On a Multiphase Vectorial Bernoulli Free Boundary Problem

TL;DR

This paper investigates the regularity of solutions to a multiphase vectorial Bernoulli free boundary problem, proving existence, local Lipschitz continuity, and smoothness of the free boundary near certain points.

Contribution

It establishes the existence and regularity of minimizers and characterizes the structure of their free boundaries in a multiphase setting.

Findings

Minimizers exist and are locally Lipschitz continuous.

Free boundaries do not have points where three or more phases meet.

Near two-phase and branching points, the free boundary is a $C^{1, heta}$ graph.

Abstract

We study the regularity of minimizers of a multiphase vectorial Bernoulli free boundary problem. This problem consists in a minimization problem for the Bernoulli functional over families of Sobolev functions with disjoint supports and non trivial grouping. We prove that minimizers exist, are locally Lipschitz continuous, and that their free boundaries do not contain points where three or more phases meet. Our main regularity result establishes that the free boundary is locally a $C^{1,\eta}$ graph near two-phase and branching points for some $\eta >0$.