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 ϕ[symbol]i ∈ H1/2.(∂D), we deal with min [ ∫D [pipe]∇υi [pipe]2 +Λ [υi ≠ 0 ] [pipe]: υi + ϕ[symbol] i on ∂ D]. We prove that, for any optimal vector U = (u1,..., uk), the free boundary ∂ (∪ki=1 [ui ≠ 0] [n-ary intersection] 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.