On the asymptotic behaviour of nonlocal perimeters
ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 48

We study a class of integral functionals known as nonlocal perimeters, which, intuitively, express a weighted interaction between a set and its complement. The weight is provided by a positive kernel K, which might be singular. In the first part of the paper, we show that these functionals are indeed perimeters in a generalised sense that has been recently introduced by A. Chambolle et al. [Archiv. Rational Mech. Anal. 218 (2015) 1263–1329]. Also, we establish existence of minimisers for the corresponding Plateau’s problem and, when K is radial and strictly decreasing, we prove that halfspaces are minimisers if we prescribe “flat” boundary conditions. A Γ-convergence result is discussed in the second part of the work. We study the limiting behaviour of the nonlocal perimeters associated with certain rescalings of a given kernel that has faster-than-L1 decay at infinity and we show that the Γ-limit is the classical perimeter, up to a multiplicative constant that we compute explicitly.

Reçu le :
Accepté le :
DOI : 10.1051/cocv/2018038
Classification : 49Q20, 53A10
Keywords: Nonlocal perimeters, nonlocal Plateau’s problem, Γ-convergence

Berendsen, Judith 1 ; Pagliari, Valerio 1

1 
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Berendsen, Judith; Pagliari, Valerio. On the asymptotic behaviour of nonlocal perimeters. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 48. doi: 10.1051/cocv/2018038

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