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Gamma-convergence of nonlocal perimeter functionals. (English) Zbl 1207.49051

Summary: Given \(\Omega\subset\mathbb R^n\) open, connected and with Lipschitz boundary, and \(s\in (0, 1)\), we consider the functional
\[ {\mathcal J}_s(E,\Omega)= \int_{E\cap \Omega}\int_{E^c\cap\Omega} \frac{dx\,dy}{|x-y|^{n+s}}+\int_{E\cap \Omega}\int_{E^c\cap \Omega^c}\frac{dx\,dy}{|x-y|^{n+s}}\,+ \int_{E\cap \Omega^c}\int_{E^c\cap \Omega} \frac{dx\,dy}{|x-y|^{n+s}}, \]
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 \(L^1_{\text{loc}}(\Omega)\) as \(s\uparrow 1\) and that
\[ \Gamma-\lim_{s\uparrow 1}(1-s){\mathcal J}_s(E,\Omega)= \omega_{n-1}P(E,\Omega),\quad \text{for every }E\subset\mathbb R^n\text{ measurable} \]
where \(P(E, \Omega )\) denotes the perimeter of \(E\) in \(\Omega \) 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 \(P(\cdot , \Omega )\).

MSC:

49Q15 Geometric measure and integration theory, integral and normal currents in optimization
49Q20 Variational problems in a geometric measure-theoretic setting
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