# Nontrivial solutions to Serrin's problem in annular domains

### Luciano Sciaraffia

Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago 7820436, Chile### Nikola Kamburov

Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago 7820436, Chile

## Abstract

We construct nontrivial bounded, real analytic domains $Ω⊆R_{n}$ of the form \( \mathrm{\Omega }_{0} \setminus \mathrm{\Omega }\limits^{‾}_{1} \), bifurcating from annuli, which admit a positive solution to the overdetermined boundary value problem

where *ν* stands for the inner unit normal to ∂Ω. From results by Reichel [1] and later by Sirakov [2], it was known that the condition $∂_{ν}u≤0$ on $∂Ω_{1}$ 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, we show that the constructed domains are self-Cheeger.

## Cite this article

Luciano Sciaraffia, Nikola Kamburov, Nontrivial solutions to Serrin's problem in annular domains. Ann. Inst. H. Poincaré Anal. Non Linéaire 38 (2021), no. 1, pp. 1–22

DOI 10.1016/J.ANIHPC.2020.05.001