Regularity of the free boundary for the vectorial Bernoulli problem

TL;DR

This paper proves the regularity and structure of the free boundary in a vector-valued Bernoulli problem, showing it consists of smooth, singular, and branching parts with specific geometric properties.

Contribution

It establishes the regularity and stratification of the free boundary for the vectorial Bernoulli problem without sign restrictions on boundary data.

Findings

The regular part of the free boundary is a smooth $C^ abla$ graph.

The singular part has Hausdorff dimension at most $d - d^*$, with $d^* \\in\\\{5,6,7\\\}$.

Branching points form a set of finite $\\mathcal{H}^{d-1}$ measure.

Abstract

In this paper we study the regularity of the free boundary for a vector-valued Bernoulli problem, with no sign assumptions on the boundary data. More precisely, given an open, smooth set of finite measure $D\subset \mathbb{R}^d$, $\Lambda>0$ and $\varphi_i\in H^{1/2}(\partial D)$, we deal with \[ \min{\left\{\sum_{i=1}^k\int_D|\nabla v_i|^2+\Lambda\Big|\bigcup_{i=1}^k\{v_i\not=0\}\Big|\;:\;v_i=\varphi_i\;\mbox{on }\partial D\right\}}. \] We prove that, for any optimal vector $U=(u_1,\dots, u_k)$, the free boundary $\partial (\cup_{i=1}^k\{u_i\not=0\})\cap D$ is made of a regular part, which is relatively open and locally the graph of a $C^\infty$ function, a singular part, which is relatively closed and has Hausdorff dimension at most $d-d^*$, for a $d^*\in\{5,6,7\}$ and by a set of branching (two-phase) points, which is relatively closed and of finite $\mathcal{H}^{d-1}$ measure. Our arguments are based on the NTA structure of the regular part of the free boundary.