Abstract Feldman The project is devoted to the study of variational problems with constraints, moving and free boundaries. The first area of the research is the study of evolution problems with pointwise gradient constraints, related to Monge-Kantorovich mass transfer problem. One such evolution problem arises as a model of collapsing sandpiles, and there are other examples, including models of compression molding and type II superconductivity. In the joint work of the proposer with L.C.Evans and R.F.Gariepy, the collapsing sandpiles model was related to a geometric evolution problem where the velocity of a moving surface at each point is determined by both local and nonlocal geometry of the surface. This geometric problem possesses some "parabolicity" properties of nonlocal nature. The main idea of the proposed research is to construct viscosity solutions of the geometric evolution problem, to study regularity and asymptotic properties of these solutions, and to study connections of the geometric evolution problem with the original variational evolution problem. The second area of the research is the study of another class of variational evolution problems with constraints - heat flows for harmonic maps. In the previous work the proposer has proved partial regularity for heat flows into spheres satisfying some stability condition - a variational condition of entropy type. The next steps include the study of existence and uniqueness of such heat flows, and the study of heat flows into compact manifolds other then sphere. The third area of research is to try to extend some of celebrated results of L.Caffarelli on regularity of free boundaries in two-phase problems for linear equations to some classes of nonlinear elliptic equations. In previous work the proposer proved regularity for anisotropic two-phase problem with Lipschitz free boundary, i.e., in the case when the solutions satisfy two different linear elliptic equations in two subregions separated by free boundary. Many problems in science and engineering lead naturally to mathematical models that have the form of variational evolution problems with constraints. One class of such problems, the evolution problems with pointwise gradient constraints, includes models of collapsing sandpiles, of compression molding, of type II superconductivity. Qualitative properties of solutions of evolution problems are of interest by itself and with respect to possible applications. Typical properties of variational evolution problems with pointwise gradient constraints include formation of moving free boundaries. The first area of the proposed research is the study of these moving boundaries by relating them to a geometric evolution problem for a moving surface and studying solutions of this geometric problem. Free boundary problems arise also in the static case. In this case the free boundaries can be viewed as a model of the interface between two different substances. The second area of the research is the study of regularity (smoothness) properties of such boundaries for some nonlinear static problems. Another class of variational evolution problems with constraints is heat flows for harmonic maps. Such problems arise in geometry and their solutions have interesting properties. These heat flows represent one of the models of liquid crystals. The third area of the research is the study of existence and uniqueness of heat flows for harmonic maps of certain classes.
