Mathematical Sciences: Nonlinear Partial Differential Equations

PI：Evans DMS-9424342 埃文斯将继续研究非线性微分方程的各种问题，主要是具有几何或物理起源的变分问题。 主要提出的新的努力包括调查新的PDE方法设计最佳的蒙格-康托洛维奇质量传递计划，平面上的高斯曲率流动下演变的表面上的斑点，可能的规律性的“最佳”Lipschitz扩展，和时间演化产生的自然拟凸能量泛函。 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建立方程，不如说是提供关于解的定性和定量信息。 这可能包括关于独特性，平滑性和增长的问题的答案。 此外，分析经常发展出解的近似方法和对这些近似的准确性的估计。

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.