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A new approach to Sobolev spaces and connections to $\mathbf\Gamma$-convergence

Authors:

Email author" target="_blank">Augusto?C?Ponce Email author

Institution:

(1) Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4 Pl. Jussieu, B.C. 187, 75252 Paris Cedex 05, France;(2) Dept. of Math., Hill Center, Rutgers University, 110 Frelinghuysen Rd, NJ 08854 Piscataway, USA

Abstract:

This is a follow-up of a paper of Bourgain, Brezis and Mironescu 2]. We study how the existence of the limit

$\int_\Omega \! \int_\Omega \omega\left( \frac{|f(x)-f(y)|}{|x-y|} \right) \rho_\varepsilon(x-y) \, dx \, dy \quad \text{as $\varepsilon \downarrow 0$},$

for $\omega : 0,\infty) \to 0,\infty)$ continuous and $(\rho_\varepsilon) \subset L^1({\mathbb R}^N)$ converging to $\delta_0$ , is related to the weak regularity of $f \in L^1_{\rm loc}(\Omega)$ . This approach gives an alternative way of defining the Sobolev spaces W ^1,p. We also briefly discuss the $\Gamma$ -convergence of (1) with respect to the $L^1(\Omega)$ -topology.Received: 12 November 2002, Accepted: 7 January 2003, Published online: 22 September 2003Mathematics Subject Classification (2000): 46E35, 49J45Augusto C. Ponce: ponce@ann.jussieu.fr

Keywords:

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