Geometric PDE's and Monge-Kantorovich Theory in Problems of Optics and Differential Geometry

DMS-0405622 P.I.：弗拉基米尔奥利克，埃默里大学题目：几何偏微分方程和蒙格-康托洛维奇理论在光学和微分几何问题摘要这项工作的目标是发展的几何方法求解非线性偏微分方程（PDE）的问题涉及控制雅可比行列式。微分几何、光学以及其他数学和工程领域的许多问题都在这门课中。最近发现的深刻联系，这些方程和蒙格-康托洛维奇最佳传质理论在欧几里德空间和流形上也将被研究。(1)发展几何和分析技术，解决需要确定反射和折射界面的问题，并以规定的方式转换强度分布。(2)研究几何问题，包括具有规定曲率函数的超曲面和几何不等式，重点是变分方法，特别是与Monge-Kantorovich理论有关的变分方法;这些变分方法在凸性问题中的应用，特别是闵可夫斯基问题及其各种推广，将作为本课程的一部分进行研究。(3)发展几何激励的，可证明收敛的和有效的多尺度数值方法，用于解决光学反射器/折射器问题中出现的非线性二阶偏微分方程，以及涉及曲率函数和控制体积映射的几何问题。在科学和工程中，用描述物理现象的映射的雅可比矩阵作为约束条件来表达能量守恒定律的非线性偏微分方程是非常常见的。例如，在光学中，当需要确定具有预定折射和/或反射特性的界面时，这些方程自然产生;在天体物理学中，当必须从间接的和有限的测量数据确定太阳系中目标的形状时，这些方程必须求解;在基于大气运动的准地转和半地转近似的天气预报模式中，这些方程描述能量守恒定律;在计算机科学中，在与辐射度估计有关的问题中也出现了同样类型的方程。2由于这些方程的高度非线性结构，它们的理论分析和数值解通常是非常困难的。幸运的是，所有这些问题共同的几何内容提供了重要的见解，从而为他们的调查和数值解提供了有效的方法。发展这样的方法是本研究的主要目标。

DMS-0405622P.I.: Vladimir Oliker, Emory UniversityTitle: Geometric PDE's and Monge-Kantorovich Theory in Problems ofOptics and Differential GeometryABSTRACTThe goal of this work is the development of geometricmethods for solving nonlinear partial differential equations (PDE's)arising in problems involving maps with controlled Jacobiandeterminant. Many problems in differential geometry, optics,and other areas of mathematics and engineering are inthis class. Recently discovered deep connections between such equationsand Monge-Kantorovich optimal mass transfer theory in Euclidean spaceand on manifolds will also be studied.Among the topics that will be considered are the following.(1) Development of geometrical and analytic techniques for solving problems requiring determination of reflecting and refracting interfaces with capabilities to transform intensitydistributions in a prescribed manner. (2) Investigation of geometric problems involving hypersurfaces with prescribed curvature functions and geometric inequalities with emphasis on variational methods, especially,those connected with Monge-Kantorovich theory; applications of these variational methods to problems in convexity, in particular, to the Minkowski problem andits various generalizations, will be studied as a part of this program.(3) Development of geometrically motivated, provably convergent and efficient multi-scale numerical methods for solving nonlinear second order PDE's arising in reflector/refractor problems of optics and in geometric problems involving curvature functions and maps with controlled volume. Nonlinear partial differentials equations expressing energy conservation laws as a constraint on the Jacobian of a map describing a physicalphenomenon are very common in science and engineering. For example,in optics such equations arise naturally in problemsrequiring determination of interfaces with prescribedrefractive and/or reflective properties; in astrophysics these equationshave to be solved when the shape of targets in the solarsystem must be determined from indirect and limited set of measurements;in weather prediction models based on quasi- and semi-geostrophicapproximations of atmospheric motion such equations describeenergy conservation laws; in computer science the same type ofequations arise in problems connected with radiosity estimates.Typically, the theoretical analysis and numerical solution ofthese equations is very difficult because of their highly nonlinearstructure. Fortunately, the geometric content common to all these problemsprovides important insights leading to effective methods for theirinvestigation and numerical solution. Development of such methodsis the main goal of this research.