class of applications needed in the sequel.

From now on we consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M140">View MathML</a> and we suppose that the following condition on ρ holds:

(h0) there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M141">View MathML</a> which is equivalent to ρ and such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M142">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M143">View MathML</a> is independent of x (see [6]).

We observe that the condition (h0) holds, for example, if Ω is an unbounded open set with the cone property, or if the open set Ω has not the cone property but the weight function ρ is equivalent to the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M144">View MathML</a> (see [6]).

Let us fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M145">View MathML</a> satisfying the conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M146">View MathML</a>

(2.9)

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M147">View MathML</a>, we put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M148">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M149">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M100">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M151">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M147">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M153">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M154">View MathML</a>

(2.10)

Moreover, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M147">View MathML</a>, it is easy to prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M156">View MathML</a>

(2.11)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M157">View MathML</a>

(2.12)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M158">View MathML</a>

(2.13)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M159">View MathML</a>

(2.14)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M160">View MathML</a> depends on k and σ, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M161">View MathML</a> depend only on n. Furthermore, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M28">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M163">View MathML</a>

(2.15)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M164">View MathML</a>

(2.16)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M165">View MathML</a> depends on s and n.

3 Hypotheses and preliminary results

Suppose that Ω has the property (2.8) and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M14">View MathML</a>. Consider in Ω the differential operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M167">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M168">View MathML</a>

with the following assumptions on the coefficients:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M169">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M170">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M171">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M172">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M173">View MathML</a>.

Fixing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M174">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M175">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M176">View MathML</a>, we put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M177">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M178">View MathML</a>.

We observe that under assumptions (h1) and (i1), the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M167">View MathML</a> from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M180">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M181">View MathML</a> is bounded and the following estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M182">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M183">View MathML</a> depends on n, p, r, ρ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M184">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M185">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M186">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M187">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M188">View MathML</a>.

Let v be a solution of the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M189">View MathML</a>

(3.1)

We want to prove a bound for the solution v of the above problem (see Lemma 3.1 below), which will be the primary technical tool in the proof of our main result (see the next Section). In order to use a classical result of Vitanza (see, Theorem 2.1 in [13]) it is necessary to make an appropriate change of variables which allows to transform the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M167">View MathML</a> into a differential operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M191">View MathML</a> whose lower order coefficients, in particular, belonging to Lebesgue spaces and their moduli of continuity can be estimated by moduli of continuity of the corresponding coefficients of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M192">View MathML</a>. To this aim, let us consider the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M193">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M194">View MathML</a>

(3.2)

Clearly

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M195">View MathML</a>

For any function g defined on B, we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M196">View MathML</a>

(3.3)

Using the equivalence between ρ and σ it is easy to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M197">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M198">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M199">View MathML</a>; moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M200">View MathML</a>

(3.4)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M201">View MathML</a>

(3.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M202">View MathML</a> depends on n, ρ, r and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M203">View MathML</a> depends on n, ρ, p.

On the other hand, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M204">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M205">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M206">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M207">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M208">View MathML</a>. Thus, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M209">View MathML</a>

(3.6)

Using again the equivalence between ρ and σ and (2.7), from (3.6) we also deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M210">View MathML</a>

(3.7)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M211">View MathML</a>

(3.8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M165">View MathML</a> depends on ρ, r and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M213">View MathML</a> depends on ρ, p.

We are now able to prove the requested a priori bound.

Lemma 3.1Suppose that the conditions (h1) and (i1) hold. Letvbe a solution of the problem (3.1). Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M214">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M215">View MathML</a>

(3.9)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M216">View MathML</a>depends onn, p, r, ρ, ν, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M217">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M218">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M219">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M220">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M221">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M222">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M223">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M224">View MathML</a>, and where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M225">View MathML</a>are the extensions of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M17">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M2">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M228">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M229">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M230">View MathML</a>. Taking into account the definitions (3.2) and (3.3), it is easily seen that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M231">View MathML</a>

and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M232">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M233">View MathML</a>

Let us denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M225">View MathML</a> the extensions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M17">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M2">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M237">View MathML</a>

(3.10)

(for the existence of such functions see Theorem 5.1 in [12]). Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M238">View MathML</a>

(3.11)

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M239">View MathML</a>

(3.12)

Moreover, from assumptions (h1), (i1), and (3.4), (3.5) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M240">View MathML</a>

(3.13)

where r and p are as in hypothesis (i1).

Consider now the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M241">View MathML</a>

(3.14)

Putting together (3.11) and (3.13) with Theorem 2.1 of [13] if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M242">View MathML</a> or with Theorem 3.5 of [14] if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M243">View MathML</a>, it follows that there exists a unique solution w of (3.14) satisfying the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M244">View MathML</a>

(3.15)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M245">View MathML</a> depends on n, p, ν, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M217">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M247">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M248">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M249">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M250">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M251">View MathML</a>.

Thus from (3.15) and classical Sobolev embedding theorems (see Lemma 5.15 in [15]) we deduce that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M252">View MathML</a>, depending on the same parameters as K, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M253">View MathML</a>

(3.16)

and hence for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M254">View MathML</a> there is a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M255">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M256">View MathML</a>) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M257">View MathML</a>

(3.17)

The map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M258">View MathML</a> is the Green function for the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M259">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M260">View MathML</a> and it has the following properties:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M261">View MathML</a>

(3.18)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M262">View MathML</a>

(3.19)

Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M263">View MathML</a> in (3.14), we find that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M264">View MathML</a>, belonging to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M265">View MathML</a>, is a solution of the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M266">View MathML</a>

(3.20)

Moreover, from (3.12), (3.13) and Lemma 3.1 of [16] (see also Lemma 3.1 of [2] for the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M242">View MathML</a>) it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M268">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M260">View MathML</a>. Finally, applying (3.17) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M270">View MathML</a> and using (3.18) and (3.19) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M271">View MathML</a>

(3.21)

From (3.21), converting back to the x-variables (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M272">View MathML</a>), we easily deduce the estimate (3.9). □

4 Main results

In this section we use the previous result to prove a bound for the solution of our main problem.

Consider in Ω the differential operator L defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M273">View MathML</a>

and put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M274">View MathML</a>

Suppose that the leading coefficients of operator L satisfy the assumption (h1) while the lower order coefficients verify the following condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M275">View MathML</a>

where r and p are as in hypothesis (i1). Moreover, assume that the following condition on ρ holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M276">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M277">View MathML</a> is defined in (2.10). For an example of function ρ whose regularizing function σ satisfy (h3) we can refer to [17].

We introduce now a class of mappings needed in the sequel. Let us fix a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M278">View MathML</a> which is equivalent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M279">View MathML</a> (for more details on the existence of such an α see, for instance, Theorem 2, Chapter IV in [18] and Lemma 3.6.1 in [19]). Hence, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M280">View MathML</a> we define the functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M281">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M282">View MathML</a> verifies (2.9). It is easy to prove that each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M283">View MathML</a> belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M98">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M285">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M286">View MathML</a>

Remark 4.1 From hypothesis (h1) and Lemma 4.2 in [12] it follows that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M280">View MathML</a> the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M288">View MathML</a> (obtained as extensions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M289">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M2">View MathML</a> with zero values out of Ω) belong to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M291">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M292">View MathML</a>

for t small enough.

Now we are able to prove our main result.

Theorem 4.2Suppose that conditions (h1), (h2), (h3) hold. Fixing<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M28">View MathML</a>, letube a solution of the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M294">View MathML</a>

(4.1)

Then there exist an open ball<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M21">View MathML</a>and a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M11">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M297">View MathML</a>

(4.2)

wherecdepends onn, p, r, ρ, ν, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M217">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M299">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M300">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M280">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M219">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M220">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M223">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M224">View MathML</a>.

Proof Without loss of generality it can be assumed that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M306">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M147">View MathML</a>, we put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M308">View MathML</a>

(4.3)

Thus, from the last two conditions of (4.1) and from (2.11), (2.12) and (4.3) it follows that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M309">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M310">View MathML</a>. Moreover, taking into account the classical Sobolev embedding theorem (see Theorem 5.4 in [15]), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M311">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M312">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M313">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M314">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M315">View MathML</a> (which will be suitably chosen later), such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M316">View MathML</a>

(4.4)

For simplicity of notation, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M147">View MathML</a>, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M318">View MathML</a> the open ball <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M319">View MathML</a>.

Let us set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M320">View MathML</a>

(4.5)

It is easily seen that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M321">View MathML</a>

(4.6)

Moreover, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M322">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M323">View MathML</a>

(4.7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M324">View MathML</a>

(4.8)

Consider now the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M325">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M326">View MathML</a>

(4.9)

Clearly

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M327">View MathML</a>

(4.10)

The first step of the proof is to show that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M328">View MathML</a> such that, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M329">View MathML</a>, each function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M325">View MathML</a> is a solution of a problem of type (3.1), where the coefficients of associated differential operator verify the assumptions of Lemma 3.1.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M147">View MathML</a>, it is easy to prove

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M332">View MathML</a>

(4.11)

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M333">View MathML</a>

(4.12)

and u is a solution of problem (4.1), from (4.11) we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M334">View MathML</a>

(4.13)

where we have put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M335">View MathML</a>

(4.14)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M336">View MathML</a>

(4.15)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M337">View MathML</a>

(4.16)

We observe that using the hypotheses (h0), (h1), (h2), the equivalence between ρ and σ, and (2.11)-(2.16), we easily get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M338">View MathML</a>

(4.17)

Using now the estimate (4.13), it is easily seen that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M339">View MathML</a>

(4.18)

This last inequality can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M340">View MathML</a>

(4.19)

where we have set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M341">View MathML</a>

(4.20)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M342">View MathML</a>

(4.21)

Hence, putting together (4.9) with (4.19) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M343">View MathML</a>

(4.22)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M344">View MathML</a>

(4.23)

Observe that using the hypotheses (h1), (h2), and (4.17), (4.5)-(4.8), it is easy to prove that, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M147">View MathML</a>, the coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M346">View MathML</a> (for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M198">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M348">View MathML</a> satisfy the first two conditions of assumption (i1) and the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M349">View MathML</a>. We show now that, for a suitable choice of the constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M350">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M328">View MathML</a> such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M329">View MathML</a> the coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M348">View MathML</a> verify also the last condition of (i1). To this aim, we firstly observe that using again hypotheses (h1), (h2), and (2.15), (2.16), from (4.15) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M354">View MathML</a>

(4.24)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M202">View MathML</a> depends on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M184">View MathML</a>, n and s.

Thus, from (4.4), (2.11)-(2.14) and hypothesis (h3) it follows that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M328">View MathML</a> such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M329">View MathML</a> we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M359">View MathML</a>

(4.25)

Now, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M329">View MathML</a>, putting together (4.25) with (4.21) and using the assumption (h1), the properties (4.6)-(4.8) and (4.4), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M361">View MathML</a>

(4.26)

Hence, fixing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M362">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M363">View MathML</a>

(4.27)

from (4.26) it follows that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M329">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M365">View MathML</a>

(4.28)

Putting together (4.28) with (4.25) and observing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M366">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M367">View MathML</a>, we deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M368">View MathML</a> a.e. in Ω. The above considerations together with (4.4), (4.9), (4.10), and (4.22) show that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M329">View MathML</a> the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M370">View MathML</a>

(4.29)

satisfy the assumptions of Lemma 3.1. Therefore, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M202">View MathML</a> depending on n, p, r, ρ, ν, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M184">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M218">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M219">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M220">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M223">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M224">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M378">View MathML</a>

(4.30)

By (4.10), the last bound with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M379">View MathML</a> becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M380">View MathML</a>

(4.31)

Now, in order to obtain the estimate (4.2), we have to provide a lower bound for the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M381">View MathML</a> in terms of the data f. First of all, we observe that, using the definitions (4.14) and (4.16), we can rewrite (4.23) as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M382">View MathML</a>

(4.32)

On the other hand, by assumption (h1), and by (2.15), (4.7), and (4.4) we easily obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M383">View MathML</a>

(4.33)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M203">View MathML</a> depends on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M184">View MathML</a>, n, s, ρ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M386">View MathML</a>. Thus, using (2.13) and hypothesis (h3) it follows that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M387">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M388">View MathML</a>, such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M389">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M390">View MathML</a>

(4.34)

Putting together (4.34) and (4.28) with (4.32) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M391">View MathML</a>

(4.35)

Taking into account (4.35), from (4.31) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M392">View MathML</a>

(4.36)

where we have put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M393">View MathML</a>

(4.37)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M394">View MathML</a>

(4.38)

To end the proof, we give some upper bounds for the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M395">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M396">View MathML</a> (with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M389">View MathML</a>). First of all, observe that using (4.7) and Hölder’s inequality in (4.37) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M398">View MathML</a>

(4.39)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M165">View MathML</a> depends on the same parameters as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M400">View MathML</a>. Using now (4.4), the equivalence on ρ and σ, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M401">View MathML</a>

(4.40)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M213">View MathML</a> depends on the same parameters as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M400">View MathML</a>. If we choose λ such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M404">View MathML</a>

(4.41)

from (4.40), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M389">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M406">View MathML</a>

(4.42)

Arguing similarly we obtain, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M389">View MathML</a>, the following bound on the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M408">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M409">View MathML</a>

(4.43)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M410">View MathML</a> depends on the same parameters as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M411">View MathML</a> and on s. Thus, using again (2.13) and assumption (h3), we see that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M412">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M413">View MathML</a>, such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M414">View MathML</a> we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M415">View MathML</a>

(4.44)

Finally, chosen <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M416">View MathML</a>, putting together (4.42) and (4.44) with (4.36) and using (4.4), (2.11), and (2.12) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M417">View MathML</a>

(4.45)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M418">View MathML</a> depends on the same parameters as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M411">View MathML</a> and on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M386">View MathML</a>. Taking into account (4.3) and using again (2.11) and (2.12), from (4.45) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M421">View MathML</a>

(4.46)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M422">View MathML</a> depends on the same parameters as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M411">View MathML</a> and on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M386">View MathML</a>.

Finally, if we choose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M425">View MathML</a>

(4.47)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M426">View MathML</a> is such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/91/mathml/M427">View MathML</a>, the estimate (4.2) follows from (4.46), (4.47), and Remark 4.1. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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