Optimal concavity of the torsion function

Abstract

In this short note we consider an unconventional overdetermined problem for the torsion function: let $n\ge 2$ and $\Omega$ be a bounded open set in ${\mathbb{R}}^n$ whose torsion function $u$ (i.e. the solution to $\Delta u=-1$ in $\Omega$, vanishing on $\partial \Omega$) satisfies the following property: $\sqrt{M-u(x)}$ is convex, where $M=\max \{u(x):x\in \overline{\Omega }\}$. Then $\Omega$ is an ellipsoid.

References in corpus (1)

• Symmetry and linear stability in Serrin's overdetermined problem via the stability of the parallel surface problem