Symmetry breaking bifurcations for two overdetermined boundary value problems with non-constant Neumann condition on exterior domains in R3

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.

Keywords:

• overdetermined problem
• Neumann boundary condition
• bifurcation

2010 MATHEMATICS SUBJECTION CLASSIFICATION.:

• 35N10; 35C20; 37G10; 58J32; 58J55