Vol. 10, No. 6, 2017

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
A class of unstable free boundary problems

Serena Dipierro, Aram Karakhanyan and Enrico Valdinoci

Vol. 10 (2017), No. 6, 1317–1359
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.

Keywords
free boundary problems, regularity, nonlinear phenomena
Mathematical Subject Classification 2010
Primary: 35R35
Milestones
Received: 11 December 2015
Accepted: 9 May 2017
Published: 14 July 2017
Authors
Serena Dipierro
School of Mathematics and Statistics
University of Melbourne
813 Swanston Street
Parkville VIC 3010
Australia
Dipartimento di Matematica
Università degli studi di Milano
Via Saldini 50
20133 Milan
Italy
Aram Karakhanyan
Maxwell Institute for Mathematical Sciences and School of Mathematics
University of Edinburgh
James Clerk Maxwell Building
Peter Guthrie Tait Road
Edinburgh EH9 3FD
United Kingdom
Enrico Valdinoci
School of Mathematics and Statistics
University of Melbourne
813 Swanston Street
Parkville VIC 3010
Australia
Istituto di Matematica Applicata e Tecnologie Informatiche
Consiglio Nazionale delle Ricerche
Via Ferrata 1
27100 Pavia
Italy
Dipartimento di Matematica
Università degli studi di Milano
Via Saldini 50
20133 Milan
Italy