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Manuscripta Mathematica

, Volume 134, Issue 3–4, pp 377–403 | Cite as

Gamma-convergence of nonlocal perimeter functionals

  • Luigi AmbrosioEmail author
  • Guido De Philippis
  • Luca Martinazzi
Article

Abstract

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).

Mathematics Subject Classification (2000)

49Q15 49Q20 

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Luigi Ambrosio
    • 1
    Email author
  • Guido De Philippis
    • 1
  • Luca Martinazzi
    • 2
  1. 1.Scuola Normale SuperiorePisaItaly
  2. 2.Centro De GiorgiPisaItaly

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