A new approach to Sobolev spaces and connections to Gamma-convergence

Ponce, Augusto
(2004) Calculus of Variations and Partial Differential Equations — Vol. 19, n° 3, p. 229-255 (2004)

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Abstract
This is a follow-up of a paper of Bourgain, Brezis and Mironescu [2]. We study how the existence of the limit integral(Omega)integral(Omega) omega((x)- f(y))/x - y) rho(epsilon)(x - y) dxdy as epsilon down arrow 0, (1) for omega: [0,infinity) --> [0, infinity) continuous and (rhoepsilon) subset of L-1(R-N) converging to delta(0), is related to the weak regularity of f is an element of L-loc(1)(Omega). This approach gives an alternative way of defining the Sobolev spaces W-1,W-p. We also briefly discuss the Gamma-convergence of (1) with respect to the L-1(Omega)-topology.
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Ponce, A. (2004). A new approach to Sobolev spaces and connections to Gamma-convergence. Calculus of Variations and Partial Differential Equations, 19(3), 229-255. https://doi.org/10.1007/s00526-003-0195-z (Original work published 2004)