Harmonic Analysis and Partial Differential Equations

9705784 Hofmann PI旨在研究时变抛物边界几何之间的关系，一方面，估计热量措施，或抛物奇异积分，另一方面。 该项目的一个主要长期目标是表明，适当的热量措施或奇异积分估计的抛物边界上的时间变化域意味着一些规律性， 边界 这些问题是一类“自由边界问题”，即， 这是一个人们希望能够对一个事先没有指定的边界说些什么的问题。 例如，在相位数学中， 过渡，其中物质的两相之间的界面（自由边界）（例如，其固相和液相）不是预先给定的，而是取决于其它物理约束随时间演变。

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.