Second Order Expansion for the Nonlocal Perimeter Functional

Hans Knüpfer, Wenhui Shi

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Abstract

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.

Original languageEnglish
Pages (from-to)1371-1402
Number of pages32
JournalCommunications in Mathematical Physics
Volume398
Issue number3
DOIs
Publication statusPublished - Mar 2023

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