Regularity of the free boundary for the vectorial Bernoulli problem

Dario Cesare Severo Mazzoleni, Susanna Terracini, Bozhidar Velichkov

Risultato della ricerca: Contributo in rivistaArticolo in rivistapeer review

Abstract

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.
Lingua originale English N/A-N/A ANALYSIS & PDE Pubblicato - 2019

Keywords

• Branching Points
• NTA domains
• Optimality conditions
• Regularity of free boundaries Branching Points

Fingerprint

Entra nei temi di ricerca di 'Regularity of the free boundary for the vectorial Bernoulli problem'. Insieme formano una fingerprint unica.