Mathematical Sciences: Partial Differential Equations and Geometric Analysis

本课题主要研究椭圆型变分问题、抛物型变分不等式、等度可测凸对称、涉及有界变差映射的变分积分以及非线性位势理论。作者最近在椭圆型变分问题领域的大部分工作都集中在涉及由不规则和光滑障碍引起的变分不等式解的正则性的问题，并且涉及退化的椭圆算子。已经发展出一种方法，在光滑障碍的情况下可以得到更多关于自由边界的信息，在椭圆系统的情况下它也可能导致另一种处理方法。这些问题和技巧导致了抛物型变分不等式中的抛物型对应，在抛物型变分不等式中，已知的较少，困难较大。椭圆型变分问题和抛物型变分不等式中的许多问题和技巧与非线性位势理论中的问题和技巧有相当大的重叠。作者发现等度凸对称化和涉及有界变差映射的变分积分问题特别引人注目，因为它们与几何分析相互作用，并且有望在应用领域的方向上走得更远。

This project focuses on elliptic variational problems, parabolic variational inequalities, equimeasurable convex symmetrization, variational integrals involving mappings of bounded variation, and nonlinear potential theory. Much of the proposer's recent work in the area of elliptic variational problems has concentrated on problems involving the regularity of solutions to variational inequalities that arise from both irregular and smooth obstacles and that involve degenerate elliptic operators. An approach has been developed that may yield more information about the free boundary in the case of smooth obstacles and it may lead also to another treatment in the case of elliptic systems. These questions and techniques lead to their parabolic counterparts in parabolic variational inequalities where less is known and the difficulties are greater. Many of the questions and techniques in elliptic variational problems, and parabolic variational inequalities have considerable overlap with those in nonlinear potential theory. The proposer finds the problems of equimeasurable convex symmetrization and variational integrals involving mappings of bounded variation especially compelling because of their interaction with geometric analysis and the promise of going further in the direction of applied areas.