Riesz fractional gradient functionals defined on partitions: nonlocal-to-local variational limits
Stefano Almi, Maicol Caponi, Manuel Friedrich, Francesco Solombrino
TL;DR
This work analyzes nonlocal energies with linear growth based on the $s$-fractional gradient acting on piecewise constant functions and establishes their $\Gamma$-convergence as $s\to 1$ to a local interfacial energy defined on $BV$-type partitions. By introducing a broad class of densities $\psi_k\in\mathcal{G}(\lambda,\Lambda)$ with a uniform approximation property and a compatibility condition with $s_k\to1$, the authors connect the nonlocal variational problem to the classical BV setting through the identity $\nabla^s u = \nabla \mathcal{I}^{1-s}u$. The main result proves compactness and $\Gamma$-convergence to a functional $\mathcal{F}_0(u) = \int_{S_u} \psi_0(y,u^+(y),u^-(y),\nu_u(y)) \,d\mathcal{H}^{n-1}(y)$ for $u\in BV(\Omega;T)$, with $\psi_0$ obtained from cell formulas and limits of the densities. This nonlocal-to-local limit advances understanding of fractional energies in BV-type problems and lays groundwork for homogenization and extensions to the full $BV^s$ framework, highlighting how interfaces encode the limiting energy.
Abstract
This paper addresses the asymptotics of functionals with linear growth depending on the Riesz $s$-fractional gradient on piecewise constant functions. We consider a general class of varying energy densities and, as $s\to 1$, we characterize their local limiting functionals in the sense of $Γ$-convergence.