2020 Regularity of the free boundary for the vectorial Bernoulli problem
Dario Mazzoleni, Susanna Terracini, Bozhidar Velichkov
Anal. PDE 13(3): 741-764 (2020). DOI: 10.2140/apde.2020.13.741


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 Dd, Λ>0, and ϕiH12(D), we deal with

min { i = 1 k D | v i | 2 + Λ | i = 1 k { v i 0 } | : v i = ϕ i 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 dd, 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.


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Dario Mazzoleni. Susanna Terracini. Bozhidar Velichkov. "Regularity of the free boundary for the vectorial Bernoulli problem." Anal. PDE 13 (3) 741 - 764, 2020. https://doi.org/10.2140/apde.2020.13.741


Received: 27 April 2018; Revised: 6 February 2019; Accepted: 3 April 2019; Published: 2020
First available in Project Euclid: 25 June 2020

zbMATH: 07190790
MathSciNet: MR4085121
Digital Object Identifier: 10.2140/apde.2020.13.741

Primary: 35R35
Secondary: 35J60 , 49K20

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

Rights: Copyright © 2020 Mathematical Sciences Publishers


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Vol.13 • No. 3 • 2020
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