Mass Transfer and Evolution Problems, Free Boundary Problems

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DMS-9970577ABSTRACTM. Feldman will work on three main topics. (1) Study of Monge-Kantorovichmass transfer problem in nonhomogeneous media, which include transporton manifolds with obstacles, and in anisotropic and nonsmooth spaces.Also, related evolution problems will be studied. Recently a progress was made by many authors in studying mass transport in the Euclidean space. Nonhomogeneous media is a natural next step. (2) Study of existence ofconvex viscosity solutions of geometric evolution equations, including nonlocal curvature motion, in the spatial dimensions three and higher. Such geometric evolutions are related to Monge-Kantorovich mass transfer. (3) Free boundary problems: existence and stability of solutions, and regularity of free boundaries.Methods of mass transfer theory were used recently in several applications including mathematical models in economics, meteorology, fluid dynamics, sandpiles growth and collapse, non-Newtonian fluids, plasticity. Free boundaries and moving interfaces arise naturally in many models.Mathematical analysis of such problems improves our understanding of qualitative properties of models, and is a necessary ingredient in developing of fast and reliable numerical algorithms. In addition, mathematical study sometimes discovers unexpected connections between different models (for example, partial differential equations and transport problems), and such interplay is useful in applications.