This project focuses on elliptic variational problems, parabolic variational inequalities, equimeasurable convex symmetrization, variational integrals involving mappings of bounded variation, and nonlinear potential theory. Much of the proposer's recent work in the area of elliptic variational problems has concentrated on problems involving the regularity of solutions to variational inequalities that arise from both irregular and smooth obstacles and that involve degenerate elliptic operators. An approach has been developed that may yield more information about the free boundary in the case of smooth obstacles and it may lead also to another treatment in the case of elliptic systems. These questions and techniques lead to their parabolic counterparts in parabolic variational inequalities where less is known and the difficulties are greater. Many of the questions and techniques in elliptic variational problems, and parabolic variational inequalities have considerable overlap with those in nonlinear potential theory. The proposer finds the problems of equimeasurable convex symmetrization and variational integrals involving mappings of bounded variation especially compelling because of their interaction with geometric analysis and the promise of going further in the direction of applied areas.
