Ahlfors-regularity for minimizers of a multiphase optimal design problem

TL;DR

This paper proves that the interfaces in a multiphase optimal design problem are Alhfors-regular, ensuring geometric regularity, by using penalization and energy decay techniques in a variational setting.

Contribution

It establishes Alhfors-regularity for minimizers in a multiphase partitioning problem, a novel geometric regularity result for such variational models.

Findings

Interfaces are proven to be (n-1)-Alhfors-regular.

The regularity is achieved through penalization and decay estimates.

The result applies to a multiphase problem with prescribed volumes.

Abstract

We establish an Alhfors-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 $\mathcal{E}(i)$ of prescribed volume that partition a given domain $\Omega\subset\mathbb{R}^n$. The cost functional associated with a configuration $\left(\{\mathcal{E}(i)\}_i,u\right)$ 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 $(n-1)$-Alhfors-regular via a penalization method and decay estimates of the energy.