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Open Access Research

Parabolic problems with data measure

Mohammmed Kbiri Alaoui

Author Affiliations

Department of Mathematics, King Khalid University, P.O. Box 9004, Abha, Kingdom of Saudi Arabia

Boundary Value Problems 2011, 2011:46  doi:10.1186/1687-2770-2011-46


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/46


Received:4 April 2011
Accepted:23 November 2011
Published:23 November 2011

© 2011 Alaoui; 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

The article deals with the existence of solutions of some unilateral problems in the Orlicz-Sobolev spaces framework when the right-hand side is a Radon measure.

Mathematics Subject Classification: 35K86.

Keywords:
unilateral problem; radon measure; Orlicz-Sobolev spaces

1 Introduction

We deal with boundary value problems

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M1">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M2">View MathML</a>

T > 0 and Ω is a bounded domain of RN, with the segment property. a : Ω × R × RN RN is a Carathéodory function (that is, measurable with respect to x in Ω for every (t, s, ξ) in R × R × RN , and continuous with respect to (s, ξ) in R × RN for almost every x in Ω) such that for all ξ, ξ* ∈ RN , ξ ξ*,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M3">View MathML</a>

(1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M4">View MathML</a>

(1.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M5">View MathML</a>

(1.3)

where c (x,t) belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M6">View MathML</a>, P is an N-function such that P M and ki (i = 1,2,3,4) belongs to R+ and α to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M7">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M8">View MathML</a>

(1.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M9">View MathML</a>

(1.5)

There have obviously been many previous studies on nonlinear differential equations with nonsmooth coefficients and measures as data. The special case was cited in the references (see [1,2]).

It is noteworthy that the articles mentioned above differ in significant way, in the terms of the structure of the equations and data. In [1], when f L1(0,T;L1(Ω)) and u0 L1(Ω). The authors have shown the existence of solutions u of the corresponding equation of the problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M10">View MathML</a> for every q such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M11">View MathML</a> which is more restrictive than the one given in the elliptic case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M12">View MathML</a>.

In this article, we are interested with an obstacle parabolic problem with measure as data. We give an improved regularity result of the study of Boccardo et al. [1].

In [1], the authors have shown the existence of a weak solutions for the corresponding equation, the function a(x, t, s, ξ) was assumed to satisfy a polynomial growth condition with respect to u and ∇u. When trying to relax this restriction on the function a(., s, ξ), we are led to replace the space Lp(0, T; W1,p()) by an inhomogeneous Sobolev space W1,xLM built from an Orlicz space LM instead of Lp, where the N-function M which defines LM is related to the actual growth of the Carathéodory's function.

For simplicity, one can suppose that there exist α > 0, β > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M13">View MathML</a>

2 Preliminaries

Let M : R+ R+ be an N-function, i.e. M is continuous, convex, with M(t) > 0 for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M14">View MathML</a> as t → 0 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M15">View MathML</a> as t → ∞. Equivalently, M admits the representation: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M16">View MathML</a> where a : R+ R+ is non-decreasing, right continuous, with a(0) = 0, a(t) > 0 for t > 0 and a(t) → ∞ as t → ∞. The N-function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M17">View MathML</a> conjugate to M is defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M18">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M19">View MathML</a> is given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M20">View MathML</a> (see [3,4]).

The N-function M is said to satisfy the Δ2 condition if, for some k > 0:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M21">View MathML</a>

(2.1)

when this inequality holds only for t t0 > 0, M is said to satisfy the Δ2 condition near infinity.

Let P and Q be two N-functions. P Q means that P grows essentially less rapidly than Q; i.e., for each ε > 0,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M22">View MathML</a>

Let Ω be an open subset of RN. The Orlicz class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M23">View MathML</a> (resp. the Orlicz space Lm(Ω)) is defined as the set of (equivalence classes of) real-valued measurable functions u on Ω such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M24">View MathML</a>

Note that LM(Ω) is a Banach space under the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M25">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M23">View MathML</a> is a convex subset of LM(Ω). The closure in LM(Ω) of the set of bounded measurable functions with compact support in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M26">View MathML</a> is denoted by EM(Ω). The equality EM(Ω) = LM(Ω) holds if and only if M satisfies the Δ2 condition, for all t or for t large according to whether Ω has infinite measure or not.

The dual of EM(Ω) can be identified with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M27">View MathML</a> by means of the pairing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M28">View MathML</a>, and the dual norm on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M27">View MathML</a> is equivalent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M29">View MathML</a>. The space LM(Ω) is reflexive if and only if M and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M17">View MathML</a> satisfy the Δ2 condition, for all t or for t large, according to whether Ω has infinite measure or not.

We now turn to the Orlicz-Sobolev space. W1LM(Ω) (resp. W1EM(Ω)) is the space of all functions u such that u and its distributional derivatives up to order 1 lie in LM(Ω) (resp. EM(Ω)). This is a Banach space under the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M30">View MathML</a>. Thus, W1LM(Ω) and W1EM(Ω) can be identified with subspaces of the product of N + 1 copies of LM(Ω). Denoting this product by ΠLM, we will use the weak topologies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M31">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M32">View MathML</a>. The space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M33">View MathML</a> is defined as the (norm) closure of the Schwartz space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M34">View MathML</a> in W1EM(Ω) and the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M35">View MathML</a> as the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M31">View MathML</a> closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M34">View MathML</a> in W1LM(Ω). We say that un converges to u for the modular convergence in W1LM(Ω) if for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M36">View MathML</a> for all |α| ≤ 1. This implies convergence for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M32">View MathML</a>. If M satisfies the Δ2 condition on R+(near infinity only when Ω has finite measure), then modular convergence coincides with norm convergence.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M37">View MathML</a> (resp. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M38">View MathML</a>) denote the space of distributions on Ω which can be written as sums of derivatives of order ≤ 1 of functions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M27">View MathML</a> (resp. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M39">View MathML</a>). It is a Banach space under the usual quotient norm.

If the open set Ω has the segment property, then the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M34">View MathML</a> is dense in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M35">View MathML</a> for the modular convergence and for the topology <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M32">View MathML</a> (cf. [5,6]). Consequently, the action of a distribution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M37">View MathML</a> on an element of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M35">View MathML</a> is well defined.

For k > 0, s R, we define the truncation at height k,Tk(s) = [k - (k - |s|)+]sign(s).

The following abstract lemmas will be applied to the truncation operators.

Lemma 2.1 [7]Let F : R R be uniformly lipschitzian, with F(0) = 0. Let M be an N-function and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M40">View MathML</a>(resp.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M33">View MathML</a>).

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M41">View MathML</a>(resp.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M33">View MathML</a> ). Moreover, if the set of discontinuity points of F' is finite, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M42">View MathML</a>

Let Ω be a bounded open subset of RN, T > 0 and set Q = Ω × ]0, T[. Let m ≥ 1 be an integer and let M be an N-function. For each α INN , denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M43">View MathML</a> the distributional derivative on Q of order α with respect to the variable x RN. The inhomogeneous Orlicz-Sobolev spaces are defined as follows <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M44">View MathML</a>.

The last space is a subspace of the first one, and both are Banach spaces under the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M45">View MathML</a>. We can easily show that they form a complementary system when Ω satisfies the segment property. These spaces are considered as subspaces of the product space ΠLm(Q) which have as many copies as there are α-order derivatives, |α| ≤ m. We shall also consider the weak topologies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M31">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M32">View MathML</a>. If u Wm, xLM(Q), then the function : t u(t) = u(t,.) is defined on [0, T] with values in WmLM(Ω). If, further, u Wm,xEM(Q) then the concerned function is a WmEM(Ω)-valued and is strongly measurable. Furthermore, the following imbedding holds: Wm,xEM(Q) ⊂ L1(0,T; WmEM(Ω)). The space Wm,xLM(Q) is not in general separable, if u Wm,xLM(Q), we cannot conclude that the function u(t) is measurable on [0,T]. However, the scalar function t ↦ ||u(t)||M, is in L1(0,T). The space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M46">View MathML</a> is defined as the (norm) closure in Wm,xEM(Q) of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M34">View MathML</a>. We can easily show as in [6] that when Ω has the segment property, then each element u of the closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M34">View MathML</a> with respect of the weak * topology <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M31">View MathML</a> is limit, in Wm,xLM(Q), of some subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M47">View MathML</a> for the modular convergence; i.e., there exists λ > 0 such that for all |α| ≤ m,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M48">View MathML</a>

this implies that (ui) converges to u in Wm,xLM(Q) for the weak topology <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M32">View MathML</a>. Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M49">View MathML</a>, and this space will be denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M50">View MathML</a>.

Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M51">View MathML</a>. Poincaré's inequality also holds in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M50">View MathML</a>, i.e., there is a constant C > 0 such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M52">View MathML</a> one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M53">View MathML</a>. Thus both sides of the last inequality are equivalent norms on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M50">View MathML</a>. We have then the following complementary system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M54">View MathML</a>

F being the dual space of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M46">View MathML</a>. It is also, except for an isomorphism, the quotient of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M55">View MathML</a> by the polar set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M56">View MathML</a>, and will be denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M57">View MathML</a>, and it is shown that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M58">View MathML</a>. This space will be equipped with the usual quotient norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M59">View MathML</a> where the infimum is taken on all possible decompositions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M60">View MathML</a>. The space F0 is then given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M61">View MathML</a> and is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M62">View MathML</a>.

We can easily check, using Lemma 4.4 of [6], that each uniformly lipschitzian mapping F, with F(0) = 0, acts in inhomogeneous Orlicz-Sobolev spaces of order 1: W1,xLM(Q) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M63">View MathML</a>.

3 Main results

First, we give the following results which will be used in our main result.

3.1 Useful results

Hereafter, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M64">View MathML</a> the real number defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M65">View MathML</a> is the measure of the unit ball of RN, and for a fixed t ∈ [0, T], we denote μ(θ) = meas{(x,t) : |u(x, t)| > θ}.

Lemma 3.1 [8]Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M66">View MathML</a>, and let fixed t ∈ [0, T], then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M67">View MathML</a>

and where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M68">View MathML</a>is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M69">View MathML</a>

Lemma 3.2 Under the hypotheses (1.1)-(1.3), if f, u0 are regular functions and f, u0 ≥ 0, then there exists at least one positive weak solution of the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M70">View MathML</a>

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M71">View MathML</a>

Proof

Let u be a continuous function, we say that u satisfies (*) if: there exists a continuous and increasing function β such that ||u(t) - u(s)||2 β(||u0||2)|t - s|, where u0(x) = u(x, 0).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M72">View MathML</a>.

Let us consider the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M73">View MathML</a>, where C is a closed convex of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M35">View MathML</a>. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M74">View MathML</a> is a closed convex (since all its elements satisfy (*) ).

We claim that the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M75">View MathML</a>

has a weak solution which is unique in the sense defined in [9].

Indeed, let us consider the approximate problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M76">View MathML</a>

where the functional Φ is defined by Φ : X R ∪ {+ ∞} such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M77">View MathML</a>

The existence of such un X was ensured by Kacur et al. [10].

Following the same proof as in [9], we can prove the existence of a solution u of the problem (E') as limit of un (for more details see [9]).

Lemma 3.3 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M78">View MathML</a>such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M79">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M80">View MathML</a>.

Then, there exists a smooth function (vj) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M81">View MathML</a>

vj v for the modular convergence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M82">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M83">View MathML</a>for the modular convergence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M84">View MathML</a>.

For the proof, we use the same technique as in [11] in the parabolic case.

3.2 Existence result

Let M be a fixed N-function, we define K as the set of N-function D satisfying the following conditions:

i) M(D-1(s)) is a convex function,

ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M85">View MathML</a>,

iii) There exists an N-function H such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M86">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M87">View MathML</a> near infinity.

Theorem 3.1 Under the hypotheses (1.1)-(1.5), The problem (P) has at least one solution u in the following sense:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M88">View MathML</a>

for all φ D(RN+1) which are zero in a neighborhood of (0, T) × ∂ Ω and {T} × Ω.

Remark 3.1 (1) If ψ = - ∞ in the problem (P), then the above theorem gives the same regularity as in the elliptic case.

(2) An improved regularity is reached for all N-function satisfying the conditions (i)-(ii)-(iii).

For example, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M89">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M90">View MathML</a>.

In the sequel and throughout the article, we will omit for simplicity the dependence on x and t in the function a(x, t, s, ξ) and denote ϵ(n, j, μ, s, m) all quantities (possibly different) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M91">View MathML</a>

and this will be in the order in which the parameters we use will tend to infinity, that is, first n, then j, μ, s, and finally m. Similarly, we will write only ϵ(n), or ϵ(n, j),... to mean that the limits are made only on the specified parameters.

3.2.1 A sequence of approximating problems

Let (fn) be a sequence in D(Q) which is bounded in L1(Q) and converge to μ in Mb(Q).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M92">View MathML</a> be a sequence in D(Ω) which is bounded in L1(Ω) and converge to u0 in Mb(Ω).

We define the following problems approximating the original (P):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M93">View MathML</a>

Lemma 3.4 Under the hypotheses (1.1)-(1.3), there exists at least one solution un of the problem (Pn) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M94">View MathML</a>a.e. in Q.

For the proof see Lemma 3.2.

3.2.2 A priori estimates

Lemma 3.5 There exists a subsequence of (un) (also denoted (un)), there exists a measurable function u such that:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M95">View MathML</a>

Proof:

Recall that un ≥ 0 since fn ≥ 0.

Let h > 0 and consider the following test function v = Th(un - Tk(un)) in (Pn), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M96">View MathML</a>

We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M97">View MathML</a>

So,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M98">View MathML</a>

Now, let us fix k > ||ψ||, we deduce the fact that: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M99">View MathML</a>.

Let h to tend to zero, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M100">View MathML</a>

(3.1)

Let us use as test function in (Pn),v = Tk(un), then as above, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M101">View MathML</a>

(3.2)

Then (Tk(un)n) is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M82">View MathML</a>, and then there exist some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M102">View MathML</a> such that

Tk(un) ⇀ ωk, weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M82">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M31">View MathML</a>, strongly in EM(Q) and a.e in Q.

Let consider the C2 function defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M103">View MathML</a>

Multiplying the approximating equation by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M104">View MathML</a>, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M105">View MathML</a> in the distributions sense. We deduce then that ηk(un) being bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M82">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M106">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M107">View MathML</a>. By Corollary 1 of [12], ηk(un) is compact in L1 (Q).

Following the same way as in [2], we obtain for every k > 0,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M108">View MathML</a>

(3.3)

Using now the estimation (3.1) and Fatou's lemma to obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M109">View MathML</a>

Let fixed a t ∈ [0, T]. We argue now as for the elliptic case, the problem becomes:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M110">View MathML</a>

We denote gn := nTn((un - ψ)-).

Let φ be a truncation defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M111">View MathML</a>

(3.4)

for all θ, h > 0.

Using v = φ (un) as a test function in the approximate elliptic problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M112">View MathML</a>, we obtain by using the same technique as in [8]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M113">View MathML</a>

(3.5)

here and below C denote positive constants not depending on n.

By using Lemma 3.1, we obtain (supposing -μ'(θ) > 0 which does not affect the proof) and following the same way as in [8], we have for D K,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M114">View MathML</a>

(3.6)

Let denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M115">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M116">View MathML</a>

Using Lemma 3.1, denoting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M117">View MathML</a> one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M118">View MathML</a>

Remark also that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M119">View MathML</a> and using Lemma 3.2, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M120">View MathML</a>.

Combining the inequalities (3.5) and (3.6) we obtain,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M121">View MathML</a>

(3.7)

and since the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M122">View MathML</a> is absolutely continuous, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M123">View MathML</a>

Then, the sequence (un) is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M124">View MathML</a> and we deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M125">View MathML</a> for all N-function D K.

3.3 Almost everywhere convergence of the gradients

Lemma 3.6 The subsequence (un) obtained in Lemma 3.5 satisfies:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M126">View MathML</a>

Proof:

Let m > 0, k > 0 such that m > k. Let ρm be a truncation defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M127">View MathML</a>

where vj D(Q) such that vj ψ and vj Tk(u) with the modular convergence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M82">View MathML</a> (for the existence of such function see [11] since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M128">View MathML</a>).

ωμ is the mollifier function defined in Landes [13], the function ωμ,j have the following properties:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M129">View MathML</a>

Set v = (Tk(un) - ωμ,j) ρm(un) as test function, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M130">View MathML</a>

(1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M131">View MathML</a>

(2)

Let us recall that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M132">View MathML</a>, there exists a smooth function u(see [14]) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M133">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M134">View MathML</a>

Remark also that,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M135">View MathML</a>

and it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M136">View MathML</a>.

Concerning I2,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M137">View MathML</a>

As in I1, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M138">View MathML</a>,

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M139">View MathML</a>

thus by using the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M140">View MathML</a>.

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M141">View MathML</a>.

About I3,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M142">View MathML</a>

Set Φ(s) = s2/2, Φ ≥ 0,then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M143">View MathML</a>

So,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M144">View MathML</a>

and easily we deduce, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M145">View MathML</a>.

Finally we conclude that: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M146">View MathML</a>.

We are interested now with the terms of (1)-(4).

About (1):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M147">View MathML</a>

recall that ρm(un) = 1 on {|un| ≤ k}.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M148">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M149">View MathML</a>

By using the inequality (1.3), we can deduce the existence of some measurable function hk such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M150">View MathML</a>

since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M151">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M152">View MathML</a> strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M153">View MathML</a>.

Then,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M154">View MathML</a>

Following the same way as in J2, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M155">View MathML</a>

Concerning the terms J4 :

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M156">View MathML</a>

Letting n → ∞, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M157">View MathML</a>

Taking now the limits j → ∞ and after μ → ∞ in the last equality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M158">View MathML</a>

Then,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M159">View MathML</a>

About (2):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M160">View MathML</a>

Since (un) is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M124">View MathML</a> and using (iii), we have (a(., un,∇un)) is bounded in LH(Q), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M161">View MathML</a>

so,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M162">View MathML</a>

About (4):

Since u ψ, then Tk(u) ≥ Tk(ψ) and there exist a smooth function vj Tk(ψ) such that vj Tk(u) for the modular convergence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M82">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M163">View MathML</a>

Taking into account now the estimation of (1), (2), (4)and (5), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M164">View MathML</a>

(3.8)

On the other hand,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M165">View MathML</a>

each term of the last right hand side is of the form ϵ(n, j, s), which gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M166">View MathML</a>

Following the same technique used by Porretta [2], we have for all r < s :

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M167">View MathML</a>

Thus, as in the elliptic case (see [7]), there exists a subsequence also denoted by un such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M168">View MathML</a>

We deduce then that,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M169">View MathML</a>

Lemma 3.7 For all k > 0,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M170">View MathML</a>

Proof:

We have proved that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M171">View MathML</a>

≤ ϵ (n, j, μ, s, m) (see (3.8)).

We can also deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M172">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M173">View MathML</a>

then,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M174">View MathML</a>

Letting n → ∞, we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M175">View MathML</a>

Using the same argument as above, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M176">View MathML</a>

and Vitali's theorem and (1.1) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M177">View MathML</a>

3.3.1 The convergence of the problems (Pn) and the completion of the proof of Theorem 3.1

The passage to the limit is an easy task by using the last steps, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M178">View MathML</a>

then,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/46/mathml/M179">View MathML</a>

for all φ D(RN+1) which are zero in a neighborhood of (0,T) × ∂Ω and {T} × Ω.

4 Conclusion

In this article, we have proved the existence of solutions of some class of unilateral problems in the Orlicz-Sobolev spaces when the right-hand side is a Radon measure.

Competing interests

The authors declare that they have no competing interests.

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