A nonlinear free boundary problem with a self-driven Bernoulli condition

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Abstract

We study a Bernoulli type free boundary problem with two phases
where is a given boundary datum. Here, and are weighted volumes of and , respectively, and Φ is a nonnegative function of two real variables.
We show that, for this problem, the Bernoulli constant, which determines the gradient jump condition across the free boundary, is of global type and it is indeed determined by the weighted volumes of the phases.
In particular, the Bernoulli condition that we obtain can be seen as a pressure prescription in terms of the volume of the two phases of the minimizer itself (and therefore it depends on the minimizer itself and not only on the structural constants of the problem).
Another property of this type of problems is that the minimizer in Ω is not necessarily a minimizer in a smaller subdomain, due to the nonlinear structure of the problem.
Due to these features, this problem is highly unstable as opposed to the classical case studied by Alt, Caffarelli and Friedman. It also interpolates the classical case, in the sense that the blow-up limits of u are minimizers of the Alt–Caffarelli–Friedman functional. Namely, the energy of the problem somehow linearizes in the blow-up limit.
As a special case, we can deal with the energy levels generated by the volume term , which interpolates the Athanasopoulos–Caffarelli–Kenig–Salsa energy, thanks to the isoperimetric inequality.
In particular, we develop a detailed optimal regularity theory for the minimizers and for their free boundaries.
Original languageEnglish
Pages (from-to)3549-3615
Number of pages67
JournalJournal of Functional Analysis
Volume273
Issue number11
DOIs
Publication statusPublished - 2017
Externally publishedYes

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Free Boundary Problem
Minimizer
Bernoulli
Free Boundary
Blow-up
Interpolate
Regularity Theory
Jump Conditions
Isoperimetric Inequality
Energy
Energy Levels
Unstable
Non-negative
Gradient
Term

Cite this

@article{8c2a0ea93b954bf6b224ceb04bfdff1c,
title = "A nonlinear free boundary problem with a self-driven Bernoulli condition",
abstract = "We study a Bernoulli type free boundary problem with two phaseswhere is a given boundary datum. Here, and are weighted volumes of and , respectively, and Φ is a nonnegative function of two real variables.We show that, for this problem, the Bernoulli constant, which determines the gradient jump condition across the free boundary, is of global type and it is indeed determined by the weighted volumes of the phases.In particular, the Bernoulli condition that we obtain can be seen as a pressure prescription in terms of the volume of the two phases of the minimizer itself (and therefore it depends on the minimizer itself and not only on the structural constants of the problem).Another property of this type of problems is that the minimizer in Ω is not necessarily a minimizer in a smaller subdomain, due to the nonlinear structure of the problem.Due to these features, this problem is highly unstable as opposed to the classical case studied by Alt, Caffarelli and Friedman. It also interpolates the classical case, in the sense that the blow-up limits of u are minimizers of the Alt–Caffarelli–Friedman functional. Namely, the energy of the problem somehow linearizes in the blow-up limit.As a special case, we can deal with the energy levels generated by the volume term , which interpolates the Athanasopoulos–Caffarelli–Kenig–Salsa energy, thanks to the isoperimetric inequality.In particular, we develop a detailed optimal regularity theory for the minimizers and for their free boundaries.",
author = "Serena Dipierro and Aram Karakhanyan and Enrico Valdinoci",
year = "2017",
doi = "10.1016/j.jfa.2017.07.014",
language = "English",
volume = "273",
pages = "3549--3615",
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A nonlinear free boundary problem with a self-driven Bernoulli condition. / Dipierro, Serena ; Karakhanyan, Aram; Valdinoci, Enrico .

In: Journal of Functional Analysis, Vol. 273, No. 11, 2017, p. 3549-3615.

Research output: Contribution to journalArticle

TY - JOUR

T1 - A nonlinear free boundary problem with a self-driven Bernoulli condition

AU - Dipierro, Serena

AU - Karakhanyan, Aram

AU - Valdinoci, Enrico

PY - 2017

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N2 - We study a Bernoulli type free boundary problem with two phaseswhere is a given boundary datum. Here, and are weighted volumes of and , respectively, and Φ is a nonnegative function of two real variables.We show that, for this problem, the Bernoulli constant, which determines the gradient jump condition across the free boundary, is of global type and it is indeed determined by the weighted volumes of the phases.In particular, the Bernoulli condition that we obtain can be seen as a pressure prescription in terms of the volume of the two phases of the minimizer itself (and therefore it depends on the minimizer itself and not only on the structural constants of the problem).Another property of this type of problems is that the minimizer in Ω is not necessarily a minimizer in a smaller subdomain, due to the nonlinear structure of the problem.Due to these features, this problem is highly unstable as opposed to the classical case studied by Alt, Caffarelli and Friedman. It also interpolates the classical case, in the sense that the blow-up limits of u are minimizers of the Alt–Caffarelli–Friedman functional. Namely, the energy of the problem somehow linearizes in the blow-up limit.As a special case, we can deal with the energy levels generated by the volume term , which interpolates the Athanasopoulos–Caffarelli–Kenig–Salsa energy, thanks to the isoperimetric inequality.In particular, we develop a detailed optimal regularity theory for the minimizers and for their free boundaries.

AB - We study a Bernoulli type free boundary problem with two phaseswhere is a given boundary datum. Here, and are weighted volumes of and , respectively, and Φ is a nonnegative function of two real variables.We show that, for this problem, the Bernoulli constant, which determines the gradient jump condition across the free boundary, is of global type and it is indeed determined by the weighted volumes of the phases.In particular, the Bernoulli condition that we obtain can be seen as a pressure prescription in terms of the volume of the two phases of the minimizer itself (and therefore it depends on the minimizer itself and not only on the structural constants of the problem).Another property of this type of problems is that the minimizer in Ω is not necessarily a minimizer in a smaller subdomain, due to the nonlinear structure of the problem.Due to these features, this problem is highly unstable as opposed to the classical case studied by Alt, Caffarelli and Friedman. It also interpolates the classical case, in the sense that the blow-up limits of u are minimizers of the Alt–Caffarelli–Friedman functional. Namely, the energy of the problem somehow linearizes in the blow-up limit.As a special case, we can deal with the energy levels generated by the volume term , which interpolates the Athanasopoulos–Caffarelli–Kenig–Salsa energy, thanks to the isoperimetric inequality.In particular, we develop a detailed optimal regularity theory for the minimizers and for their free boundaries.

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