Inverse Problems for Nonlinear Quasi-Variational Inequalities with an Application to Implicit Obstacle Problems of pp-Laplacian Type

The primary objective of this research is to investigate an inverse problem of parameter identification in nonlinear mixed quasi-variational inequalities posed in a Banach space setting. By using a fixed point theorem, we explore properties of the solution set of the considered quasi-variational inequality. We develop a general regularization framework to give an existence result for the inverse problem. Finally, we apply the abstract framework to a concrete inverse problem of identifying the material parameter in an implicit obstacle problem given by an operator of $p$-Laplacian type

Existence of solutions for implicit obstacle problems of fractional laplacian type involving set-valued operators

The paper is devoted to a new kind of implicit obstacle problem given by a fractional Laplacian-type operator and a set-valued term, which is described by a generalized gradient. An existence theorem for the considered implicit obstacle problem is established, using a surjectivity theorem for set-valued mappings, Kluge’s fixed point principle and nonsmooth analysis

Some Stationary and Evolution Problems Governed by Various Notions of Monotone Operators

The purpose of this work is to explore some notions of monotonicity for operators between Banach spaces and the applications to the study of boundary value problems (BVPs) and initial boundary value problems (IBVPs) for partial differential equations (PDEs), with the possibility in the end to examine new problems and provide some solutions. Variational approach will be used to reformulate these problems into stationary equations (in the case of BVPs) and evolution equations (in the case of IBVPs), where the underlined operators constructed as realizations of those problems in appropriate function spaces. This is known as weak formulation, which allows us to find weak solutions of the problems in a larger functions space rather than classical solutions that are sufficiently smooth. The theory of monotone and pseudomonotone operators will be applied to find existence theorems for stationary equations and evolution equations. In addition, the existence theorem for evolution equations with locally monotone operator will also be presented as a generalisation of the one with monotone operators. Another type of monotonicity so-called strict p-quasimonotonicity, which is defined in term of Young measures. This type of weaker, integrated version of monotonicity is directly applied in the study of elliptic and parabolic system of PDEs, the difficulty arises from dealing with this monotonicity is overcome by the theory of Young measures. The application of these monotonicity in the study of variational inequality will also be discussed. In particular, there is a new setting for strict p-quasimonotonicity in a particular type of elliptic variational inequalities, the proof of the new existence theorem will also be presented. Some open problems on the application of strict p-quasimonotonicity in the study of parabolic variational inequalities will also be discussed. Finally, we mention the theory of monotone and pseudomonotone operators in the study of second order evolution equations. A new setting of the local monotonicity in the second order evolution equations will be presented as well as the new existence theorem