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A class of unstable
free boundary problems
Serena Dipierro, Aram Karakhanyan and Enrico Valdinoci |
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Vol. 10 (2017), No. 6, 1317–1359
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Abstract |
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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
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Mathematical Subject Classification 2010
Primary: 35R35
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Milestones
Received: 11 December 2015
Accepted: 9 May 2017
Published: 14 July 2017
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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 |
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| 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 |
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| 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 |
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