Mathematical Sciences: Partial Differential Equations and Geometric Analysis

这个项目的重点是椭圆变分问题， 抛物型变分不等式，等距凸 对称化，有界映射的变分积分 变分和非线性势理论。 提案人的大部分 最近在椭圆变分问题领域的工作， 专注于涉及解决方案的规律性问题， 变分不等式产生于不规则和光滑 障碍，并涉及退化椭圆算子。 一个 已经开发了一种方法，可以产生更多关于 在光滑障碍物的情况下的自由边界，它可能导致 也涉及椭圆系统的另一种处理方法。 这些 问题和技术导致他们的抛物线对应， 抛物型变分不等式，其中已知较少， 困难更大。 书中的许多问题和技巧 椭圆变分问题和抛物变分问题 不等式与非线性不等式有相当大的重叠， 潜力理论 提案人认为， 等测凸对称化与变分积分 涉及有界变差的映射的，特别引人注目的 由于它们与几何分析的相互作用， 在应用领域进一步发展的前景。

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.