Gamma-convergence as $s\to1^-$ of anisotropic nonlocal fractional perimeter functionals

Abstract

We investigate the asymptotic behavior in the sense of $\Gamma(L^1_{loc})$-convergence as $s\to1^-$ of anisotropic non local $s$-fractional perimeters defined with respect to general anisotropic integration kernels $k_s(\cdot)$, under the hypothesis of pointwise convergence of such kernels. In particular, we prove the $\Gamma(L^1_{loc})$-convergence as $s\to1^-$ of the rescaled anisotropic nonlocal $s$-fractional perimeters defined with respect to the kernels $k_s(\cdot)$ to a suitable anisotropic perimeter. We do so both in $\mathbb{R}^n$ and on a bounded domain $\Omega\subset\mathbb{R}^n$ with Lipschitz boundary.