Second variation of domain functionals and applications to problems with Robin boundary conditions

TL;DR

This paper computes the first and second variations of functionals related to elliptic problems with Robin boundary conditions, investigates optimal shapes like the ball, and explores eigenvalues of Steklov problems.

Contribution

It introduces new calculations of domain variations for Robin boundary problems and characterizes optimal shapes, including the ball, among nearly circular domains.

Findings

The ball is optimal for certain shape functionals.

Derived conditions for minimality and maximality of domains.

Connected eigenvalue problems with shape optimization.

Abstract

In this paper the first and second domain variation for functionals related to elliptic boundary and eigenvalue problems with Robin boundary conditions is computed. Minimality and maximality properties of the ball among nearly circular domains of given volume are derived. The discussion leads to the investigation of the eigenvalues of a Steklov eigenvalue problem. As a byproduct a general characterization of the optimal shapes is obtained.