TY - JOUR

T1 - Second Order Expansion for the Nonlocal Perimeter Functional

AU - Knüpfer, Hans

AU - Shi, Wenhui

N1 - Funding Information:
HK and WS are very grateful to Felix Otto who suggested the usage of the autocorrelation function. This project started in discussions with him. HK acknowledges support by the German Research Foundation (DFG) by the project #392124319 and by the Germany’s Excellence Strategy EXC-2181/1 – 390900948. WS is partially supported by the German Research Foundation (DFG) by the project SH 1403/1-1. We would also like to thank both referees for the careful reading and helpful comments.
Publisher Copyright:
© 2022, The Author(s).

PY - 2022/11/19

Y1 - 2022/11/19

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85142296090&partnerID=8YFLogxK

U2 - 10.1007/s00220-022-04549-w

DO - 10.1007/s00220-022-04549-w

M3 - Article

AN - SCOPUS:85142296090

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

ER -