Abstract
Given \({\Omega\subset\mathbb{R}^{n}}\) open, connected and with Lipschitz boundary, and \({s\in (0, 1)}\), we consider the functional
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
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).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ambrosio L., Fusco N., Pallara D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000)
Bourgain, J., Brézis, H., Mironescu, P.: Another look at Sobolev spaces. In: Menaldi, J.L., Rofman, E., Sulem, A. (eds.) Optimal Control and Partial Differential Equations, pp. 439–455. IOS Press (2001)
Brézis, H.: How to recognize constant functions. A connection with Sobolev spaces, Uspekhi Mat. Nauk, 57, 59–74 (2002), transl. in Russian Math. Surveys 57, 693–708 (2002)
Caffarelli, L., Valdinoci, E.: Uniform estimates and limiting arguments for nonlocal minimal surfaces, preprint (2009)
Caffarelli, L., Roquejoffre, J.-M., Savin, O.: Non-local minimal surfaces, preprint (2009)
Dal Maso G.: An Introduction to Γ-Convergence. Birkhäuser, Basel (1993)
De Giorgi E.: Nuovi teoremi relativi alle misure (r − 1)-dimensionali in uno spazio a r dimensioni. Ricerche Mat. 4, 95–113 (1955)
De Giorgi E., Letta E.: Une notion générale de convergence faible pour des fonctions croissantes d’ensemble. Ann. Scuola Norm. Sup. Pisa 4, 61–99 (1977)
Fonseca I., Müller S.: Quasi-convex integrands and lower semicontinuity in L 1. SIAM J. Math. Anal. 23, 1081–1098 (1992)
Giusti E.: Minimal Surfaces and Functions of Bounded Variation. Monographs In Mathematics. Brickhauser, Basel (1984)
Maz’ya V., Shaposhnikova T.: Erratum to: On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces. J. Funct. Anal. 201, 298–300 (2003)
Morse A.P.: Perfect blankets. Trans. Am. Math. Soc. 61, 418–422 (1947)
Nguyen H.-M.: Further characterizations of Sobolev spaces. J. Eur. Math. Soc. 10, 191–229 (2008)
Nguyen, H.-M.: Γ-convergence, Sobolev norms and BV functions, preprint (2009)
Visintin A.: Generalized coarea formula and fractal sets. Jpn. J. Indust. Appl. Math. 8, 175–201 (1991)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ambrosio, L., Philippis, G.D. & Martinazzi, L. Gamma-convergence of nonlocal perimeter functionals. manuscripta math. 134, 377–403 (2011). https://doi.org/10.1007/s00229-010-0399-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-010-0399-4