A Sequential Minimization Technique for Elliptic Quasi-variational Inequalities with Gradient Constraints

Abstract

A class of nonlinear elliptic quasi-variational inequality (QVI) problems with gradient constraints in function space is considered. Problems of this type arise, for instance, in the mathematical description of the magnetization of superconductors, in problems in elastoplasticity, or in electrostatics as well as in game theory. The paper addresses the iterative solution of the QVIs by a sequential minimization technique relying on the repeated solution of variational inequality--type problems. A monotone operator theoretic approach is developed which does not resort to Mosco convergence as is often done in connection with existence analysis for QVIs. For the numerical solution of the QVIs a penalty approach combined with a semismooth Newton iteration is proposed. The paper ends with a report on numerical tests involving the $p$-Laplace operator and various types of nonlinear constraints.