Mathematical Sciences: Nonlinear Partial Differential Equations

埃文斯将继续他对非线性微分方程的各种问题的研究，主要是具有几何或物理根源的变分问题。主要提出的新努力包括研究设计最优Monge-Kantorovich传质方案的新PDE方法，在高斯曲率流下表面上的扁平点，“最优”Lipschitz扩展的可能规律，以及自然准凸能量泛函产生的时间演化。偏微分方程是物理世界数学建模的基础。数学分析的作用与其说是创建方程，不如说是提供关于解的定性和定量信息。这可能包括回答关于独特性、平滑性和增长性的问题。此外，分析常常发展出解的近似方法和对这些近似精度的估计。

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.