A class of unstable free boundary problems

S. Dipierro, A. Karakhanyan, E. Valdinoci

Research output: Contribution to journalArticle

2 Citations (Scopus)

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.
Original languageEnglish
Pages (from-to)1317-1359
Number of pages43
JournalAnalysis and PDE
Volume10
Issue number6
DOIs
Publication statusPublished - 2017
Externally publishedYes

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Free Boundary Problem
Perimeter
Free Boundary
Unstable
Energy Functional
Minimizer
Nonlinear Function
Energy
Surface tension
Density Estimates
Minimal Solution
Nonlocality
Boundary conditions
Surface Tension
Dirichlet
Superposition
Geometry
Fractional
Regularity
Class

Cite this

Dipierro, S. ; Karakhanyan, A. ; Valdinoci, E. / A class of unstable free boundary problems. In: Analysis and PDE. 2017 ; Vol. 10, No. 6. pp. 1317-1359.
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A class of unstable free boundary problems. / Dipierro, S.; Karakhanyan, A.; Valdinoci, E.

In: Analysis and PDE, Vol. 10, No. 6, 2017, p. 1317-1359.

Research output: Contribution to journalArticle

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AU - Valdinoci, E.

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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.

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