# Optimal concavity of the torsion function

## TL;DR

This paper proves that if the torsion function of a bounded domain in Euclidean space has a convex square root of its maximum minus the function, then the domain must be an ellipsoid, revealing a geometric characterization based on concavity properties.

## Contribution

It establishes a new geometric characterization of ellipsoids via the convexity of the square root of the torsion function's difference from its maximum.

## Key findings

- The torsion function's specific concavity condition characterizes ellipsoids.
- The result applies to domains in any dimension n ≥ 2.
- This provides a novel link between PDE solutions and geometric shape.

## Abstract

In this short note we consider an unconventional overdetermined problem for the torsion function: let $n\geq 2$ 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.