Regularity of the free boundary for the vectorial Bernoulli problem

Dario Cesare Severo Mazzoleni, Susanna Terracini, Bozhidar Velichkov

Research output: Contribution to journalArticlepeer-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.
Original languageEnglish
Pages (from-to)N/A-N/A
JournalANALYSIS & PDE
Publication statusPublished - 2019

Keywords

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

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