Regularity of solutions to nonlinear elliptic PDEs

非线性椭圆偏微分方程解的正则性

• 批准号: 0701037
• 负责人: Ovidiu Savin
• 金额: $ 24.53万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701037&HistoricalAwards=false
• 关键词: Regularity; solutions; nonlinear; elliptic; PDEs

Regularity of Solutions to Nonlinear Elliptic PDEs.Abstract of Proposed Research Ovidiu SavinThe focus of this project will be on proving properties of certain solutions of nonlinear elliptic partial differential equations. The equations include various semi-linear and fully nonlinear equations; especially variants of the Monge-Ampere equation and the infinity Laplacian. Many of the problems have a variational description. The solutions may be entire functions, solutions of obstacle problems or free boundary problems. The issues include questions about the regularity and symmetries of solutions, their level sets and possible singularities. The problems to be studied in this project arise from simple models of concrete phenomena in elasticity, geometry and statistical mechanics and related areas. The motivation for one of the problems comes from the theory of optimal transportation. Another problem involves the regularity of free boundaries originating as limits of certain random surfaces and arises in statistical mechanics. For all of these problems there are many open mathematical questions that will be investigated. Particular concentration will be put on developing regularity theories for the solutions of these problems.

非线性椭圆偏微分方程解的正则性。建议研究摘要Ovidiu Savin本项目的重点将是证明非线性椭圆偏微分方程某些解的性质。这些方程包括各种半线性和完全非线性方程;特别是蒙赫-安培方程和无穷拉普拉斯算子的变体。许多问题都有变分描述。这些解可以是整函数、障碍问题的解或自由边界问题的解。这些问题包括关于解的正则性和对称性、它们的水平集和可能的奇点的问题。在这个项目中要研究的问题来自弹性，几何和统计力学及相关领域的具体现象的简单模型。其中一个问题的动机来自于最优运输理论。另一个问题涉及的规则性的自由边界起源于某些随机表面的限制，并出现在统计力学。对于所有这些问题，有许多开放的数学问题，将被调查。特别是集中将放在发展规律性理论，这些问题的解决方案。