### Abstract

The main difference with the existing literature is that the total energy is here a nonlinear superposition of the either local or nonlocal surface tension effect with the elastic energy.

In sharp contrast with the linear case, the problem considered in this paper is unstable; namely a minimizer in a given domain is not necessarily a minimizer in a smaller domain.

We provide an explicit example for this instability. We also give a free boundary condition, which emphasizes the role played by the domain in the geometry of the free boundary. In addition, we provide density estimates for the free boundary and regularity results for the minimal solution.

As far as we know, this is the first case in which a nonlinear function of the perimeter is studied in this type of problem. Also, the results obtained in this nonlinear setting are new even in the case of the local perimeter, and indeed the instability feature is not a consequence of the possible nonlocality of the problem, but it is due to the nonlinear character of the energy functional.

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

Pages (from-to) | 1317-1359 |

Number of pages | 43 |

Journal | Analysis and PDE |

Volume | 10 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2017 |

Externally published | Yes |

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### Cite this

*Analysis and PDE*,

*10*(6), 1317-1359. https://doi.org/10.2140/apde.2017.10.1317

}

*Analysis and PDE*, vol. 10, no. 6, pp. 1317-1359. https://doi.org/10.2140/apde.2017.10.1317

**A class of unstable free boundary problems.** / Dipierro, S.; Karakhanyan, A.; Valdinoci, E.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A class of unstable free boundary problems

AU - Dipierro, S.

AU - Karakhanyan, A.

AU - Valdinoci, E.

PY - 2017

Y1 - 2017

N2 - We consider the free boundary problem arising from an energy functional which is the sum of a Dirichlet energy and a nonlinear function of either the classical or the fractional perimeter.The main difference with the existing literature is that the total energy is here a nonlinear superposition of the either local or nonlocal surface tension effect with the elastic energy.In sharp contrast with the linear case, the problem considered in this paper is unstable; namely a minimizer in a given domain is not necessarily a minimizer in a smaller domain.We provide an explicit example for this instability. We also give a free boundary condition, which emphasizes the role played by the domain in the geometry of the free boundary. In addition, we provide density estimates for the free boundary and regularity results for the minimal solution.As far as we know, this is the first case in which a nonlinear function of the perimeter is studied in this type of problem. Also, the results obtained in this nonlinear setting are new even in the case of the local perimeter, and indeed the instability feature is not a consequence of the possible nonlocality of the problem, but it is due to the nonlinear character of the energy functional.

AB - We consider the free boundary problem arising from an energy functional which is the sum of a Dirichlet energy and a nonlinear function of either the classical or the fractional perimeter.The main difference with the existing literature is that the total energy is here a nonlinear superposition of the either local or nonlocal surface tension effect with the elastic energy.In sharp contrast with the linear case, the problem considered in this paper is unstable; namely a minimizer in a given domain is not necessarily a minimizer in a smaller domain.We provide an explicit example for this instability. We also give a free boundary condition, which emphasizes the role played by the domain in the geometry of the free boundary. In addition, we provide density estimates for the free boundary and regularity results for the minimal solution.As far as we know, this is the first case in which a nonlinear function of the perimeter is studied in this type of problem. Also, the results obtained in this nonlinear setting are new even in the case of the local perimeter, and indeed the instability feature is not a consequence of the possible nonlocality of the problem, but it is due to the nonlinear character of the energy functional.

U2 - 10.2140/apde.2017.10.1317

DO - 10.2140/apde.2017.10.1317

M3 - Article

VL - 10

SP - 1317

EP - 1359

JO - ANALYSIS & PDE

JF - ANALYSIS & PDE

SN - 1948-206X

IS - 6

ER -