This proposal focuses on regularity issues for stationary and evolution free boundary problems. Part of this project deals with two-phase free boundary problems with distributed sources, driven by uniformly elliptic operators. We plan to develop new techniques to settle the fundamental question of whether flat or Lipschitz free boundaries are smooth. We also propose to investigate similar issues in the context of free boundary problems in the Heisenberg group, which cannot be treated within the classical framework. Appropriate strategies will be developed to take into account the subtle geometric features of the problem. Another problem under investigation is the regularity of the moving free boundary originating in a non-homogeneous evolution problem of Stefan-type. Finally, we propose to complete the analysis of thin one phase free boundaries by investigating the question of their higher regularity (analyticity).
The free boundary problems under investigation arise in different areas of mathematics and science in general: shape optimization, combustion theory, probability and statistics among others. Typical examples of two-phase free boundary problems with distributed sources are the Prandtl-Bachelor model in fluid dynamics or the eigenvalue problem in magneto hydrodynamics. Stefan-type evolution problems describe the melting (or solidification) of a material with a solid-liquid interphase. Finally, thin free boundaries are relevant in classical physical models in mediums where long range (non-local) interactions are present, as in the propagation of surfaces of discontinuities, like planar crack expansion. A better understanding of these problems is thus of great interest to a wide scientific community.
