Gamma-convergence of nonlocal perimeter functionals

By Luigi Ambrosio, Guido De Philippis, L. Martinazzi. 2010

230 citations recorded by the source corpus.

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).

Subfields: Computational Theory and Mathematics, Geometry and Topology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/107d2d1780b1ed0528450a51efe200c95729f921

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Citations in the corpus

Regularity properties of nonlocal minimal surfaces via limiting arguments (2011, background): “The functional Js has been recently introduced in [4] as a model for nonlocal minimal surfaces, and its relation with the classical minimal surfaces has been established in [6, 1], both in the geometric sense and in the Gamma–convergence framework.”
Anisotropic fractional perimeters (2013, unknown): “Fractional s -perimeters have attracted increased attention in recent years (see [3, 6–8, 11, 14, 15, 36] and the references therein).”
A critical Kirchhoff type problem involving a nonlocal operator (2013, background): “It is known that (up to a suitable rescaling) `σ tends to the classical length of the string as σ → 1 (see [1, 5]).”
Isoperimetry and Stability Properties of Balls with Respect to Nonlocal Energies (2014, background): “A regularity theory for such nonlocal minimal surfaces has been developed by several authors [10, 4, 18, 35, 12], while the relation of nonlocal perimeters with their local counterpart has been investigated in [8, 3].”
Existence of solutions for Kirchhoff type problem involving the non-local fractional p-Laplacian (2015, background): “The literature on non-local operators and their applications is very interesting and quite large, we refer the interested reader to [2,5,8,9,19,21,22,28,31,42] and the references therein.”
Graph properties for nonlocal minimal surfaces (2015, background): “Several analytic and geometric properties of s-minimal sets have been recently investigated, in terms, for instance, of asymptotics [18, 3, 9, 1, 13], regularity [10, 21, 15] and classification [4, 11].”
Γ-convergence for nonlocal phase transitions (2012, unknown): “In the case when s ∈ ( 0 , 1 / 2 ) , the limiting functional F : X → R ∪ {+∞} is deﬁned as In this case, F agrees with the nonlocal area functional of ∂E in Ω that was studied in [8,10,6].”