2014, Dipierro, Serena, Savin, Ovidiu, Valdinoci, Enrico

We show that we can approximate every function f Ck (B1) with a s-harmonic function in B1 that vanishes outside a compact set. That is, s-harmonic functions are dense in Ck 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.

2016, Caffarelli, Luis, Dipierro, Serena, Outrata, Jir̆í

We consider here a logistic equation, modeling processes of nonlocal character both in the diffusion and proliferation terms. More precisely, for populations that propagate according to a Levy process and can reach resources in a neighborhood of their position, we compare (and find explicit threshold for survival) the local and nonlocal case. As ambient space, we can consider: bounded domains, periodic environments, transition problems, where the environment consists of a block of infinitesimal diffusion and an adjacent nonlocal one. In each of these cases, we analyze the existence/nonexistence of solutions in terms of the spectral properties of the domain. In particular, we give a detailed description of the fact that nonlocal populations may better adapt to sparse resources and small environments.

2015, Dipierro, Serena, Savin, Ovidiu, Valdinoci, Enrico

In this paper we show that a nonlocal minimal surface which is a graph outside a cylinder is in fact a graph in the whole of the space. As a consequence, in dimension 3, we show that the graph is smooth. The proofs rely on convolution techniques and appropriate integral estimates which show the pointwise validity of an Euler-Lagrange equation related to the nonlocal mean curvature.

2014, Dipierro, Serena, Ros-Oton, Xavier, Valdinoci, Enrico

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, decreasing energy, and convergence to a constant as time flows. 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.

2015, Davila, Juan, Pino, Manuel del, Dipierro, Serena, Valdinoci, Enrico

We construct codimension 1 surfaces of any dimension that minimize a nonlocal perimeter functional among surfaces that are periodic, cylindrically symmetric and decreasing. These surfaces may be seen as a nonlocal analogue of the classical Delaunay surfaces (onduloids). For small volume, most of their mass tends to be concentrated in a periodic array and the surfaces are close to a periodic array of balls (in fact, we give explicit quantitative bounds on these facts).

2015, Dipierro, Serena, Valdinoci, Enrico

In this paper we show a density property for fractional weighted Sobolev spaces. That is, we prove that any function in a fractional weighted Sobolev space can be approximated by a smooth function with compact support. The additional difficulty in this nonlocal setting is caused by the fact that the weights are not necessarily translation invariant.

2016, Dipierro, Serena, Valdinoci, Enrico

We consider surfaces which minimize a nonlocal perimeter functional and we discuss their interior regularity and rigidity properties, in a quantitative and qualitative way, and their (perhaps rather surprising) boundary behavior. We present at least a sketch of the proofs of these results, in a way that aims to be as elementary and self contained as possible, referring to the papers [CRS10, SV13, CV13, BFV14,FV,DSV15,CSV16] for full details.

2016, Cozzi, Matteo, Dipierro, Serena, Valdinoci, Enrico

This paper contains three types of results: the construction of ground state solutions for a long-range Ising model whose interfaces stay at a bounded distance from any given hyperplane, the construction of nonlocal minimal surfaces which stay at a bounded distance from any given hyperplane, the reciprocal approximation of ground states for long-range Ising models and nonlocal minimal surfaces. In particular, we establish the existence of ground state solutions for long-range Ising models with planelike interfaces, which possess scale invariant properties with respect to the periodicity size of the environment. The range of interaction of the Hamiltonian is not necessarily assumed to be finite and also polynomial tails are taken into account (i.e. particles can interact even if they are very far apart the one from the other). In addition, we provide a rigorous bridge between the theory of long-range Ising models and that of nonlocal minimal surfaces, via some precise limit result.

2016, Dipierro, Serena, Karakhanyan, Aram, Valdinoci, Enrico

We study a Bernoulli type free boundary problem with two phases and a nonlinear energy superposition. We show that, for this problem, the Bernoulli constant, which determines the gradient jump condition across the free boundary, is of global type and it is indeed determined by the weighted volumes of the phases. In particular, the Bernoulli condition that we obtain can be seen as a pressure prescription in terms of the volume of the two phases of the minimizer itself (and therefore it depends on the minimizer itself and not only on the structural constants of the problem). Another property of this type of problems is that the minimizer in a given domain is not necessarily a minimizer in a smaller subdomain, due to the nonlinear structure of the problem. Due to these features, this problem is highly unstable as opposed to the classical case studied by Alt, Caffarelli and Friedman. It also interpolates the classical case, in the sense that the blow-up limits are minimizers of the Alt-Caffarelli-Friedman functional. Namely, the energy of the problem somehow linearizes in the blow-up limit. We also develop a detailed optimal regularity theory for the minimizers and for their free boundaries.

2013, Dipierro, Serena, Palatucci, Giampiero, Valdinoci, Enrico

We consider an evolution equation arising in the PeierlsNabarro 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.