Nonsmooth Variational Problems and Their Inequalities

Abstract

A.), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 9 8 7 6 5 4 3 2 1 springer.com (TXQ/SB) PrefaceNonsmooth variational problems have their origin in the study of nondifferentiable energy functionals, and they arise as necessary conditions of critical points of such functionals. In this way, variational inequalities are related with convex energy or potential functionals, whereas the new class of hemivariational inequalities arise in the study of nonconvex potential functionals that are, in general, merely locally Lipschitz. The foundation of variational inequalities is from Fichera, Lions, and Stampacchia, and it dates back to the 1960s. Hemivariational inequalities were first introduced by Panagiotopoulos about two decades ago and are closely related with the development of the new concept of Clarke's generalized gradient. By using this new type of inequalities, Panagiotopoulos was able to solve various open questions in mechanics and engineering. This book focuses on nonsmooth variational problems not necessarily related with some potential or energy functional, which arise, e.g., in the study of boundary value problems with nonsmooth data and/or nonsmooth constraints such as multivalued elliptic problems with multifunctions of Clarke's subgradient type, variational inequalities, hemivariational inequalities, and their corresponding evolutionary counterparts. The main purpose is to provide a systematic and unified exposition of comparison principles based on a suitably extended sub-supersolution method. This method manifests as an effective and flexible technique to obtain existence and comparison results of solutions. Moreover, it can be employed for the investigation of various qualitative properties such as location, multiplicity, and extremality of solutions. In the treatment of the problems under consideration, a wide range of methods and techniques from nonlinear and nonsmooth analysis are applied; a brief outline of which has been provided in a preliminary chapter to make the book self-contained. The book is an outgrowth of the authors' research on the subject during the past 10 years. A great deal of the material presented here has been obtained only in recent years and appears for the first time in book form.

On the Boundedness of Solutions to Elliptic Variational Inequalities

“…Regarding existence and multiplicity results for problems of the form (1.1) we refer, without guarantee of completeness, to the papers of Carl [6], [27] and the references therein. An overview about results to nonsmooth analysis and variational-hemivariational inequalities can be found in the monographs of Carl-Le-Motreanu [7] and Motreanu-Rȃdulescu [26]. We also point out a recent work of Carl [4] in which the class of variational-hemivariational inequalities has been extended to a more general class of inequalities.…”

The Sub-Supersolution Method and Extremal Solutions of Quasilinear Elliptic Equations in Orlicz-Sobolev Spaces

“…In [1,Chapter 3], the differential part of (1) is a LerayLions operator in Sobolev spaces and the nonlinearity ( , , ) satisfies the growth condition:…”

“…The choice ( ) = | | , > 1 leads to [1,Theorem 3.17]. A nonstandard example is ( ) = ∫ | | 0 log(1 + | | )| | −2 (see, e.g., [2,3]).…”

“…A nonstandard example is ( ) = ∫ | | 0 log(1 + | | )| | −2 (see, e.g., [2,3]). Many papers used the surjectivity result for pseudomonotone operators (see, e.g., [1, Theorem 2.99]) defined on reflexive spaces to prove the existence of the solution (see, e.g., [1,2,4,5]). Our method does not need the reflexivity of the spaces.…”

“…In this paper, we get rid of the restriction of the reflexivity of the spaces and get a weak solution for (1) in Orlicz-Sobolev spaces by using a linear functional analysis method. We also give the enclosure of solutions and prove the existence of extremal solutions.…”