A Rigidity Result for the Robin Torsion Problem

TL;DR

This paper proves that for the Robin torsion problem in two dimensions, the only domain where the Talenti-type comparison equality holds is a disk with a radial torsion function, establishing a rigidity result.

Contribution

It establishes a new rigidity theorem showing that equality in a Talenti-type comparison for Robin torsion problems characterizes disks and radial solutions.

Findings

Equality in the Talenti-type comparison occurs only for disks.

The torsion function must be radial in the equality case.

The result extends previous understanding of symmetry in Robin boundary problems.

Abstract

Let $\Omega \subset \mathbb{R}^2$ be an open, bounded and Lipschitz set. We consider the torsion problem for the Laplace operator associated to $\Omega$ with Robin boundary conditions. In this setting, we study the equality case in the Talenti-type comparison, proved in arXiv:1909.11950. We prove that the equality is achieved only if $\Omega$ is a disk and the torsion function $u$ is radial.