Abstract

We examine Serrin’s classical overdetermined problem under a perturbation of the Neumann boundary condition. The solution of the problem for constant Neumann boundary condition exists provided that the underlying set is a ball. The question arises whether for a perturbation of the constant there still are sets admitting solutions to the problem. Furthermore, one may ask whether solutions to the perturbed problem are close to the original solution, i.e. whether the solution for constant Neumann boundary condition is a stable one. We prove the existence and uniqueness of solutions for small perturbations using a modified implicit function theorem. Furthermore, we arrive at linear stability estimates. This is work in progress and joint work with Michiaki Onodera.