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

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{vi0}| : 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.

regularity of free boundaries, NTA domains, optimality conditions, viscosity solutions, branching points
Mathematical Subject Classification 2010
Primary: 35R35
Secondary: 35J60, 49K20
Received: 27 April 2018
Revised: 6 February 2019
Accepted: 3 April 2019
Published: 15 April 2020
Dario Mazzoleni
Dipartimento di Matematica e Fisica “N. Tartaglia”
Università Cattolica
Susanna Terracini
Dipartimento di Matematica “Giuseppe Peano”
Università di Torino
Bozhidar Velichkov
Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”
Università degli Studi di Napoli Federico II