Regularity of the free boundary for the vectorial Bernoulli problem

Abstract

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 $D\subset {ℝ}^d$, $\mathrm{\Lambda }>0$, and ${\varphi }_i\in H^{1∕2}\left(\partial D\right)$, we deal with

$$min\left\{\sum _{i=1}^k{\int }_D|\nabla v_i{|}^2+\mathrm{\Lambda }\left|\bigcup _{i=1}^k\left\{v_i\ne 0\right\}\right|:v_i={\varphi }_i\text{on }\partial D\right\}.$$

We prove that, for any optimal vector $U=\left(u_1,\dots ,u_k\right)$, the free boundary $\partial \left({\bigcup }_{i=1}^k\left\{u_i\ne 0\right\}\right)\cap D$ is made of 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 $d-d^{\ast }$, for a $d^{\ast }\in \left\{5,6,7\right\}$, and by a set of branching (two-phase) points, which is relatively closed and of finite ${\mathcal{\mathscr{H}}}^{d-1}$ measure. For this purpose we shall exploit the NTA property of the regular part to reduce ourselves to a scalar one-phase Bernoulli problem.