Regularity of the free boundary for the vectorial Bernoulli problem

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⊂ℝd, Λ>0, and ϕi∈H1∕2(∂D), we deal withmin{∑i=1k ∫ D|∇vi|2+Λ|⋃i=1k{vi≠0}|:vi=ϕi  on ∂D}.We prove that, for any optimal vector U=(u1,…,uk), the free boundary ∂(⋃i=1k{ui≠0})∩D is made of a regular part, which is relatively open and locally the graph of a C∞ function, a (one-phase) singular part, of Hausdorff dimension at most d−d∗, for a d∗∈{5,6,7}, and by a set of branching (two-phase) points, which is relatively closed and of finite ℋd−1 measure. For this purpose we shall exploit the NTA property of the regular part to reduce ourselves to a scalar one-phase Bernoulli problem.