Second Order Expansion for the Nonlocal Perimeter Functional

The seminal results of Bourgain et al. (Optimal Control and Partial Differential Equations, IOS, Amsterdam, 2001) and Dávila (Calc Var Partial Differ Equ 15(4):519–527, 2002) show that the classical perimeter can be approximated by a family of nonlocal perimeter functionals. We consider a correspond...

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Bibliographic Details
Published in:Communications in mathematical physics 2023-03, Vol.398 (3), p.1371-1402
Main Authors: Knüpfer, Hans, Shi, Wenhui
Format: Article
Language:English
Subjects:
Autocorrelation functions
Classical and Quantum Gravitation
Complex Systems
Ferromagnetic films
Ferromagnetic materials
Mathematical and Computational Physics
Mathematical Physics
Optimal control
Partial differential equations
Physics
Physics and Astronomy
Quantum Physics
Relativity Theory
Theoretical
Thin films
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Summary:The seminal results of Bourgain et al. (Optimal Control and Partial Differential Equations, IOS, Amsterdam, 2001) and Dávila (Calc Var Partial Differ Equ 15(4):519–527, 2002) show that the classical perimeter can be approximated by a family of nonlocal perimeter functionals. We consider a corresponding second order expansion for the nonlocal perimeter functional. In a special case, the considered family of energies is also relevant for a variational model for thin ferromagnetic films. We derive the Γ –limit of these functionals as ϵ → 0 . We also show existence for minimizers with prescribed volume fraction. For small volume fraction, the unique, up to translation, minimizer of the limit energy is given by the ball. The analysis is based on a systematic exploitation of the associated symmetrized autocorrelation function.
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-022-04549-w