Regularity of the free boundary for the vectorial Bernoulli problem

Mazzoleni, Dario; Terracini, Susanna; Velichkov, Bozhidar

doi:10.2140/apde.2020.13.741

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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

DOI: 10.2140/apde.2020.13.741

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 $D \subset ℝ^{d}$ , $Λ > 0$ , and $ϕ_{i} \in H^{1 ∕ 2} (\partial D)$ , we deal with

$min \{\sum_{i = 1}^{k} \int_{D} | \nabla v_{i} |^{2} + Λ | ⋃_{i = 1}^{k} {v_{i} \neq 0} | : v_{i} = ϕ_{i} on \partial D \} .$

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

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Keywords

regularity of free boundaries, NTA domains, optimality conditions, viscosity solutions, branching points

Mathematical Subject Classification 2010

Primary: 35R35

Secondary: 35J60, 49K20

Milestones

Received: 27 April 2018

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

Vol. 13, No. 3, 2020