The aim of this proposal is to study certain free boundary problems with the lack of "classical" properties such as the uniform ellipticity (parabolicity) or the nonnegativity of the solutions. More specifically, the PI proposes to study the following questions that will help to understand the essence of the new difficulties and the possible ways of their resolution: 1) the continuity of the derivatives in the obstacle problem for the sub-Laplacian in the Heisenberg group; 2) the classification of homogeneous global solutions of the obstacle problem for the p-Laplacian in the plane; 3) uniform convergence of the level sets in a degenerate phase-transition model; 4) smoothness of the free boundary in a degenerate Bernoulli-type problem in the plane; and 5) one-phase obstacle and Stefan type problems with no sign restrictions on solutions. Some of the problems can be attacked by using recently developed methods, while a revitalization of some older methods might prove successful in other cases. Partial results as well as empirical evidence (in some cases) are provided to support the expected and conjectured results.
Free boundaries are apriori unknown sets, coming up in solutions of partial differential equations and variational problems. Typical examples are the interfaces and moving boundaries in problems on phase transitions and fluid mechanics. Main questions of interest are the regularity (smoothness) of free boundaries and their structure. Thanks to the contributions by many mathematicians the theory of free boundaries has developed over the last decades to a very deep and beautiful part of mathematics. However, in a number of free boundary problems that arise in applications, ranging from geometry and optimal control to robotics and superconductivity, certain traditional assumptions may break down. The aim of this proposal is to understand which results could be carried over from the classical theory of free boundaries to the case of those problems.
