# Limiting behaviour of rescaled nonlocal perimeters and of their first variations. – Valerio Pagliari (Universita’ di Pisa)

#### Abstract

We introduce a class of integral functionals known as nonlocal perimeters, which can be thought as interactions between a set and its complement that are weighted by a positive kernel. In the first part of the talk, we summarise the main features of these functionals and then we study the asymptotic behaviour of the family associated with mass-preserving rescalings of a given kernel. Namely, we prove that when the scaling parameter approaches $0$, the rescaled non local perimeters $Gamma$-converge to De Giorgi’s perimeter, up to a multiplicative constant. In the second part of the talk, we show that a similar result holds for nonlocal curvatures, i.e. for the first variations of the nonlocal perimeters; time permitting, we shall hint at possible applications of this to dislocation dynamics.

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