Free Boundary Problems with a Degenerate Phase: Regularity of Solutions and Interphases

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Under this award the investigator will study of the regularity of solutions and the discontinuity sets for several free boundary problems. These problems involve systems where there are 2 distinct phases together with a "mushy region" where the phase is "degenerate". Examples of such problems include a mixed-type parabolic-conservation law equation, the ill-posed Hele-Shaw problem, the space-time regularity of the free boundary in the Hele-Shaw problem, and a forward backward heat equation. Previous methods of analyzing the free boundaries using monotonicity formulae are no longer applicable when a degenerate phase is present. The PI has developed an approach to the regularity of free boundaries in the spirit of kinetic solutions that allows to study the free boundary measures directly. This is done by finding expressions of some auxiliary measures and approximating the free boundary with level sets from within the regular region. In these degenerate problems exceptional sets will be null for the free boundary measure, rather then the usual n-dimensional Hausdorff measure. This approach is based on available regularity or integrabilities for solutions across the set of discontinuities. There are many problems in science which feature a "mushy region" where two phases of a system coexist. This project involves the mathematical characterization, description and properties of the solutions of the equations that describe such systems.