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
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 d , Λ > 0, and ϕi H12(D), we deal with

min{ i=1kD|vi|2 + Λ| i=1k{v i0}| : vi = ϕi  on D}.

We prove that, for any optimal vector U = (u1,,uk), the free boundary ( i=1k{ui0}) D is made of a regular part, which is relatively open and locally the graph of a C function, a (one-phase) singular part, of Hausdorff dimension at most d d , for a d{5,6,7}, and by a set of branching (two-phase) points, which is relatively closed and of finite d1 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
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