Inserted: 24 nov 2017
Last Updated: 24 nov 2017
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 an generalised sense and we establish existence of minimisers for the corresponding Plateau’s problem; also, when $K$ is radial and strictly decreasing, we prove that halfspaces are minimisers if we prescribe “flat” boundary conditions.
A $\Gamma$-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-$L^1$ decay at infinity and we show that the $\Gamma$-limit is the classical perimeter, up to a multiplicative constant that we compute explicitly.