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 , , and , we deal with
We prove that, for any optimal vector , the free boundary is made of a regular part, which is relatively open and locally the graph of a function, a (one-phase) singular part, of Hausdorff dimension at most , for a , and by a set of branching (two-phase) points, which is relatively closed and of finite measure. For this purpose we shall exploit the NTA property of the regular part to reduce ourselves to a scalar one-phase Bernoulli problem.
"Regularity of the free boundary for the vectorial Bernoulli problem." Anal. PDE 13 (3) 741 - 764, 2020. https://doi.org/10.2140/apde.2020.13.741