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2017 A class of unstable free boundary problems
Serena Dipierro, Aram Karakhanyan, Enrico Valdinoci
Anal. PDE 10(6): 1317-1359 (2017). DOI: 10.2140/apde.2017.10.1317


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.


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Serena Dipierro. Aram Karakhanyan. Enrico Valdinoci. "A class of unstable free boundary problems." Anal. PDE 10 (6) 1317 - 1359, 2017.


Received: 11 December 2015; Accepted: 9 May 2017; Published: 2017
First available in Project Euclid: 16 November 2017

zbMATH: 1368.35279
MathSciNet: MR3678490
Digital Object Identifier: 10.2140/apde.2017.10.1317

Primary: 35R35

Keywords: free boundary problems , nonlinear phenomena , regularity

Rights: Copyright © 2017 Mathematical Sciences Publishers


Vol.10 • No. 6 • 2017
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