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

By Giampiero Palatucci, A. Pisante. 2013

354 citations recorded by the source corpus.

We obtain an improved Sobolev inequality in $$\dot{H}^s$$H˙s spaces involving Morrey norms. This refinement yields a direct proof of the existence of optimizers and the compactness up to symmetry of optimizing sequences for the usual Sobolev embedding. More generally, it allows to derive an alternative, more transparent proof of the profile decomposition in $$\dot{H}^s$$H˙s obtained in Gérard (ESAIM Control Optim Calc Var 3:213–233, 1998) using the abstract approach of dislocation spaces developed in Tintarev and Fieseler (Concentration compactness. Functional-analytic grounds and applications. Imperial College Press, London, 2007). We also analyze directly the local defect of compactness of the Sobolev embedding in terms of measures in the spirit of Lions (Rev Mat Iberoamericana 1:145–201, 1985, Rev Mat Iberoamericana 1:45–121, 1985). As a model application, we study the asymptotic limit of a family of subcritical problems, obtaining concentration results for the corresponding optimizers which are well known when $$s$$s is an integer (Rey in Manuscr Math 65:19–37, 1989, Han in Ann Inst Henri Poincaré Anal Non Linéaire 8:159–174, 1991, Chou and Geng in Differ Integral Equ 13:921–940, 2000).

Subfields: Applied Mathematics, Computational Mechanics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/b7b0cca9a5b3a735acb6f6a8facbf9c3023deb37

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Citations in the corpus

The concentration-compactness principle for fractional order Sobolev spaces in unbounded domains and applications to the generalized fractional Brezis–Nirenberg problem (2018, neutral): “In the fractional setting such extension was recently obtained by Palatucci and Pisante [29] for p = 2 and then by Mosconi et al for any 1 p ns , see [28], to deal with problems in bounded domains.”
Multiplicity and concentration of positive solutions for the fractional Schrödinger–Poisson systems with critical growth (2017, neutral): “The lemma below provides a way to manipulate smooth truncations for the Laplacian see [31].”
Fractional elliptic problems with critical growth in the whole of $\R^n$ (2015, background): “See also, [3, 43], where this principle was proved for different problems.”
On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions (2011, result): “Related concentration-compactness results have been obtained by Palatucci and Pisante [85].”
Fractional superharmonic functions and the Perron method for nonlinear integro-differential equations (2016, background): “, the strong three-commutators estimates introduced in [6,7] to deduce the regularity of weak fractional harmonic maps (see also [40]), the strong barriers and density estimates in [37, 39], the pseudo-differential commutator and energy estimates in [35, 36], and many other successful techniques whi”
Local behavior of fractional $p$-minimizers (2015, positive): “…s -harmonic extension [4] nor of various tools as, e.g., the sharp 3-commutators estimates introduced in [5] to deduce the regularity of weak fractional harmonic maps, the strong barriers and density estimates in [26,28,29], the commutator and energy estimates in [25,27], and so on.”