Gamma-convergence of nonlocal perimeter functionals

Given {\Omega\subset\mathbb{R}^{n}} open, connected and with Lipschitz boundary, and {s\in (0, 1)}, we consider the functional
\mathcal{J}_s(E,\Omega)\,=\, \int_{E\cap \Omega}\int_{E^c\cap\Omega}\frac{dxdy}{|x-y|^{n+s}}+\int_{E\cap \Omega}\int_{E^c\cap \Omega^c}\frac{dxdy}{|x-y|^{n+s}}\,+ \int_{E\cap \Omega^c}\int_{E^c\cap \Omega}\frac{dxdy}{|x-y|^{n+s}},
where {E\subset\mathbb{R}^{n}} is an arbitrary measurable set. We prove that the functionals {(1-s)\mathcal{J}_s(\cdot, \Omega)} are equi-coercive in {L^1_{\rm loc}(\Omega)} as {s\uparrow 1} and that
\Gamma-\lim_{s\uparrow 1}(1-s)\mathcal{J}_s(E,\Omega)=\omega_{n-1}P(E,\Omega),\quad \text{for every }E\subset\mathbb{R}^{n}\,{\rm measurable}
where P(E, Ω) denotes the perimeter of E in Ω in the sense of De Giorgi. We also prove that as {s\uparrow 1} limit points of local minimizers of {(1-s)\mathcal{J}_s(\cdot,\Omega)} are local minimizers of P(·, Ω).