Optimal design problems with fractional diffusions

TL;DR

This paper investigates optimization problems involving fractional diffusion operators, establishing existence, regularity of solutions, and smoothness of free boundaries using penalization techniques.

Contribution

It introduces new existence proofs and regularity results for solutions and free boundaries in fractional diffusion optimization problems.

Findings

Solutions exist under volume constraints.

Solutions are locally $C^{0,eta}$ regular.

Free boundaries are $C^{1,eta}$ surfaces.

Abstract

In this article we study optimization problems ruled by $\alpha$-fractional diffusion operators with volume constraints. By means of penalization techniques we prove existence of solutions. We also show that every solution is locally of class $C^{0,\alpha}$ (optimal regularity), and that the free boundary is a $C^{1,\gamma}$ surface, up to a $\mathcal{H}^{n-1}$-negligible set.