Mathematical Sciences: The Geometry of Harmonic Measure

在这个项目中，首席研究员将继续他在调和分析和偏微分方程之间的边界问题上的精细工作。他的主要工作目标是描述凸域外部的调和测度。一个关键问题是提供一个关于静电容量的第二次变化的界，它类似于体积的布伦-闵可夫斯基不等式。其他的研究课题是自由边界问题的特征函数的节点线的行为，Lipschitz域中的混合边值问题和airy型算子的L-p估计。调和分析和偏微分方程这两个学科在被称为经典分析的数学分支中占有突出地位。事实上，谐波分析是从傅立叶分析中发展出来的，傅立叶分析是为了帮助解决各种类型的偏微分方程。在这个项目中，首席研究员将研究跨越这两个领域的问题。特别是，他将尝试描述凸域外部的调和测度，并将使用调和函数理论中的技术研究各种类型的边值问题。

In this project the principal investigator will continue his fine work on problems that lie on the border between harmonic analysis and partial differential equations. The main goal of his work is the description of harmonic measure for the exterior of a convex domain. A key question is to provide a bound on the second variation of the electrostatic capacity that is analgous to the Brunn-Minkowski inequality for the volume. Other topics for study are the behavior of nodal lines of eigenfunctions of free boundary problems, mixed boundary value problems in Lipschitz domains and L-p estimates for Airy-type operators. The two subjects of harmonic analysis and partial differential equations figure prominently in the branch of mathematics known as classical analysis. Indeed, harmonic analysis grew out of Fourier analysis, which was developed to help solve various types of partial differential equations. In this project the principal investigator will work on problems that span both of these areas. In particular, he will try to describe the harmonic measure of the exterior of a convex domain and he will study various types of boundary value problems using techniques from the theory of harmonic functions.