*Accepted Paper*

**Inserted:** 20 apr 2018

**Last Updated:** 4 nov 2019

**Journal:** Analysis & PDE

**Year:** 2018

**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 consider the free boundary problem \[ \min{\Big\{\sum_{i=1}^k\int_D\vert\nabla v_i\vert^2+\Lambda\,\mathcal L^d\left(\bigcup_{i=1}^k\{v_i\not=0\}\right)\;:\;v_i=\varphi_i\;on \;\partial D\Big\}}. \] 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 by a regular part, which is relatively open and locally the graph of a $C^\infty$ function, a (one-phase) singular part, of 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 $(d-1)$-dimensional Hausdorff measure. Our arguments are based on the NTA structure of the regular part of the free boundary.

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