Harmonic Analysis and Partial Differential Equations

点击翻译按钮获取中文摘要

9705784 Hofmann The PI intends to investigate the relationship between the geometry of time varying parabolic boundaries, on the one hand, and estimates for caloric measure, or parabolic singular integrals, on the other. A primary long-term goal of the project is to show that appropriate caloric measure or singular integral estimates on the parabolic boundary of a time varying domain imply some regularity of the boundary. These questions are a type of "free boundary problem", that is, a problem where one would like to be able to say something about a boundary which has not been specified in advance. These sorts of questions arise, for example, in the mathematics of phase transitions, where the interface (the free boundary) between two phases of a substance (e.g., its solid and liquid phases) is not given in advance, but rather evolves with time depending on other physical constraints.