Open Access Research

An anisotropic quasilinear problem with perturbations

Jie Rui12 and Jianguo Si1*

Author Affiliations

1 School of Mathematics, Shandong University, Jinan, Shandong, 250100, P.R. China

2 College of Science, China University of Petroleum (East China), Qingdao, Shandong, 266555, P.R. China

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Boundary Value Problems 2013, 2013:147  doi:10.1186/1687-2770-2013-147


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/147


Received:7 December 2012
Accepted:30 April 2013
Published:18 June 2013

© 2013 Rui and Si; 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

This work focuses on proving the existence and uniqueness of strong solutions of perturbed anisotropic total variation flow with the Neumann boundary condition when the initial data is an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1">View MathML</a> function.

MSC: 35K65, 35K55.

Keywords:
anisotropic total variation flow; semigroups; strong solution

1 Introduction

Problems of general anisotropic total variation flow arise in a number of areas of science. The parabolic equations represent what Giga et al. called a very singular diffusivity (see [1]) and are a natural generalization of the total variation flow in the presence of an anisotropy. In the isotropic case, the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M2">View MathML</a> becomes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M3">View MathML</a> when the Lagrangian <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M4">View MathML</a> is given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M5">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M6">View MathML</a> is the usual <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M7">View MathML</a>-norm; i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M8">View MathML</a>. Let us recall that this PDE appears when one uses the steepest decent method to minimize the total variation. This method was introduced by Rudin and Osher (see [2,3]) in the context of image denoising and reconstruction. In the last years, its applications have been studied by many authors (see [4-7]).

Let Ω be an open bounded subset in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M10">View MathML</a>, with boundary Ω of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M11">View MathML</a>. In this paper, we are interested in the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M12">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M14">View MathML</a> is a 1-homogeneous convex function with linear growth as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M16">View MathML</a> is the Neumann boundary operator associated to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M17">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M18">View MathML</a> with ν the unit outward normal on Ω, and the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M19">View MathML</a> satisfies the following assumptions, which we shall refer to collectively as (M):

(M1) For almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M21">View MathML</a> is continuous nondecreasing, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M22">View MathML</a>;

(M2) For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M24">View MathML</a> is in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1">View MathML</a>.

As argued in [8], the choice of Neumann boundary conditions is a natural choice in image processing. It corresponds to the reflection of the picture across the boundary and has the advantage of not imposing any value on boundary and not creating edges on it. For instance, in [9], Andreu, Caselles and Mazón considered the elliptic problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M26">View MathML</a> with Neumann boundary conditions. In [7], Andreu et al. obtained the existence and uniqueness of entropy solutions of quasilinear parabolic equation with the Neumann boundary, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M27">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M29">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M30">View MathML</a> satisfies some additional assumptions. Our problem is closely related to motion under anisotropic mean curvature flow (see [10]) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M31">View MathML</a>. If we take the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M32">View MathML</a>-distance to give a set E as an initial condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M32">View MathML</a> being the polar function of f), then each sublevel set of the anisotropic mean curvature motion behaves instantaneously as the solution of Cauchy problem (1.1) where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M34">View MathML</a>. Recently Moll [11] proved the existence and uniqueness of the solutions of Dirichlet problem (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M31">View MathML</a>. As we all know, it is possible that the solution of (1.1) will blow up with perturbations. Therefore, in this paper, we extend the problem introduced in Moll [11] and obtain the existence and uniqueness of strong solutions of (1.1) when perturbation term satisfies assumption (M).

This paper is organized as follows. In Section 2 we recall some notions and basic facts. In Section 3 we define the notion of a strong solution for the Neumann problem of (1.1), and give the basic results in this paper. In Section 4 we prove the existence and uniqueness of solutions of an auxiliary equation, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M36">View MathML</a>

(1.3)

and for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M37">View MathML</a> we obtain the existence and uniqueness of a strong solution of problem (1.1).

2 Preliminaries

To make precise our notions, let us recall some preliminary facts.

Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M38">View MathML</a>, Du decomposes into absolutely continuous and singular parts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M39">View MathML</a>, where ∇u denotes the Radon-Nikodým derivative with respect to the Lebesgue measure and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M40">View MathML</a> is its singular part. There is also the polar decomposition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M41">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M42">View MathML</a> is the total variation measure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M40">View MathML</a>. For further information concerning functions of bounded variation, we refer to [12].

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M44">View MathML</a> we denote the space of weakly measurable functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M45">View MathML</a> (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M46">View MathML</a> is measurable for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M47">View MathML</a>) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M48">View MathML</a>. Observe that since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M49">View MathML</a> has separable predual, it follows easily that the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M50">View MathML</a> is measurable.

We shall need several results from [13] in order to give sense to the integrals of bounded vector fields with divergence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M51">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M52">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M53">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M54">View MathML</a>. Following [13], let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M55">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M56">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M57">View MathML</a>, the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M58">View MathML</a> is defined by the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M59">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M60">View MathML</a> is a Radon measure in Ω, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M61">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M62">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M63">View MathML</a> is absolutely continuous with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M64">View MathML</a> with the Radon-Nikodým derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M65">View MathML</a> which is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M64">View MathML</a> measurable function from Ω to ℝ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M67">View MathML</a> for any Borel set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M68">View MathML</a>. We also have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M69">View MathML</a>.

In [13], a weak trace on Ω of the normal component of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M56">View MathML</a> is defined. Concretely, it is proved that there exists a linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M71">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M72">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M73">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M74">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M75">View MathML</a>.

Next, let us introduce the concept of generalized total variation of a BV function with respect to a Finsler metric [14]. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M76">View MathML</a> be a Borel function not identically +∞. The function f will be called convex if for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M77">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M78">View MathML</a> is convex on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M9">View MathML</a>. We shall say that f is lower semicontinuous (in short l.s.c.) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M78">View MathML</a> is lower semicontinuous for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M77">View MathML</a>. The function will be called positively homogeneous of degree 1 (in short 1-homogeneous) if it satisfies the following property:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M82">View MathML</a>

(2.1)

f is a sublinear growth if there exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M83">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M84">View MathML</a>

(2.2)

Let us recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M85">View MathML</a> is a Finsler metric if it is a Borel function and it satisfies (2.1) and (2.2). If f satisfies (2.1), then the dual function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M86">View MathML</a> is defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M87">View MathML</a>. It is easy to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M88">View MathML</a> is convex, l.s.c. and satisfies (2.1). Then, if we adopt the following conventions: for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M89">View MathML</a>, we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M90">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M91">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M92">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M93">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M94">View MathML</a>

We say that f is coercive if there exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M95">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M96">View MathML</a>

(2.3)

It is easy to see that f is convex and has a sublinear growth, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M78">View MathML</a> is continuous for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M20">View MathML</a>.

We introduce the classes of vector fields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M99">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M100">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M38">View MathML</a>, the generalized total variation of u with respect to f in Ω is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M102">View MathML</a>

(2.4)

As a direct consequence of the definition, we have that the generalized total variation of u with respect to f in Ω is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M103">View MathML</a>-lower semicontinuous in Ω.

Now, we introduce the relaxed functional, which plays a basic role in proving the existence and uniqueness of the problem.

We define the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M104">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M105">View MathML</a>

(2.5)

We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M106">View MathML</a> the relaxed functional of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M107">View MathML</a>; i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M108">View MathML</a>

In [14], Amer and Belletini obtained the following result:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M109">View MathML</a>

(2.6)

Moreover, in [11], Moll proved the representation result:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M110">View MathML</a>

(2.7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M111">View MathML</a> is a representative of the equivalence class of homogeneous integrands <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M112">View MathML</a> associated to sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M113">View MathML</a> which are countable and sequentially weakly-dense in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M114">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M115">View MathML</a>

(2.8)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M38">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M117">View MathML</a>. The following useful inequality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M118">View MathML</a>

(2.9)

The equality holds if and only if the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M107">View MathML</a> defined by (2.5) is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M120">View MathML</a>-lower semicontinuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M121">View MathML</a>. By the inequality (2.9), we have the measure <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M122">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M123">View MathML</a>

(2.10)

for every Borel set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M124">View MathML</a>.

In this paper, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M125">View MathML</a> is a convex homogeneous integrand, i.e., for some constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M126">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M127">View MathML</a>

(2.11)

Let us define the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M128">View MathML</a> by the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M129">View MathML</a>

(2.12)

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M130">View MathML</a> on Ω, and Theorem 4 in [11], it is easy to obtain that the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M131">View MathML</a> is the relaxed functional of F defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M132">View MathML</a>

(2.13)

3 Strong solutions and main results

In this section we give the main concepts and results of Neumann problems (1.3) and (1.1).

Definition 3.1 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M133">View MathML</a> is a strong solution of (1.3) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M134">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M135">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M136">View MathML</a> a.e. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M9">View MathML</a> and a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M138">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M139">View MathML</a>

(3.1)

and a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M138">View MathML</a> it holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M141">View MathML</a>

(3.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M142">View MathML</a>

(3.3)

Next we give the main definition in this paper that is the strong solution of problem (1.1).

Definition 3.2 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M133">View MathML</a> is a strong solution of (1.1) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M144">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M145">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M136">View MathML</a> a.e. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M9">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M148">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M149">View MathML</a> a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M138">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M151">View MathML</a>

(3.4)

for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M152">View MathML</a> and a.e. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M153">View MathML</a>.

The main results of this paper are the following.

Theorem 3.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M13">View MathML</a>. Assume thatfsatisfies (2.11), then there exists a unique strong solution of (1.3) in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M155">View MathML</a>for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M156">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M157">View MathML</a>. Moreover, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M158">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M159">View MathML</a>are the strong solutions of (1.3) corresponding to initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M160">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M161">View MathML</a>, respectively, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M162">View MathML</a>

(3.5)

Theorem 3.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M13">View MathML</a>. Assume thatfsatisfies (2.11) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M37">View MathML</a>satisfies (M), then there exists a unique strong solution of (1.1) in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M155">View MathML</a>for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M156">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M157">View MathML</a>. Moreover, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M168">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M169">View MathML</a>are the strong solutions of (1.1) corresponding to initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M170">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M171">View MathML</a>, respectively, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M172">View MathML</a>

(3.6)

4 Proof of the main results

In this section we prove Theorem 3.3 by using the techniques of completely accretive operators [15] and Crandall-Liggett’s semigroup generation theorem [16].

Let us recall the notion of completely accretive operators introduced in [15]. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M173">View MathML</a> be the space of measurable functions in Ω. Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M174">View MathML</a>, we shall write that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M175">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M176">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M177">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M178">View MathML</a>. Let A be an operator (possibly multivalued) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M173">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M180">View MathML</a>. We shall say that A is completely accretive if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M181">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M182">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M183">View MathML</a>, then A is completely accretive if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M184">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M185">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M186">View MathML</a>. A completely accretive operator in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1">View MathML</a> is said to be m-completely accretive if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M188">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M189">View MathML</a>. In that case, by Crandall-Liggett’s theorem, A generates a contraction semigroup denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M190">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1">View MathML</a>, which is given by the exponential formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M192">View MathML</a>

Let us write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M193">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M194">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M195">View MathML</a>, and it is a mild solution (a solution in the sense of semigroups [15]) of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M196">View MathML</a>

(4.1)

such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M197">View MathML</a>.

We shall use a stronger notion of the solution of (4.1). We say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M198">View MathML</a> is a strong solution of (4.1) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M153">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M200">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M201">View MathML</a> for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M202">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M203">View MathML</a> (the domain of A) and A is m-completely accretive, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M204">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M159">View MathML</a> is a strong solution of (4.1) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M206">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M207">View MathML</a>.

To obtain the solution of problem (1.1), we need the result of problem (1.3). Thus, at first, we will prove the existence and uniqueness of a strong solution of problem (1.3). Let us introduce the following operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M208">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1">View MathML</a> associated to problem (1.3).

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M210">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M211">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M212">View MathML</a> and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M213">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M214">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M215">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M216">View MathML</a>

(4.2)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M217">View MathML</a>.

Proposition 4.1The operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M208">View MathML</a>is m-completely accretive with dense domain. For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M219">View MathML</a>, the semigroup solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M220">View MathML</a>is a mild solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M221">View MathML</a>

(4.3)

To prove Proposition 4.1, we need to prove the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M208">View MathML</a> has some characterization, satisfies the range condition and has dense domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1">View MathML</a>.

By the results of Section 2, the relaxed functional ℱ is convex and lower semicontinuous. Therefore, the subdifferential <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M224">View MathML</a> of ℱ is a maximal monotone operator in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1">View MathML</a>, and consequently, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M226">View MathML</a> is the semigroup solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1">View MathML</a> generated by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M228">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M229">View MathML</a> is a strong solution of the problem (see [15])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M230">View MathML</a>

(4.4)

Recall that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M224">View MathML</a> is defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M232">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M233">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M234">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M235">View MathML</a>.

To prove the existence and uniqueness of a strong solution of problem (1.3), we also need the next proposition.

Proposition 4.2The operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M224">View MathML</a>has dense domain in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M238">View MathML</a>.

The following lemmas will be used to prove Proposition 4.1 and Proposition 4.2.

Lemma 4.3We have the following characterization of the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M208">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M210">View MathML</a>if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M211">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M212">View MathML</a>and there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M213">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M214">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M215">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M246">View MathML</a>

(4.5)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M247">View MathML</a>. Moreover, we have that

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M248">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M249">View MathML</a>,

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M250">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M251">View MathML</a>.

Proof We denote the operator by ℬ defined in the statement of the lemma. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M252">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M253">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M254">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M211">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M212">View MathML</a> and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M257">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M214">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M259">View MathML</a> and (4.2). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M251">View MathML</a>, applying results from [13], we have that there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M261">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M262">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M103">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M264">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M265">View MathML</a>. Using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M266">View MathML</a> as a test function in (4.2) and letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M267">View MathML</a>, we obtain (4.5), then we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M268">View MathML</a>, therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M269">View MathML</a>.

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M270">View MathML</a> in (4.5), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M271">View MathML</a>

Moreover, using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M272">View MathML</a>, we obtain (i).

We take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M273">View MathML</a> in (4.5) to obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M274">View MathML</a>. Using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M275">View MathML</a> in (4.5) and (i), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M276">View MathML</a>

Thus, (ii) holds.

Using (ii) in (4.5) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M277">View MathML</a>

Since the same inequality holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M278">View MathML</a>, (iii) is obtained. □

We consider the following possibly multi-valued functions: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M279">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M280">View MathML</a>. By the convexity of f, it follows that A is a monotone function satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M281">View MathML</a>

(4.6)

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M77">View MathML</a>, we consider the Moreau-Yosida approximation to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M283">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M284">View MathML</a>

(4.7)

and the Yosida approximation of the multi-valued function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M285">View MathML</a> is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M286">View MathML</a>

We have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M287">View MathML</a> is a convex Fréchet differentiable function (see [17]) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M288">View MathML</a> pointwise and a.e. in Ω when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M289">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M290">View MathML</a>. Moreover, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M291">View MathML</a>, we get the minimum in (4.6). In [11], Moll gave the following estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M292">View MathML</a>

(4.8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M293">View MathML</a>

(4.9)

We consider the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M294">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1">View MathML</a> to prove Proposition 4.1. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M296">View MathML</a>, we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M297">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M298">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M299">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M300">View MathML</a>

(4.10)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M301">View MathML</a>.

The operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M302">View MathML</a> satisfies the classical Leray-Lions assumption [18]. Hence, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M303">View MathML</a>, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M302">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M305">View MathML</a>.

Moreover, we need the following characterization of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M302">View MathML</a>.

Lemma 4.4For every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M303">View MathML</a>, the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M302">View MathML</a>is completely accretive in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M310">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M311">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M297">View MathML</a>, taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M313">View MathML</a> as a test function in (4.10), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M314">View MathML</a>

(4.11)

Similarly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M315">View MathML</a>, we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M316">View MathML</a> as a test function in (4.10) and obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M317">View MathML</a>

(4.12)

Using (4.11) + (4.12), we may write that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M318">View MathML</a>

According to (4.6) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M319">View MathML</a>, we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M320">View MathML</a>. Moreover, by Lemma 4.3 and Theorem 2 in [4], we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M321">View MathML</a>

It follows that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M302">View MathML</a> is completely accretive in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1">View MathML</a>. □

Lemma 4.5The operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M208">View MathML</a>satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M325">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M326">View MathML</a>is dense in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1">View MathML</a>.

Proof We divide the proof into two steps.

Step 1. We first prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M328">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M329">View MathML</a>, we shall find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M330">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M331">View MathML</a>, i.e., there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M332">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M333">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M334">View MathML</a>-a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M335">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M336">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M337">View MathML</a>

(4.13)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M217">View MathML</a>.

Using (4.10) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M305">View MathML</a>, we have that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M303">View MathML</a> there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M341">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M342">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M343">View MathML</a>

(4.14)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M344">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M302">View MathML</a> is completely accretive, it is obtained that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M346">View MathML</a>

(4.15)

Now taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M347">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M348">View MathML</a> in (4.14), respectively, we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M349">View MathML</a>

Using the estimate (4.9), (4.8) and Ω being a bounded subset in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M9">View MathML</a>, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M351">View MathML</a>

By (4.15), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M352">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M353">View MathML</a> depends on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M354">View MathML</a>. Moreover, we obtain that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M303">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M356">View MathML</a>

(4.16)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M357">View MathML</a>

(4.17)

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M358">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M121">View MathML</a> and we may extract a subsequence such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M360">View MathML</a> converges in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M103">View MathML</a>. Now, by (4.15) and (4.16), we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M362">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M211">View MathML</a>.

Observe that by (4.6) and (4.17), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M365">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M366">View MathML</a> and weakly relatively compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M366">View MathML</a>. We may assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M368">View MathML</a>

By (4.17) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M369">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1">View MathML</a>, we also have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M371">View MathML</a>

(4.18)

Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M372">View MathML</a> and taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M373">View MathML</a> in (4.14), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M374">View MathML</a>

and letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M375">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M376">View MathML</a>

that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M336">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M259">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M379">View MathML</a>

(4.19)

From the proof of Proposition 4 in [11], we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M380">View MathML</a>. Moreover, by (4.17) and (4.18), it implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M381">View MathML</a>. Now, we prove that u, v and z verify (4.13). Applying the Lebesgue convergence theorem in (4.14), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M211">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M383">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M384">View MathML</a>

(4.20)

To prove (4.13), we assume that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M385">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M217">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M387">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M262">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M267">View MathML</a>. Using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M266">View MathML</a> as a test function in (4.20) and letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M267">View MathML</a>, we obtain (4.13). That is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M393">View MathML</a>.

Step 2. Now let us prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M326">View MathML</a> is dense in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1">View MathML</a>. We only need to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M396">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M397">View MathML</a>. By Step 1, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M398">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M399">View MathML</a>. Thus, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M303">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M401">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M402">View MathML</a> and, in consequence, there exists some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M403">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M404">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M259">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M406">View MathML</a>

for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M251">View MathML</a>. Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M408">View MathML</a> in the above inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M409">View MathML</a>

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M410">View MathML</a>, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M411">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1">View MathML</a>. This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M413">View MathML</a>. □

Proof of Proposition 4.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M310">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M415">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M416">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M417">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M418">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M419">View MathML</a>

(4.21)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M420">View MathML</a>

(4.22)

for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M251">View MathML</a>. Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M422">View MathML</a> as a test function in (4.21), taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M423">View MathML</a> as a test function in (4.22), and by Theorem 2 in [4], we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M424">View MathML</a>

We get the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M208">View MathML</a> is completely accretive in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1">View MathML</a>.

Now, we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M208">View MathML</a> is closed. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M428">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M429">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M430">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M431">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M432">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M433">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M259">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M435">View MathML</a>

(4.23)

for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M251">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M437">View MathML</a>, we may assume that

(4.24)

Working as before, it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M381">View MathML</a>. Moreover, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M440">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M442">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M215">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M444">View MathML</a>

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M267">View MathML</a> in (4.23), and having in mind the lower semicontinuity of the functional ℱ defined in (2.13), we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M446">View MathML</a>

Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M447">View MathML</a>. By Lemma 4.5, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M448">View MathML</a> is m-completely accretive in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1">View MathML</a>. By Crandall-Liggett’s theorem, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M448">View MathML</a> generates a contraction semigroup in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1">View MathML</a> given by the exponential formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M452">View MathML</a>

The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M453">View MathML</a> is a mild solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M454">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M197">View MathML</a>. □

Proof of Proposition 4.2 We first prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M456">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M447">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M251">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M381">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M460">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M215">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M462">View MathML</a>

for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M251">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M464">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M465">View MathML</a>.

Next, by the proof of Proposition 4.1, we have that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M208">View MathML</a> is closed. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M467">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M468">View MathML</a>, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M469">View MathML</a>. □

Proof of Theorem 3.3 As a consequence of Proposition 4.2, the semigroups generated by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M208">View MathML</a> and by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M224">View MathML</a> coincide, and therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M472">View MathML</a> is a strong solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M473">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M197">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M475">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M476">View MathML</a> for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M477">View MathML</a>. Then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M478">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M149">View MathML</a>. By the characterization (i) in Lemma 4.3, we have (3.2) and (3.3) hold. The contractivity estimate (3.5) follows directly from the nonlinear semigroup theory. □

Let us define several operators that will be needed in this section. The single-valued operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M480">View MathML</a> is defined in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M482">View MathML</a>

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M483">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M484">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M485">View MathML</a>. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M486">View MathML</a> and H is convex. Moreover, by Fatou’s lemma, H is lower semicontinuous. Hence, ∂H is a maximal monotone graph in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M1">View MathML</a>.

Thus, to prove Theorem 3.4, we only need to obtain the following result.

Lemma 4.6<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M488">View MathML</a>.

Proof From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M489">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M211">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M212">View MathML</a> and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M257">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M493">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M259">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M495">View MathML</a>

(4.25)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M496">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M37">View MathML</a> satisfies (M1), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M498">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M499">View MathML</a>

By the above inequality and ℱ being lower semicontinuous in (4.25), we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M500">View MathML</a>

We have proved that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M501">View MathML</a>.

By Proposition 14 in [19], we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M502">View MathML</a>, and the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M503">View MathML</a> is closed. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/147/mathml/M504">View MathML</a>. □

Using Crandall-Liggett’s theorem and a similar proof of Theorem 3.3 again, we obtain that Theorem 3.4 holds.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and typed, read and approved the final manuscript.

Acknowledgements

We would like to thank the anonymous referees for their constructive comments, which were very helpful for improving this paper. The authors acknowledge the financial support of this research by the National Natural Science Foundation of China (Grant No. 10871117), NSFSP (Grant No. ZR2010AM013) and Fundamental Research Funds for the Central Universities (12CX04081A, 11CX04058A).

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