Regularity Problems in the Calculus of Variations and Elliptic Partial Differential Equations

Columbia University, New York NY

Investigators

Abstract

This project focuses on central problems in classical fields such as the calculus of variations and free boundary problems. Some problems find their motivation in the applied sciences and therefore offer opportunities for collaborations among mathematicians and other researchers. For example, the so-called "one-phase problem" describes the motion of a fluid in which a cavity is present that is a mixture of vapor and gas and the pressure on the cavity is constant. Other problems under investigation are related to issues of optimal transportation and allocation of resources. All such problems require the development of new sophisticated techniques, and progress will be disseminated to the scientific community to invigorate the advancement of the theory. The PI will study the classification of cones in low dimensions for the "one-phase" free boundary problem. As a tool in the analysis of free boundaries, the PI will investigate some boundary Harnack type results. In the context of optimal transportation, the PI is interested in the higher regularity for a class of degenerate Monge-Ampere equations in which the right hand side vanishes on the boundary. Finally, parts of this project deal with the fundamental question of local regularity of minimizers in the calculus of variations. In particular, in the scalar case the PI will investigate situations in which the integrand becomes highly degenerate on some compact set. In the vectorial case, the PI plans to construct some interesting singular minimizers in low dimensions and will also explore a viscosity approach to regularity for certain types of functionals.