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. These notes grew out of a few lectures given in an undergraduate class held at the Università di Roma "Tor Vergata". It is a pleasure to thank the students for their warm interest, their sharp observations and their precious feedback.

Abstract. We extend the De Giorgi-Nash-Moser theory to nonlocal, possibly degenerate integro-differential operators.

Abstract. We state and prove a general Harnack inequality for minimizers of nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional p-Laplacian.

We obtain an improved Sobolev inequality in 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 obtained in G,rard (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 is an integer (Rey in Manuscr Math 65:19-37, 1989, Han in Ann Inst Henri Poincar, Anal Non Lin,aire 8:159-174, 1991, Chou and Geng in Differ Integral Equ 13:921-940, 2000)

We study existence, uniqueness and other geometric properties of the minimizers of the energy functionalwhere u H s (Ω) denotes the total contribution from Ω in the H s norm of u and W is a double-well potential. We also deal with the solutions of the related fractional elliptic Allen-Cahn equation on the entire space R n . The results collected here will also be useful for forthcoming papers, where the second and the third author will study the Γ -convergence and the density estimates for level sets of minimizers.

Abstract. We consider an evolution equation arising in the Peierls-Nabarro model for crystal dislocation. We study the evolution of such dislocation function and show that, at a macroscopic scale, the dislocations have the tendency to concentrate at single points of the crystal, where the size of the slip coincides with the natural periodicity of the medium. These dislocation points evolve according to the external stress and an interior repulsive potential.

No abstract

Qualche tempo dopo Stampacchia, partendo sempre dalla sua disequazione variazionale, aperse un nuovo campo di ricerche che si rivelò importante e fecondo. Si tratta di quello che oggiè chiamato "il problema dell'ostacolo".[ Sandro Faedo, 1987 ] Abstract We investigate the obstacle problem for a class of nonlinear equations driven by nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional p-Laplacian operator with measurable coefficients. Amongst other results, we will prove both the existence and uniqueness of the solutions to the obstacle problem, and that these solutions inherit regularity properties, such as boundedness, continuity and Hölder continuity (up to the boundary), from the obstacle.