56 research outputs found
Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces
We obtain an improved Sobolev inequality in 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 H^s obtained in [P. Gerard, ESAIM 1998]
using the abstract approach of dislocation spaces developed in [K. Tintarev &
K. H. Fieseler, Imperial College Press 2007]. We also analyze directly the
local defect of compactness of the Sobolev embedding in terms of measures in
the spirit of [P. L. Lions, Rev. Mat. Iberoamericana 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 is an integer ([O. Rey, Manuscripta math. 1989; Z.-C. Han, Ann.
Inst. H. Poincare Anal. Non Lineaire 1991], [K. S. Chou & D. Geng, Differential
Integral Equations 2000]).Comment: 33 page
Fractional superharmonic functions and the Perron method for nonlinear integro-differential equations
We deal with a class of equations driven by nonlocal, possibly degenerate,
integro-differential operators of differentiability order and
summability growth , whose model is the fractional -Laplacian with
measurable coefficients. We state and prove several results for the
corresponding weak supersolutions, as comparison principles, a priori bounds,
lower semicontinuity, and many others. We then discuss the good definition of
-superharmonic functions, by also proving some related properties. We
finally introduce the nonlocal counterpart of the celebrated Perron method in
nonlinear Potential Theory.Comment: To appear in Math. An
Nonlocal Harnack inequalities
We state and prove a general Harnack inequality for minimizers of nonlocal,
possibly degenerate, integro-differential operators, whose model is the
fractional p-Laplacian.Comment: To appear in J. Funct. Ana
Hitchhiker's guide to the fractional Sobolev spaces
This paper deals with the fractional Sobolev spaces W^[s,p]. We analyze the
relations among some of their possible definitions and their role in the trace
theory. We prove continuous and compact embeddings, investigating the problem
of the extension domains and other regularity results. Most of the results we
present here are probably well known to the experts, but we believe that our
proofs are original and we do not make use of any interpolation techniques nor
pass through the theory of Besov spaces. We also present some counterexamples
in non-Lipschitz domains
Global estimates for nonlinear parabolic equations
We consider nonlinear parabolic equations of the type under standard growth
conditions on , with only assumed to be integrable. We prove general
decay estimates up to the boundary for level sets of the solutions and the
gradient which imply very general estimates in Lebesgue and Lorentz
spaces. Assuming only that the involved domains satisfy a mild exterior
capacity density condition, we provide global regularity results.Comment: To appear in J. Evol. Equation
Local behavior of fractional -minimizers
We extend the De Giorgi-Nash-Moser theory to nonlocal, possibly degenerate
integro-differential operators.Comment: 26 pages. To appear in Ann. Inst. H. Poincare Anal. Non Lineaire.
arXiv admin note: text overlap with arXiv:1405.784
Equazioni frazionarie non lineari nel gruppo di Heisenberg
We deal with a wide class of nonlinear nonlocal equations led by integro-differential operators of order (s,p), with summability exponent p in (1,∞) and differentiability order s in (0,1), whose prototype is the fractional subLaplacian in the Heisenberg group. We present very recent boundedness and regularity estimates (up to the boundary) for the involved weak solutions, and we introduce the nonlocal counterpart of the Perron Method in the Heisenberg group, by recalling some results on the fractional obstacle problem. Throughout the paper we also list various related open problems.Investighiamo una ampia classe di equazioni non lineari e non locali guidate da operatori integro-differenziali di ordine (s,p), con esponente di sommabilità p in (1,∞) e ordine di differenziabilitàs in (0,1), il cui prototipo è il subLaplaciano frazionario nel gruppo di Heisenberg. Presentiamo recenti stime di limitatezza e di regolarità (fino al bordo) per le relative soluzioni deboli, e introduciamo l'analogo non locale del Metodo di Perron nel gruppo di Heisenberg, richiamando anche alcuni risultati sul problema dell'ostacolo frazionario. Diversi problemi aperti sono inoltre menzionati
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