Gamma-convergence of nonlocal energies for partitions

TL;DR

This paper demonstrates that specific nonlocal partition energies Gamma-converge to a local perimeter-based functional, revealing new relaxation effects and behaviors distinct from fractional perimeter convergence.

Contribution

It establishes Gamma-convergence of nonlocal energies with generalized surface tensions to local perimeter functionals, including cases with non-triangular coefficients.

Findings

Nonlocal functionals Gamma-converge to perimeter-based functionals.

Lower semicontinuity holds even without triangle inequality.

Reveals a relaxation process and novel effects in the limit.

Abstract

We prove that certain nonlocal functionals defined on partitions made of measurable sets Gamma-converge to a local functional modeled on the perimeter in the sense of De Giorgi. Those nonlocal functionals involve generalized surface tension coefficients, and are lower semicontinuous even if the coefficients do not satisfy the triangular inequality. It implies a relaxation process in the limit, and provides a novel effect compare to the known gamma-convergence of the fractional perimeter towards the standard perimeter.