Inverse problems for nonlinear quasi-hemivariational inequalities with application to mixed boundary value problems**Project supported by the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie Grant agreement No. 823731 CONMECH, and National Science Center of Poland under Preludium Project No. 2017/25/N/ST1/00611. It is also supported by the Natural Science Foundation of Guangxi under Grant No. 2018JJA110006, and

Abstract

This paper is devoted to studying an inverse problem of parameter identification in a nonlinear quasi-hemivariational inequality posed in a Banach space. We employ the Kluge’s fixed point theorem for the set-valued selection map, use the Minty approach and some properties of the Clarke subgradient to prove that the quasi-hemivariational inequality associated to the inverse problem has a nonempty, bounded, and weakly compact solution set. We develop a general regularization framework to provide an existence result for the inverse problem. As an illustrative application, we study an identification inverse problem in a complicated mixed elliptic boundary value problem with p -Laplace operator and an implicit obstacle.