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Dirichlet problem for divergence form elliptic equations with discontinuous coefficients

Sara Monsurrò and Maria Transirico*

Author Affiliations

Dipartimento di Matematica, Università di Salerno, via Ponte Don Melillo, Fisciano, (SA), 84084, Italy

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Boundary Value Problems 2012, 2012:67  doi:10.1186/1687-2770-2012-67


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/67


Received:27 February 2012
Accepted:15 June 2012
Published:28 June 2012

© 2012 Monsurrò and Transirico; licensee Springer

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

Abstract

We study the Dirichlet problem for linear elliptic second order partial differential equations with discontinuous coefficients in divergence form in unbounded domains. We establish an existence and uniqueness result and we prove an a priori bound in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M2">View MathML</a>.

MSC: 35J25, 35B45, 35R05.

Keywords:
elliptic equations; discontinuous coefficients; a priori bounds

1 Introduction

We are interested in the Dirichlet problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M3">View MathML</a>

(1.1)

where Ω is an unbounded open subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M4">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M5">View MathML</a>, and L is a linear uniformly elliptic second order differential operator with discontinuous coefficients in divergence form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M6">View MathML</a>

(1.2)

If Ω is bounded, this problem is classical in literature and has been deeply analyzed taking into account various kinds of hypotheses on the coefficients (for more details see, for instance, [1-6]).

Considering unbounded domains, as far as we know, the first work on this subject goes back to [7], where Bottaro and Marina provide, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M7">View MathML</a>, an existence and uniqueness result for the solution of problem (1.1) assuming that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M8">View MathML</a>

(1.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M9">View MathML</a>

(1.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M10">View MathML</a>

(1.5)

In this order of ideas, various generalizations have been performed still maintaining hypotheses (1.3) and (1.5) but weakening the condition (1.4). Indeed in [8], where the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M5">View MathML</a> is considered, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M12">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M13">View MathML</a> and c are supposed to satisfy assumptions as those in (1.4), but just locally. Successively in [9], for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M7">View MathML</a>, further improvements have been carried on since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M12">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M13">View MathML</a> and c are in suitable Morrey-type spaces with lower summabilities.

In [7-9] we also find the bound

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M17">View MathML</a>

(1.6)

where the dependence of the constant C on the data of the problem is fully determined.

More recently, in [10], supposing that the coefficients of lower-order terms are as in [9] for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M18">View MathML</a> and as in [8] for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M19">View MathML</a>, we showed that, for a sufficiently regular set Ω, and if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M20">View MathML</a>, then there exists a constant C, whose dependence is completely described, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M21">View MathML</a>

(1.7)

for any bounded solution u of (1.1) and for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M22">View MathML</a>.

Here, in the same framework but replacing the classical hypothesis of sign (1.5) by the less common one

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M23">View MathML</a>

(1.8)

we establish two kinds of results for the solution of (1.1). First of all, we provide an existence and uniqueness theorem, then, taking into account an additional assumption on the regularity of the boundary of Ω, we prove the analogue of (1.7).

Let us briefly survey the way these results are achieved. In Section 2, we introduce the tools needed in the sequel. The definitions and some features of the Morrey-type spaces are given and some functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M24">View MathML</a>, related somehow to the solution of the problem and to the coefficients of the operator, are described, together with some specific properties. Section 3 is devoted to the solvability of problem (1.1). We start proving, by means of the above mentioned functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M24">View MathML</a>, the estimate in (1.6) that leads also to the uniqueness at once. Then, in view of well-known results of the operator theory, we get the existence verifying that L is a Fredholm operator with zero index. In the last section, we prove the claimed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M1">View MathML</a>-estimate. This is done by means of a technical lemma, exploiting again the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M24">View MathML</a>, which allows us to conclude.

Considering the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M28">View MathML</a>, we notice that, as a consequence of (1.6), the bound (1.7) is true under both sign hypotheses even supposing no regularity on the boundary of Ω.

We believe that the two estimates (1.7), obtained under the different sign assumptions, combined together should permit to prove, by means of a duality argument, that (1.7) holds true actually for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M29">View MathML</a>, considering one of the hypotheses (1.5) or (1.8) at a time.

For further studies of the Dirichlet problem for linear elliptic second order differential equations with discontinuous coefficients in divergence form in unbounded domains we refer the reader also to [11-13].

2 Tools

This section is devoted to the definitions and to some fundamental properties of the Morrey-type spaces where the coefficients of lower-order terms of our operator belong, and of some functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M24">View MathML</a> related to the solution of the problem and to all the coefficients of the operator (see the proofs of Theorem 3.1 and Lemma 4.1 for more details on this aspect) that are indispensable tools in the sequel.

Given an unbounded open subset Ω of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M5">View MathML</a>, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M33">View MathML</a> the σ-algebra of all Lebesgue measurable subsets of Ω. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M35">View MathML</a> is its characteristic function and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M36">View MathML</a> is the intersection <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M37">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M39">View MathML</a>), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M40">View MathML</a> is the open ball centered in x and with radius r.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M41">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M42">View MathML</a>, the space of Morrey type <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M43">View MathML</a> is the set of all the functions g in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M44">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M45">View MathML</a>

endowed with the norm above defined. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M46">View MathML</a> denotes the closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M47">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M43">View MathML</a>. These functional spaces generalize the classical notion of Morrey spaces to the case of unbounded domains and were introduced in [9] (we refer also to [14] where further characteristics are considered).

For the reader’s convenience, in the next lemma we recall some results of [15] and [8,9] concerning the multiplication operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M49">View MathML</a>

(2.1)

where the function g belongs to suitable spaces of Morrey type.

Lemma 2.1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M50">View MathML</a>, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M51">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M52">View MathML</a>if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M19">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M54">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M55">View MathML</a>if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M56">View MathML</a>, then the operator in (2.1) is bounded and there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M57">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M58">View MathML</a>

(2.2)

with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M59">View MathML</a>.

Moreover, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M60">View MathML</a>, then the operator in (2.1) is also compact.

Now, let us deal with the above mentioned functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M24">View MathML</a>. They were employed for the first time in [7] and were studied in the framework of Morrey-type spaces in [9].

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M62">View MathML</a>, we define the functions of the real variable t

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M63">View MathML</a>

(2.3)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M64">View MathML</a>

(2.4)

Lemma 2.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M66">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M67">View MathML</a>. Then there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M68">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M69">View MathML</a>, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M70">View MathML</a>, such that set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M71">View MathML</a>

(2.5)

one has<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M72">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M73">View MathML</a>

(2.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M74">View MathML</a>

(2.7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M75">View MathML</a>

(2.8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M76">View MathML</a>

(2.9)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M77">View MathML</a>

(2.10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M78">View MathML</a>

(2.11)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M79">View MathML</a>

(2.12)

with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M80">View MathML</a>positive constant.

Proof The proofs of the properties (2.6), (2.7), (2.9), (2.11) and (2.12) can be found in [9].

Inequality (2.8) is an immediate consequence of (2.7).

Considering (2.10), observe that in the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M81">View MathML</a> it is a trivial consequence of (2.6).

Thus let us fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M82">View MathML</a> and such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M83">View MathML</a>. As already proved in [16] and in [7], in the case of unbounded domains, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M84">View MathML</a>

This, together with (2.3) and (2.4), gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M85">View MathML</a>

(2.13)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M86">View MathML</a>.

On the other hand, by definition,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M87">View MathML</a>

(2.14)

Combining (2.14) and (2.13), we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M88">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M89">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M86">View MathML</a>. Hence by (2.6) we get (2.10). □

3 Existence and uniqueness result

Let Ω be an unbounded open subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M5">View MathML</a>.

We are interested in the study of the following Dirichlet problem in Ω:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M93">View MathML</a>

(3.1)

where L is a second order linear differential operator in divergence form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M94">View MathML</a>

(3.2)

satisfying the following hypotheses on the leading coefficients: Considering the coefficients of lower-order terms, we suppose that We associate to L the bilinear form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M95">View MathML</a>

(3.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M96">View MathML</a>.

As a consequence of Lemma 2.1, a is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M97">View MathML</a>; and therefore, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M98">View MathML</a> is continuous too.

Theorem 3.1Under hypotheses (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M99">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M100">View MathML</a>), problem (3.1) is uniquely solvable and its solutionusatisfies the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M101">View MathML</a>

(3.4)

whereCis a constant depending onn, t, ν, μ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M102">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M103">View MathML</a>.

Proof We start proving estimate (3.4) that yields also to the uniqueness of the solution at once. Successively, in view of classical results concerning operator theory, to get the existence, it will be enough to verify that L is a Fredholm operator with zero index.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M24">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M105">View MathML</a>, be the functions of Lemma 2.2 corresponding to a solution u of (3.1), to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M106">View MathML</a> and to a positive real number ε that will be specified in the sequel.

By a well-known characterization of the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M107">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M108">View MathML</a>

Thus, if we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M24">View MathML</a> as a test function in the variational formulation of problem (3.1), by simple calculations and (2.9) and (2.10), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M110">View MathML</a>

Hypotheses (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M99">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M100">View MathML</a>) together with (2.7) give then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M113">View MathML</a>

(3.5)

On the other hand, by the Hölder inequality, the embedding results contained in Lemma 2.1 and using hypothesis (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M114">View MathML</a>) and (2.11), one has that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M115">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M116">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M117">View MathML</a>.

Hence, set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M118">View MathML</a>

by (3.5) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M119">View MathML</a>

Thus, choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M120">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M121">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M122">View MathML</a>.

If we rewrite the last inequality for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M123">View MathML</a> and we estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M124">View MathML</a>, then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M125">View MathML</a> and we estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M126">View MathML</a> and so on, we get by substituting that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M127">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M122">View MathML</a>.

Therefore, taking into account (2.6), we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M129">View MathML</a>

This, together with (2.12), ends the proof of the bound in (3.4).

Now, as it was already mentioned, it only remains to show that the operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M130">View MathML</a>

is a Fredholm operator with zero index.

To this aim, set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M131">View MathML</a> and denote by γu, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M66">View MathML</a>, the element of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M107">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M134">View MathML</a>

which is well defined in view of Lemma 2.1.

Then, consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M135">View MathML</a>

(3.6)

Clearly, if we show that (3.6) has a unique solution, we end our proof, since in this case the operator L can be seen as a sum between a Fredholm operator with zero index and a compact operator; and therefore, it is a Fredholm operator with zero index itself.

Indeed, we explicitly observe that the operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M136">View MathML</a>

is compact, since, by hypothesis (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M114">View MathML</a>) and Lemma 2.1, it is obtained as a composition between the compact operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M138">View MathML</a>

and the bounded one

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M139">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M140">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M141">View MathML</a>, is the element of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M107">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M143">View MathML</a>

To get the existence and uniqueness of the solution of problem (3.6), we want to make use of Lax-Milgram Lemma. Thus let us consider the bilinear form associated to it

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M144">View MathML</a>

(3.7)

The continuity of the form (3.7) can be easily obtained by Lemma 2.1. Considering the coercivity, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M66">View MathML</a>, in view of hypotheses (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M99">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M100">View MathML</a>), one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M148">View MathML</a>

On the other hand, Hölder and Young inequalities give that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M149">View MathML</a>

and therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M150">View MathML</a>

This concludes the proof of Theorem 3.1. □

4 An a priori bound in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M1">View MathML</a>

Here we want to prove, for a sufficiently regular datum f, a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M1">View MathML</a>-a priori estimate, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M2">View MathML</a>, for a bounded solution of problem (3.1).

To this aim, we require a further assumption on the boundary of Ω:

Moreover, a technical lemma below is needed. We note that the proof of Lemma 4.1 follows the idea of the one of the estimate (3.4). However, in this case, there are some specific arguments that need to be explicitly treated.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M24">View MathML</a> be the functions of Lemma 2.2 corresponding to a fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M155">View MathML</a>, to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M106">View MathML</a> and to a positive real number ε to be specified in the proof of Lemma 4.1. The following result holds true:

Lemma 4.1Letabe the bilinear form in (3.3). Under hypotheses (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M99">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M158">View MathML</a>), there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M159">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M160">View MathML</a>

(4.1)

whereCdepends ons, r, ν, μ.

Proof Let ugε and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M24">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M122">View MathML</a>, be as above specified. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M163">View MathML</a>, by definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M24">View MathML</a> and by Lemma 2.2, the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M165">View MathML</a>. Therefore, in view of hypothesis (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M158">View MathML</a>), Lemma 3.2 in [17] applies giving that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M167">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M2">View MathML</a>.

Thus, we can take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M169">View MathML</a> as a test function in (3.3), obtaining by (2.9) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M170">View MathML</a>

If we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M171">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M172">View MathML</a>

(4.2)

by hypotheses (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M99">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M100">View MathML</a>) and in view of (2.7), one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M175">View MathML</a>

(4.3)

On the other hand, by (2.6), (2.8) and (2.10), using the Hölder inequality, we get that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M115">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M177">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M178">View MathML</a>.

Thus, using hypothesis (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M114">View MathML</a>), by Lemma 2.1 and (2.11), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M180">View MathML</a>

(4.4)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M181">View MathML</a>.

Now, we observe that explicit calculations give

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M182">View MathML</a>

(4.5)

Hence, putting together (4.3), (4.4) and (4.5), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M183">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M184">View MathML</a>.

Thus, by Young inequality,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M185">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M186">View MathML</a>.

Choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M187">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M188">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M189">View MathML</a>

(4.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M122">View MathML</a>.

If we rewrite the last inequality for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M123">View MathML</a>, then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M125">View MathML</a> and take into account the estimate of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M193">View MathML</a> obtained in the previous step, and so on, we conclude our proof. Indeed, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M194">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M195">View MathML</a>. □

We are finally in position to prove the above mentioned <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M1">View MathML</a>-bound.

Theorem 4.2Assume that the hypotheses (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M99">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M158">View MathML</a>) are satisfied. Iffis in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M199">View MathML</a>and the solutionuof (3.1) is in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M200">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M201">View MathML</a>

whereCis a constant depending onn, t, p, ν, μ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M102">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M103">View MathML</a>.

Proof Fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M204">View MathML</a>. If we consider the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M206">View MathML</a>, corresponding to the solution u, to g and ε as in Lemma 4.1, easy computations together with (2.6) give that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M207">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M208">View MathML</a>.

Thus, by (4.1), one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M209">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M210">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M211">View MathML</a>.

Hence by (2.8) and Hölder inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/67/mathml/M212">View MathML</a>

This concludes the proof, in view of (2.12). □

Competing interests

The authors declare that they have no competing interests.

Author’s contributions

The authors conceived and wrote this article in collaboration and with the same responsibility. Both of them read and approved the final manuscript.

Acknowledgement

The authors would like to thank anonymous referees for a careful reading of this article and for valuable suggestions and comments.

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