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

在这个奖项下，研究人员将研究几个自由边界问题解的正则性和间断集。这些问题涉及系统，其中有两个不同的阶段，以及一个“糊状区域”，其中阶段是“退化”的。这类问题的例子包括混合型抛物守恒律方程，不适定的Hele-Shaw问题，Hele-Shaw问题中自由边界的时空正则性，以及前向后向热方程。以前使用单调性公式分析自由边界的方法在存在简并相时不再适用。在动力学解的精神下，PI发展了一种自由边界正则性的方法，允许直接研究自由边界措施。这是通过寻找一些辅助度量的表达式，并用正则区域内的水平集来逼近自由边界来实现的。在这些退化问题中，自由边界度量的例外集将为零，而不是通常的n维Hausdorff度量。这种方法是基于一组不连续面上的解的可用正则性或可积性。在科学中，有许多问题的特点是一个系统的两个阶段共存的“糊状区域”。这个项目涉及描述这种系统的方程的数学表征、描述和解的性质。

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 Previous methods of analyzing the free boundaries using monotonicity formulae are no longer applicable when a degenerate phase is present.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.