#### Vol. 13, No. 3, 2020

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Regularity of the free boundary for the vectorial Bernoulli problem

### Dario Mazzoleni, Susanna Terracini and Bozhidar Velichkov

Vol. 13 (2020), No. 3, 741–764
##### 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 $D\subset {ℝ}^{d}$, $\Lambda >0$, and ${\varphi }_{i}\in {H}^{1∕2}\left(\partial D\right)$, we deal with

We prove that, for any optimal vector $U=\left({u}_{1},\dots ,{u}_{k}\right)$, the free boundary $\partial \left(\right{\bigcup }_{i=1}^{k}\left\{{u}_{i}\ne 0\right\}\left)\right\cap D$ 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 ${\mathsc{ℋ}}^{d-1}$ measure. For this purpose we shall exploit the NTA property of the regular part to reduce ourselves to a scalar one-phase Bernoulli problem.

##### Keywords
regularity of free boundaries, NTA domains, optimality conditions, viscosity solutions, branching points
##### Mathematical Subject Classification 2010
Primary: 35R35
Secondary: 35J60, 49K20
##### Milestones
Revised: 6 February 2019
Accepted: 3 April 2019
Published: 15 April 2020
##### Authors
 Dario Mazzoleni Dipartimento di Matematica e Fisica “N. Tartaglia” Università Cattolica Brescia Italy Susanna Terracini Dipartimento di Matematica “Giuseppe Peano” Università di Torino Torino Italy Bozhidar Velichkov Dipartimento di Matematica e Applicazioni “Renato Caccioppoli” Università degli Studi di Napoli Federico II Napoli Italy