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

¹ Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine, BP 70239, 54506 Vandœuvre-lès-Nancy Cedex, France.
² Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France.
³ IRMA, Université de Strasbourg, CNRS UMR 7501, 7 rue René Descartes, 67084 Strasbourg, France.

^* Corresponding author: mazari@math.cnrs.fr; idrissmazari@gmail.com

Received: 6 November 2019
Accepted: 22 July 2020

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.