### Abstract

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 language | English |
---|---|

Pages (from-to) | 3549-3615 |

Number of pages | 67 |

Journal | Journal of Functional Analysis |

Volume | 273 |

Issue number | 11 |

DOIs | |

Publication status | Published - 2017 |

Externally published | Yes |

### Fingerprint

### Cite this

*Journal of Functional Analysis*,

*273*(11), 3549-3615. https://doi.org/10.1016/j.jfa.2017.07.014

}

*Journal of Functional Analysis*, vol. 273, no. 11, pp. 3549-3615. https://doi.org/10.1016/j.jfa.2017.07.014

**A nonlinear free boundary problem with a self-driven Bernoulli condition.** / Dipierro, Serena ; Karakhanyan, Aram; Valdinoci, Enrico .

Research output: Contribution to journal › Article

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

Y1 - 2017

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.

U2 - 10.1016/j.jfa.2017.07.014

DO - 10.1016/j.jfa.2017.07.014

M3 - Article

VL - 273

SP - 3549

EP - 3615

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 11

ER -