无穷Laplace方程的超定边值问题

Overdetermined Boundary Value Problems for the Infinity Laplace Equation

在有界环形区域上,研究一类无穷Laplace方程的超定边值问题,证明方程解的对称性及环形区域的对称性。首先构造与点到边界距离有关的web函数作为方程特解,此特解的存在性等价于Ω为Stadium-like区域,通过对Stadium-like区域的性质分析,证明Ω为一个同心球环。该结论可以推广到Laplace方程与p-Laplace方程。

The aim of this paper is to study a class of overdetermined boundary value problems of ∞-Laplace equations in bounded annular domains, and prove the symmetry of both the solutions and the annular domains. Firstly, we construct a web function which is related with the distance to the boundary as a special solution of ∞-Laplace equations. Then by analyzing the properties of stadi-um-like domains, we prove that Ω is a spherical ring with same center via the fact that the existence of special solutions is equivalent to that Ω is a stadium-like domain. Finally, we show that the conclusion can be extended to Laplace equations and p-Laplace equations.