Abstract

We study two overdetermined elliptic boundary value problems on exterior domains (the complement of a ball and the complement of a solid cylinder in ${\mathbb{R}}^3$ respectively). The Neumann condition is non-constant and involves the mean curvature of the boundary. We show there exists a family of bifurcation branches of domains which are small deformations of the complement of a ball and of the complement of a solid cylinder, respectively, and which support the solution of the overdetermined boundary value problem.

中文翻译：

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R3外域上具有非恒定Neumann条件的两个超定边值问题的对称破坏分岔

摘要

我们研究了两个外域上的超定椭圆边值问题（球的补和实心圆柱的补 ${\mathbb{电阻}}^3$分别）。诺依曼条件是非常量的，涉及边界的平均曲率。我们表明存在一族域的分岔分支，它们分别是球的补体和实心圆柱体的补体的小变形，并且支持超定边界值问题的解决方案。