Analytical and Geometrical Problems in Non Linear Partial Differential Equations

拟研究摘要Luis caffarelli本研究旨在研究用非线性偏微分方程建模的一些科学现象的数学分析。具体主题包括涉及反常（特别是积分）扩散过程的非线性问题的解的性质，例如相变，流体动力学和最优控制。也可用于全非线性方程的非线性随机均匀化或随机穿孔域中的约束问题。一般信号传播规律的天线设计，以及其他闵可夫斯基型问题正在研究的现象类型的具体例子包括建模边界控制的方程分析（优化表面的绝缘形状，或半透膜的行为），表面火焰传播和过程高度不连续时的期权定价（Levi过程）。还有一组小的、随机分布的热源的有效加热，火焰在随机遮挡的介质上的传播或水滴在粗糙表面上的滑动，以及周期性介质中最佳反射表面的设计。

Analytical and Geometrical Problems in Nonlinear Partial Differential EquationsAbstract of Proposed Research Luis CaffarelliThis research is to study the mathematical analysis of a number of scientific phenomena that are modeled by nonlinear partial differential equations. Specific topics include the properties of solutions of nonlinear problems involving anomalous (in particular integral) diffusion processes, such as phase transitions, fluid dynamics and optimal control. Also nonlinear random homogenization of fully non linear equations or constrained problems in randomly perforated domains. Antenna design for general signal propagation laws, and other Minkowski type problemsSpecific examples of the type of phenomena being studied include the analysis of equations modeling boundary control (optimizing insulation shape across a surface, or the behavior of semi permeable membranes), surface flame propagation and the pricing of options when processes are highly discontinuous ( Levi processes). Also the effective heating of a family of small, randomly distributed, heating sources, the propagation of a flame on a medium with random occlusions or the sliding of a drop on a rough surface and the design of an optimal reflecting surface in a periodic medium.