Domain perturbations for elliptic problems with Robin boundary conditions of opposite sign

TL;DR

This paper investigates how the energy of the torsion problem with Robin boundary conditions varies with domain shape and size, revealing that the ball is not always optimal and highlighting the role of Steklov eigenvalues.

Contribution

It introduces a detailed analysis of energy dependence on domain perturbations for Robin boundary problems, using Steklov eigenfunctions and shape derivatives, especially for non-minimizing solutions.

Findings

The ball does not minimize energy for certain Robin boundary conditions.

Energy is maximized for nearly spherical domains with small elasticity constants.

The analysis links domain shape, Steklov eigenvalues, and torsional energy.

Abstract

We consider the energy of the torsion problem with Robin boundary conditions in the case where the solution is not a minimizer. Its dependence on the volume of the domain and the surface area of the boundary is discussed. In contrast to the case of positive elasticity constants, the ball does not provide a minimum. For nearly spherical domains and elasticity constants close to zero the energy is largest for the ball. This result is true for general domains in the plane under an additional condition on the first non-trivial Steklov eigenvalue. For more general elasticity constants the situation is more involved and it is strongly related to the particular domain perturbation. The methods used in this paper are the series representation of the solution in terms of Steklov eigenfunctions, the first and second shape derivatives and an isoperimetric inequality of Payne and Weinberger \cite{PaWe61} for the torsional rigidity.