Some properties of the torsion function with Robin boundary conditions

TL;DR

This paper investigates the properties of the torsion function with Robin boundary conditions, focusing on shape derivatives of its norms and identifying critical shapes under volume-preserving deformations.

Contribution

It derives the shape derivatives of the $L^{ abla}$ and $L^p$ norms of the torsion function and proves that balls are critical shapes for these functionals under volume constraints.

Findings

Balls are critical shapes for the torsion function norms under volume preservation.

Shape derivatives of the $L^{ abla}$ and $L^p$ norms are explicitly computed.

The study extends understanding of shape optimization for PDE-related functionals.

Abstract

In this paper we study some properties of the torsion function with Robin boundary conditions. Here we write the shape derivative of the $L^{\infty}$ and $L^p$ norms, for $p\ge 1$, of the torsion function, seen as a functional on a bounded simply connected open set $\Omega \subset \mathbb{R}^n$, and prove that the balls are critical shapes for these functionals, when the volume of $\Omega$ is preserved.