Improved Sobolev embeddings, profile decomposition, and\n concentration-compactness for fractional Sobolev spaces

2013/02/24 by Giampiero Palatucci, Palatucci, Giampiero, Adriano Pisante +1 · 4 citations
Mathematics · Computer Science · Engineering · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics

paper · pdf · doi:10.48550/arxiv.1302.5923

Abstract

We obtain an improved Sobolev inequality in H^s spaces involving Morrey\nnorms. This refinement yields a direct proof of the existence of optimizers and\nthe compactness up to symmetry of optimizing sequences for the usual Sobolev\nembedding. More generally, it allows to derive an alternative, more transparent\nproof of the profile decomposition in H^s obtained in [P. Gerard, ESAIM 1998]\nusing the abstract approach of dislocation spaces developed in [K. Tintarev &\nK. H. Fieseler, Imperial College Press 2007]. We also analyze directly the\nlocal defect of compactness of the Sobolev embedding in terms of measures in\nthe spirit of [P. L. Lions, Rev. Mat. Iberoamericana 1985]. As a model\napplication, we study the asymptotic limit of a family of subcritical problems,\nobtaining concentration results for the corresponding optimizers which are well\nknown when s is an integer ([O. Rey, Manuscripta math. 1989; Z.-C. Han, Ann.\nInst. H. Poincare Anal. Non Lineaire 1991], [K. S. Chou & D. Geng, Differential\nIntegral Equations 2000]).\n

Citations

• A relation between pointwise convergence of functions and convergence of functionals

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