Mathematical Sciences: Nonlinear Partial Differential Equations

PI：埃文斯DMS-9424342埃文斯将继续研究来自非线性微分方程的各种问题，主要是有几何或物理根源的变分问题。主要的新工作包括研究新的偏微分方程组方法来设计最优的Monge-Kantorovich传质方案，高斯曲率流下表面上的平斑演化，最优Lipschitz延拓的可能的正则性，以及自然拟凸能量泛函产生的时间演化。偏微分方程式是建立物理世界数学模型的基础。数学分析的作用与其说是创建方程，不如说是提供有关解的定性和定量信息。这可能包括回答有关唯一性、平稳性和成长性的问题。此外，分析经常开发出近似解的方法和对这些近似的精度的估计。

PI: Evans DMS-9424342 Evans will continue his research on a variety of questions from nonlinear differential equations, mostly variational problems having geometric or physical origin. The primary proposed new efforts include investigations of new PDE methods for designing optimal Monge-Kantorovich mass transfer schemes, flat spots on surfaces evolving under Gauss curvature flow, the possible regularity of "optimal" Lipschitz extensions, and time evolutions generated by natural quasiconvex energy functionals. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations.