Mathematical Sciences: The Geometry of Harmonic Measure

在这个项目中，首席研究员将继续他的 在谐波和谐波之间的边界上的问题上的出色工作 分析和偏微分方程。 的主要目标 他的工作是描述外调和测度 一个凸域。 一个关键问题是提供 静电容量的第二种变化， Brunn-Minkowski不等式。 其他主题 的本征函数的节线的行为， 自由边值问题，混合边值问题 Lipschitz域和Airy型算子的L-p估计 调和分析和偏微分方程这两门学科 微分方程在数学的分支中占有重要地位， 这就是所谓的经典分析。 事实上，Harmonic 傅立叶分析是从傅立叶分析发展而来的， 帮助解决各种类型的偏微分方程。 在 这个项目的首席研究员将工作的问题 跨越这两个领域。 特别是，他将尝试描述调和测度 一个凸域的外部，他将研究各种 边值问题的类型使用的技术从理论 调和函数。

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.