## Analysis & PDE

• Anal. PDE
• Volume 10, Number 6 (2017), 1317-1359.

### A class of unstable free boundary problems

#### Abstract

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.

#### Article information

Source
Anal. PDE, Volume 10, Number 6 (2017), 1317-1359.

Dates
Accepted: 9 May 2017
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.apde/1510843525

Digital Object Identifier
doi:10.2140/apde.2017.10.1317

Mathematical Reviews number (MathSciNet)
MR3678490

Zentralblatt MATH identifier
1368.35279

Subjects
Primary: 35R35: Free boundary problems

#### Citation

Dipierro, Serena; Karakhanyan, Aram; Valdinoci, Enrico. A class of unstable free boundary problems. Anal. PDE 10 (2017), no. 6, 1317--1359. doi:10.2140/apde.2017.10.1317. https://projecteuclid.org/euclid.apde/1510843525

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