Regularity of solutions to nonlinear elliptic PDEs

Columbia University, New York NY

Investigators

Abstract

Regularity of Solutions to Nonlinear Elliptic PDEs. Abstract of Proposed Research Ovidiu Savin The focus of this project will be on proving properties of certain solutions of nonlinear elliptic partial differential equations. The equations include various semi-linear and fully nonlinear equations; especially variants of the Monge-Ampere equation and the infinity Laplacian. Many of the problems have a variational description. The solutions may be entire functions, solutions of obstacle problems or free boundary problems. The issues include questions about the regularity and symmetries of solutions, their level sets and possible singularities. The problems to be studied in this project arise from simple models of concrete phenomena in elasticity, geometry and statistical mechanics and related areas. The motivation for one of the problems comes from the theory of optimal transportation. Another problem involves the regularity of free boundaries originating as limits of certain random surfaces and arises in statistical mechanics. For all of these problems there are many open mathematical questions that will be investigated. Particular concentration will be put on developing regularity theories for the solutions of these problems.