PI: Evans DMS-9424342 Evans will continue his research on a variety of questions from nonlinear differential equations, mostly variational problems having geometric or physical origin. The primary proposed new efforts include investigations of new PDE methods for designing optimal Monge-Kantorovich mass transfer schemes, flat spots on surfaces evolving under Gauss curvature flow, the possible regularity of "optimal" Lipschitz extensions, and time evolutions generated by natural quasiconvex energy functionals. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations.
