Research On Doubly Nonlinear Evolution Equations And Hemivariational Inequalities

In this thesis, we first introduce a class of monotone operators, namely, pseudomonotone operator, which plays an important role in the study of nonlinear problems. We present its various generalization forms and discuss their relations to each other. Then, several types of doubly nonlinear evolution equations and hemivariational inequali-ties with pseudomonotone operators are studied seriously by means of the theory of partial differential equations, nonlinear monotone oper-ators, convex analysis, nonsmooth analysis and multivalued analysis, the method of implicit time-discretization and the compact embed-ding theorems in Lebesgue-Bochner spaces. The existence theorems are established at the first time and our results generalize the exist-ing ones concerning doubly nonlinear evolution equations, parabolic, hyperbolic evolutional equations and hemivariational inequalities.Chapter One is a general introduction. It introduces the two branches of inequalities, i.e., variational inequalities and hemivari-ational inequalities, and presents the main difficulty, methods and techniques regarding nonlinear problems. Then, we recall the main methods and results concerning nonlinear evolution equations, ellip-tic, parabolic and hyperbolic equations and hemivariational inequal-ities. Finally, we show the main point of this paper and the work of the following chapters.Chapter Two is concerned with the preliminaries necessary for the study of this paper, including the theory of functional spaces, convex analysis, nonsmooth analysis and monotone operators, sev-eral compactness lemmas and some conclusions in nonlinear functional analysis.In the first section of Chapter Three, we introduce some gener- alized definitions of multivalued pseudomonotone mapping and the existence theorems for parabolic problems governed by them. Par-ticularly, we show that the Wλ0pseudomonotone mapping given by professor Kasyanov is not the generalization of its previous one. The second section deals with a class of nonlinear evolution equations with pseudomonotone operators. The existence theorems of solutions for these problems are established under a weaker coerciveness condition compared to the previous result.As we know, much concern is devoted to doubly nonlinear evolu-tion equations and elliptic, parabolic, hyperbolic hemivariational in-equalities and the theory result is fruitful. However, on the one hand, the study of doubly nonlinear equations requires that both the two nonlinear operators be maximal monotone and the subdifferential of convex functional. On the other hand, the evolution hemivariational inequalities studied by mathematicians now are mainly parabolic and hyperbolic. Therefore, considering these situations, in this thesis we deal with several classes of doubly nonlinear evolution equations and hemivariational inequalities in the following fourth to sixth chapter.Chapter Four and Five are concerned with a class of single-valued doubly nonlinear evolutional hemivariational inequalities and a class of doubly nonlinear evolutional hemivariational inequalities of bound-ary type, respectively. In Chapter Six, we deal with the doubly non-linear evolutional inclusions and hemivariational inequalities of first and second orders. In this chapter, the nonlinear operator may be multivalued and the results are more general.Unlike the methods to parabolic hemivariational inequalities such as the method of Faedo-Galerkin, regularity method of duality opera-tor, and upper and lower solution method, in this paper we adopt the implicit time-discretization technique to transform the evolution prob-lems into elliptic ones, and then construct the approximate solutions by the existence theorems of elliptic problems. After establishing a series of a priori estimates and appropriate compact lemmas, we com-plete the limit process of approximate solutions to true solutions by using the properties of Clarke's generalized gradient and the mono-tonicity of the operators. Our results extend the existing ones and represent the latest breakthrough in this field.