On the penalization by the perimeter in shape optimization applied to Dirichlet inverse obstacle problem

¹ Université de Pau et des Pays de l’Adour, E2S UPPA, CNRS, LMAP, UMR 5142, 64000 Pau, France
² Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, UPS, F-31062 Toulouse Cedex 9, France

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 15 May 2025
Accepted: 1 December 2025

Abstract

This paper is devoted to the understanding of regularization process in the shape optimization approach to the so-called Dirichlet inverse obstacle problem for elliptic operators. More precisely, we study two different regularizations of the very classical shape optimization approach consisting in minimizing a mismatched functional. The first one is an implicit regularization when working in the class of inclusion having a uniform ε-cone property, a natural class in shape optimization. As this regularity is not trivial to guarantee numerically, we discuss the regularization by perimeter penalization. We show that this second regularization provides a stability gain in the minimization process.