Regularity of solutions to nonlinear elliptic PDEs

非线性椭圆偏微分方程解的正则性。本课题的重点是证明非线性椭圆型偏微分方程若干解的性质。方程包括各种半线性和全非线性方程；尤其是蒙日-安培方程和无穷拉普拉斯方程的变体。许多问题都有变分描述。解可以是整个函数、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规律性和对称性，它们的水平集和可能的奇点的问题。本课题研究的问题来自于弹性、几何、统计力学及相关领域中混凝土现象的简单模型。其中一个问题的动机来自最优交通理论。另一个问题涉及自由边界的规律性，它起源于某些随机曲面的极限，并出现在统计力学中。对于所有这些问题，有许多开放的数学问题将被研究。特别的重点将放在发展规则理论的解决这些问题。

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.