Inverse problems for generalized quasi-variational inequalities with application to elliptic mixed boundary value systems

Abstract

Abstract This paper investigates the inverse problem of estimating a discontinuous parameter in a quasi-variational inequality involving multi-valued terms. We prove that a well-defined parameter-to-solution map admits weakly compact values under some quite general assumptions. The Kakutani–Ky Fan fixed point principle for multi-valued maps is the primary technical tool for this result. Inspired by the total variation regularization for estimating discontinuous parameters, we develop an abstract regularization framework for the inverse problem and provide a new existence result. The theoretical results are applied to identify a parameter in an elliptic mixed boundary value system with the p-Laplace operator, an implicit obstacle, and multi-valued terms involving convex subdifferentials and the generalized subdifferentials in the sense of Clarke.