Volume 25, 2019
|Number of page(s)||27|
|Published online||25 September 2019|
On the asymptotic behaviour of nonlocal perimeters
Institut für Analysis und Numerik, Westfälische Wilhelms-Universität Münster,
2 Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy.
* Corresponding author: firstname.lastname@example.org
Accepted: 18 June 2018
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.
Mathematics Subject Classification: 49Q20 / 53A10
Key words: Nonlocal perimeters / nonlocal Plateau’s problem / Γ-convergence
© EDP Sciences, SMAI 2019
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