Nonlinear Elliptic Quasi-variational Hemivariational Inequalities And Their Inverse Problems

As an important part of modern partial differential equation theory,variational inequality has been widely used in physics,mechanics,engineering science,economics and other fields,and has important research significance.If energy function or constraint set in real problems is directly dependent on the state variable,the quasi-variational or quasi-hemivariational inequality problem is derived.In this dissertation,we mainly study a class of nonlinear elliptic quasi-variational hemivariational inequality with parameters and their inverse problems for parameters identification.When the parameters are fixed,the existence and properties of solutions of quasi-variational hemivariational inequalities are proved by combining monotone operator theory,nonsmooth analysis,convex analysis and nonlinear functional analysis.Based on a regularization optimization method,Weierstrass type theorem,Rellich-Kondrachov tight embedding theorem and lower semicontinuity theory,we obtain an existence result of inverse problem.This dissertation is divided into six chapters.They are organized as follows.Chapter one is a general introduction.It mainly introduces the research background and significance of variational and hemivariational inequality problems,summarizes their existing research achievements and methods and introduces the main research content and theoretical achievements of this dissertation.In chapter two,we focus on summarizing the preliminaries necessary for the study of this dissertation,including basic concepts and important theorems of nonsmooth analysis,convex analysis and nonlinear functional analysis.In chapter three,we consider a class of nonlinear quasi-variational hemivariational inequality with parameters.This inequality mainly has two characteristics: the first one is that it contains two nondifferentiable functions which are explicitly dependent on the solution;The second is that of unknown parameters appearing on the principal operator and constraint sets,respectively.By using KKM lemma,monotone operator theory and the properties of Clarke's generalized gradient,we prove the existence and weak closeness of solutions for nonlinear quasi-variational hemivariational inequality with parameters.Moreover,the existence result of solutions for this inequality develops some theoretical results in literature.This is a new breakthrough and development.In chapter four,on the basis of chapter three,we continue to study the inverse problem of nonlinear quasi-variational hemivariational inequality.Under appropriate conditions,we obtain uniformly boundedness of solutions for the direct problem.The inverse problem for parameter identification on the principal operator and constraint sets is studied by using regularization optimization method.In chapter five,the theoretical results obtained in chapter three and chapter four are applied to elliptic mixed boundary value obstacle problems.existence results of weak solutions for obstacle problems and its inverse problems are obtained.Finally,several examples are given to satisfying the conditions of convex and nonconvex functions.In chapter six,we summarize the main research results of this dissertation and the prospect of future research.