Ahlfors-regularity for minimizers of a multiphase optimal design problem

¹ Dipartimento di Matematica, Università degli Studi di Salerno, Via Giovanni Paolo II 132, Fisciano 84084, Italy
² Institut Èlie Cartan, Université de Lorraine, CNRS, IECL, F-54000 Nancy, France
³ Dipartimento di Matematica e Fisica, Università della Campania Luigi Vanvitelli, viale Lincoln 5, Caserta 81100, Italy

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

Received: 18 June 2025
Accepted: 13 May 2026

Abstract

We establish an Ahlfors-regularity result for minimizers of a multiphase optimal design problem. It is a variant of the classical variational problem which involves a finite number of chambers ℰ (i) of prescribed volume that partition a given domain Ω ⊂ ℝⁿ. The cost functional associated with a configuration ({ℰ(i)]_i,u) is made up of the perimeter of the partition interfaces and a Dirichlet energy term, which is discontinuous across the interfaces. We prove that the union of the optimal interfaces is (n – 1)-Ahlfors-regular via a penalization method and decay estimates of the energy.