The Existence Of Solutions To Obstacle Problems For Elliptic Hemivariational Inequalities

This dissertation collects the main results obtained by the author during the period when he has applied for the M.D.The major contents are presented as follows:In chapter 2, by using the penalty method from optimization theory, we consider the existence of solutions to a class of quasilinear elliptic variational inequality involv-ing p-Laplacian as follows: whereλ1 is the first eigenvalue of negative p-Laplacian. For the convenience of using penalty method, some appropriate assumptions needed to be made on the potential function f(x, u). In order to obtain the existence of solutions to the problem above, we also use the Clarke generalized subdifferential, the property of the first eigenvalue of negative p-Laplacian and penalty operator theorem to acquire an approximation sequence, and then on the basis of properties of variational inequalities, we can obtain the existence of solutions to the inequality above.In chapter 3, we consider the existence of solutions of an obstacle problem for a class of nonlinear hemivariational inequality defined on closed convex sets as follows: where w∈Lq(Ω), w(x)∈(?)j(x, u(x)) a.e. onΩ. Our approach is mainly variational and it is based on the nonsmooth critical point theory for locally Lipschitz functionals defined on a closed convex set. After making some appropriate assumptions on the po-tential function j(x, u), the important compact condition is satisfied for hemivariational inequality above, and so nonsmooth critical point theorem on closed convex sets, prop-erties of Clarke subdifferential and so on, can be used, hence, the existence of solutions to this problem is acquired.In chapter 4, by using nonsmooth version of three points critical theory, we con-sider a class of obstacle problems for variational-hemivariational inequalities involv-ing p-Laplacian as follows: (?) v∈K, where J0(x,u) is Clarke generalized directional derivative of J(x,u)=∫0uj(x, t)dt, j(x,u) is locally Lipschitz with respect to u a.e. onΩand G0(x,u) is the directional derivative of G(x,u)=∫0g(x,t)dt in the sense of convex analysis,-g(x, u) is proper, convex, lower semicontinuous with respect to u a.e. onΩ. For the sake of applying nonsmooth three points critical theory, we make some appropriate assumptions on the potential function j(x, u) and g(x, u), and according to properties of Sobolev space W01,p(Ω), we make some estimates of inequalities so that the compact condition and Min-max inequality can be satisfied, and finally, we prove the existence of three solutions for the problem above. In chapter 5, we consider a class of obstacle problems for Neumann-type variational-hemivariational inequalities involving the p(x)-Laplacian as follows:Find u∈K={w∈W01,p(x)(Ω), w(x)≥0 a.e. onΩ} such that (?) v∈K, where p∈C(Ω),1< p-:= infx∈Ωp(x)≤p+:= supx∈Ωp(x)＜∞; v is the outward normal vector of (?)Ω;λ,μare two-parameters andλ,μ≥0. We know that the p(x)-Laplacian is a generalization of the p-Laplacian(i.e. p(x)= p is a constant), and the p(x)-Laplacian operator possesses more complicated nonlinear properties, for example, it is not homogeneous, and usually it does not have the so-called first eigenvalue, since the infimum of its principal eigenvalue is zero. This causes many problems, and some classical theories and methods, such as the theory of Sobolev spaces, the properties of the first eigenvalue of negative p-Laplacian and so on, are not applicable. To overcome these difficulties, some appropriate assumptions need to be made on the potential function j(x, u) and g(x,u), and on the basis of the properties of Lebesgue-Sobolev space W01,p(x) (Ω), we obtain some estimates of inequalities so that nonsmooth (PS)-condition, and Min-max inequality can be satisfied, and the existence of multiple solutions to variational-hemivariational inequality above can be proved by applying nonsmooth three points critical theory.