Nontrivial solutions to Serrin's problem in annular domains

Abstract

We construct nontrivial smooth bounded domains $\Omega \subseteq {\mathbb{R}}^n$ of the form ${\Omega }_0\setminus {\overline{\Omega }}_1$, bifurcating from annuli, for which there exists a positive solution to the overdetermined boundary value problem \[ -Δu = 1, \; u>0 \quad \text{in } Ω, \qquad u = 0 ,\; \partial_νu = \text{const} \quad \text{on } \partialΩ_0, \qquad u = \text{const} ,\; \partial_νu = \text{const} \quad \text{on } \partial Ω_1, \] where $\nu$ stands for the inner unit normal to $\partial \Omega$. From results by Reichel and later by Sirakov, it was known that the condition ${\partial }_{\nu }u\le 0$ on $\partial {\Omega }_1$ is sufficient for rigidity to hold, namely, the only domains which admit such a solution are annuli and solutions are radially symmetric. Our construction shows that the condition is also necessary. In addition, the constructed domains are shown to be self-Cheeger.

22 pages, 1 figure (updated). We have slightly modified our original method to yield nontrivial domains, admitting solutions to Serrin's problem that satisfy the same global constant Neumann condition. We have added a discussion of the Cheeger problem and a section in which we prove the constructed domains are self-Cheeger

References in corpus (1)

• On one phase free boundary problem in ${\mathbb{R}}^N$