Gamma-convergence as $s\to 1^{-}$ of anisotropic nonlocal fractional perimeter functionals

We investigate the asymptotic behavior in the sense of $\mathrm{\Gamma }(L_{loc}^1)$-convergence as $s\to 1^{-}$ 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 $\mathrm{\Gamma }(L_{loc}^1)$-convergence as $s\to 1^{-}$ 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 $\mathrm{\Omega }\subset {\mathbb{R}}^n$ with Lipschitz boundary.