Inverse Problems for Evolutionary Hemi-Quasi -Variational Inequalities with Applications

This paper develops a comprehensive framework for estimating discontinuous or rapidly varying coefficients in evolutionary hemi-quasi -variational inequalities involving multivalued monotone, semi-monotone, and pseudo-monotone maps. To establish that the coefficient-to-solution map is well-defined, we present new solvability results and demonstrate the weak compactness of the solution set for the considered hemi-quasi -variational inequalities.We introduce a novel variational selection to circumvent the commonly adopted but highly restrictive assumption that the sum of a monotone map and a pseudo-monotone map is monotone. Additionally, we relax the compactness assumption on the involved embedding operators to make the results readily applicable to evolutionary problems. Subsequently, we establish the existence of solutions for the inverse problem by developing a general regularization framework to counter the ill-posedness of such problems. The feasibility and efficacy of the developed framework are tested on three applied models: nonlinear implicit obstacle problems, a variational model with nonlocal constraints, and contact problems.