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Lack of compactness in the 2D critical Sobolev embedding, the general case

Abstract : This Note is devoted to the description of the lack of compactness of the Sobolev embedding of H-1 (R-2) in the critical Orlicz space L(R-2). It turns out that up to cores our result is expressed in terms of the concentration-type examples derived by J. Moser (1971) in [16] as in the radial setting investigated in Bahouri et al. (2011) [5]. However, the analysis we used in this work is strikingly different from the one conducted in the radial case which is based on an L-infinity estimate far away from the origin and which is no longer valid in the general frame work. The strategy we adopted to build the profile decomposition in terms of examples by Moser concentrated around cores is based on capacity arguments and relies on an extraction process of mass concentrations. (C) 2012 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.
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Submitted on : Tuesday, May 1, 2012 - 11:42:55 AM
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Hajer Bahouri, Mohamed Majdoub, Nader Masmoudi. Lack of compactness in the 2D critical Sobolev embedding, the general case. Comptes Rendus. Mathématique, Académie des sciences (Paris), 2012, 350 (3-4), pp.177--181. ⟨10.1016/j.crma.2012.01.016⟩. ⟨hal-00692661⟩



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