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 $s\in (0,1)$ and summability growth $p>1$, whose model is the fractional $p$-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 $(s,p)$-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

Global estimates for nonlinear parabolic equations

We consider nonlinear parabolic equations of the type $$u_t - div a(x, t, Du)= f(x,t) on \Omega_T = \Omega\times (-T,0),$$ under standard growth conditions on $a$, with $f$ only assumed to be integrable. We prove general decay estimates up to the boundary for level sets of the solutions $u$ and the gradient $Du$ 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

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

Local behavior of fractional pp-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

Gamma-Convergence for one-dimensional nonlocal phase transition energies

We study the asymptotic behavior as epsilon goes to 0 of an appropriate scaling of the following nonlocal Allen-Cahn energy,E-epsilon(s)(u) = epsilon(2s) integral integral(IxI) vertical bar u(x) - u(y)vertical bar(2)/vertical bar x - y vertical bar(1+2s) dxdy + integral(I) W(u) dx,where I is an interval in R, and W is a double-well potential. We provide a Gamma-convergence result for any s is an element of (0, 1), by extending the case when s = 1/2 studied by Alberti, Bouchitte and Seppecher in [2]. We also investigate the convergence as s NE arrow 1 of the related optimal profile problem to the local counterpart