Calculus of Variations and Geometric Measure Theory
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A $\Gamma$-convergence result for nonlocal perimeters

Valerio Pagliari (Dipartimento di matematica, Universita' di Pisa)

created by pagliari on 10 Nov 2017

20 nov 2017 -- 12:30   [open in google calendar]

Sala Riunioni, VII Piano, Dipartimento di Matematica, Università di Padova

Abstract.

We focus on a class of integral functionals known as nonlocal perimeters. Intuitively, these functionals express a weighted interaction between a set and its complement, possibly localised to some reference set; the weight is provided by a positive kernel $K$ which might be singular. Firstly, we describe some general properties of these functionals and we show that they are indeed perimeters in a generalised sense. Then, 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. Finally, in the second part of the talk, the asymptotic behaviour of the functionals is discussed: we show that suitable rescalings of the nonlocal perimeters associated with certain kernels $\Gamma$-converge to the classical perimeter, up to a multiplicative constant that we exhibit explicitly. This is a joint work with J. Berendsen.

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