We introduce a new Neumann problem for the fractional Laplacian arising from a simple probabilistic consideration, and we discuss the basic properties of this model. We can consider both elliptic and parabolic equations in any domain. In addition, we formulate problems with nonhomogeneous Neumann conditions, and also with mixed Dirichlet and Neumann conditions, all of them having a clear probabilistic interpretation.We prove that solutions to the fractional heat equation with homogeneous Neumann conditions have the following natural properties: conservation of mass inside Ω, decreasing energy, and convergence to a constant as t → ∞. Moreover, for the elliptic case we give the variational formulation of the problem, and establish existence of solutions.We also study the limit properties and the boundary behavior induced by this nonlocal Neumann condition.For concreteness, one may think that our nonlocal analogue of the classical Neumann condition ∂νu = 0 on ∂Ω consists in the nonlocal prescription Ω u(x) − u(y) |x − y| n+2s dy = 0 for x ∈ R n \ Ω. We made an effort to keep all the arguments at the simplest possible technical level, in order to clarify the conncetions between the different scientific fields that are naturally involved in the problem, and make the paper accessible also to a wide, non-specialistic public (for this scope, we also tried to use and compare different concepts and notations in a somehow more unified way).2010 Mathematics Subject Classification. 35R11, 60G22.

Chapter 1. Introduction 1.1. The fractional Laplacian 1.2. The Mountain Pass Theorem 1.3. The Concentration-Compactness Principle Chapter 2. The problem studied in this monograph 2.1. Fractional critical problems 2.2. An extended problem and statement of the main results Chapter 3. Functional analytical setting 3.1. Weighted Sobolev embeddings 3.2. A Concentration-Compactness Principle Chapter 4. Existence of a minimal solution and proof of Theorem 2.2.2 4.1. Some convergence results in view of Theorem 2.2.2 4.2. Palais-Smale condition for F ε 4.3. Proof of Theorem 2.2.2 Chapter 5. Regularity and positivity of the solution 5.1. A regularity result 5.2. A strong maximum principle and positivity of the solutions Chapter 6. Existence of a second solution and proof of Theorem 2.2.4 6.1. Existence of a local minimum for I ε 6.2. Some preliminary lemmata towards the proof of Theorem 2.2.4 6.3. Some convergence results in view of Theorem 2.2.4 6.4. Palais-Smale condition for I ε 6.5. Bound on the minimax value 6.6. Proof of Theorem 2.2.4 Bibliography

Abstract. We show that we can approximate every function f ∈ C k (B 1 ) with a s-harmonic function in B 1 that vanishes outside a compact set.That is, s-harmonic functions are dense in C k loc . This result is clearly in contrast with the rigidity of harmonic functions in the classical case and can be viewed as a purely nonlocal feature. IntroductionIt is a well-known fact that harmonic functions are very rigid. For instance, in dimension 1, they reduce to a linear function and, in any dimension, they never possess local extrema.The goal of this paper is to show that the situation for fractional harmonic functions is completely different, namely one can fix any function in a given domain and find a s-harmonic function arbitrarily close to it.Heuristically speaking, the reason for this phenomenon is that while classical harmonic functions are determined once their trace on the boundary is fixed, in the fractional setting the operator sees all the data outside the domain. Hence, a careful choice of these data allows a s-harmonic function to "bend up and down" basically without any restriction.The rigorous statement of this fact is in the following Theorem 1.1. For this, we recall that, given s ∈ (0, 1), the fractional Laplace operator of a function u is defined (up to a normalizing constant) asWe refer to [4,7,9,10] for other equivalent definitions, motivations and applications.Theorem 1.1. Fix k ∈ N. Then, given any function f ∈ C k (B 1 ) and any ǫ > 0,2010 Mathematics Subject Classification. 35R11, 60G22, 35A35, 34A08.

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.