TY - JOUR

T1 - Regularity of the free boundary for the vectorial Bernoulli problem

AU - Mazzoleni, Dario Cesare Severo

AU - Terracini, Susanna

AU - Velichkov, Bozhidar

PY - 2019

Y1 - 2019

N2 - In this paper 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 $\varphi_i\in H^{\sfrac12}(\partial D)$, we deal with
\[
\min{\left\{\sum_{i=1}^k\int_D|\nabla v_i|^2+\Lambda\Big|\bigcup_{i=1}^k\{v_i\not=0\}\Big|\;:\;v_i=\varphi_i\;\mbox{on }\partial D\right\}}.
\]
We prove that, for any optimal vector $U=(u_1,\dots, u_k)$, the free boundary $\partial (\cup_{i=1}^k\{u_i\not=0\})\cap D$ 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 $d-d^*$, for a $d^*\in\{5,6,7\}$, and by a set of branching (two-phase) points, which is relatively closed and of finite $\HH^{d-1}$ measure. For this purpose we shall exploit the NTA property of the regular part to reduce ourselves to a scalar one-phase Bernoulli problem.

AB - In this paper 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 $\varphi_i\in H^{\sfrac12}(\partial D)$, we deal with
\[
\min{\left\{\sum_{i=1}^k\int_D|\nabla v_i|^2+\Lambda\Big|\bigcup_{i=1}^k\{v_i\not=0\}\Big|\;:\;v_i=\varphi_i\;\mbox{on }\partial D\right\}}.
\]
We prove that, for any optimal vector $U=(u_1,\dots, u_k)$, the free boundary $\partial (\cup_{i=1}^k\{u_i\not=0\})\cap D$ 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 $d-d^*$, for a $d^*\in\{5,6,7\}$, and by a set of branching (two-phase) points, which is relatively closed and of finite $\HH^{d-1}$ measure. For this purpose we shall exploit the NTA property of the regular part to reduce ourselves to a scalar one-phase Bernoulli problem.

KW - Branching Points

KW - NTA domains

KW - Optimality conditions

KW - Regularity of free boundaries Branching Points

KW - Branching Points

KW - NTA domains

KW - Optimality conditions

KW - Regularity of free boundaries Branching Points

UR - http://hdl.handle.net/10807/133201

M3 - Article

SN - 2157-5045

SP - N/A-N/A

JO - ANALYSIS & PDE

JF - ANALYSIS & PDE

ER -