Regularity Results for an Optimal Design Problem with lower order terms

TL;DR

This paper investigates the regularity of interfaces in optimal energy configurations with perimeter penalties, providing bounds on the size of the singular set and extending known regularity results for the function u.

Contribution

It offers new estimates on the Hausdorff dimension of the singular set of the boundary in optimal design problems with lower order terms.

Findings

Hausdorff dimension of the singular set is less than n-1

Hölder continuity of u is established for minimal configurations

Provides bounds on the size of the singular boundary set

Abstract

We study the regularity of the interface for optimal energy configurations of functionals involving bulk energies with an additional perimeter penalization of the interface. It is allowed the dependence on $(x,u)$ for the bulk energy. For a minimal configuration $(E,u)$, the H\"{o}lder continuity of $u$ is well known. We give an estimate for the singular set of the boundary $\partial E$. Namely we show that the Hausdorff dimension of the singular set is strictly smaller than $n-1$.