Shape optimization of a Dirichlet type energy for semilinear elliptic partial differential equations

TL;DR

This paper investigates a nonlinear shape optimization problem involving a semilinear elliptic PDE, proving existence of optimal shapes and analyzing the stability of the ball as a candidate solution under symmetric conditions.

Contribution

It introduces a relaxed formulation for a nonlinear Dirichlet energy minimization problem and establishes existence and stability results for optimal shapes.

Findings

Existence of optimal shapes under certain conditions.

Stability analysis of the ball as an optimal shape.

Extension of shape optimization techniques to nonlinear PDEs.

Abstract

Minimizing the so-called "Dirichlet energy" with respect to the domain under a volume constraint is a standard problem in shape optimization which is now well understood. This article is devoted to a prototypal non-linear version of the problem, where one aims at minimizing a Dirichlet-type energy involving the solution to a semilinear elliptic PDE with respect to the domain, under a volume constraint. One of the main differences with the standard version of this problem rests upon the fact that the criterion to minimize does not write as the minimum of an energy, and thus most of the usual tools to analyze this problem cannot be used. By using a relaxed version of this problem, we first prove the existence of optimal shapes under several assumptions on the problem parameters. We then analyze the stability of the ball, expected to be a good candidate for solving the shape optimization problem, when the coefficients of the involved PDE are radially symmetric.