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Regularity of the free boundary for the vectorial Bernoulli problem

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 $D\subset \mathbb{R}^d$, $Λ>0$ and $φ_i\in H^{1/2}(\partial D)$, we deal with \[ \min{\left\{\sum_{i=1}^k\int_D|\nabla v_i|^2+Λ\Big|\bigcup_{i=1}^k\{v_i\not=0\}\Big|\;:\;v_i=φ_i\;\mbox{on }\partial D\right\}}. \] 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.