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- ItemThe Dirichlet problem for nonlocal operators with kernels: Convex and nonconvex domains(Berlin : WeierstraĆ-Institut fĆ¼r Angewandte Analysis und Stochastik, 2015) Ros-Oton, Xavier; Valdinoci, EnricoWe study the interior regularity of solutions to a Dirichlet problem for anisotropic operators of fractional type. A prototype example is given by the sum of one-dimensional fractional Laplacians in fixed, given directions. We prove here that an interior differentiable regularity theory holds in convex domains. When the spectral measure is a bounded function and the domain is smooth, the same regularity theory applies. In particular, solutions always possess a classical first derivative. The assumptions on the domain are sharp, since if the domain is not convex and the spectral measure is singular, we construct an explicit counterexample.
- ItemGevrey regularity for integro-differential operators(Berlin : WeierstraĆ-Institut fĆ¼r Angewandte Analysis und Stochastik, 2013) Albanese, Guglielmo; Fiscella, Alessio; Valdinoci, EnricoWe prove a regularity theory in the Gevrey class for a family of nonlocal differential equations of elliptic type.
- ItemA nonlocal concave-convex problem with nonlocal mixed boundary data(Berlin : WeierstraĆ-Institut fĆ¼r Angewandte Analysis und Stochastik, 2016) Abdellaoui, Boumediene; Dieb, Abdelrazek; Valdinoci, EnricoThe aim of this paper is to study a nonlocal equation with mixed Neumann and Dirichlet external conditions. We prove existence, nonexistence and multiplicity of positive energy solutions and analyze the interaction between the concave-convex nonlinearity and the Dirichlet-Neumann data.
- ItemStrongly nonlocal dislocation dynamics in crystals(Berlin : WeierstraĆ-Institut fĆ¼r Angewandte Analysis und Stochastik, 2013) Dipierro, Serena; Figalli, Alessio; Valdinoci, EnricoWe consider an equation motivated by crystal dynamics and driven by a strongly nonlocal elliptic operator of fractional type. We study the evolution of the dislocation function for macroscopic space and time scales, by showing that 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. We also prove that the motion of these dislocation points is governed by an interior repulsive potential that is superposed to an elastic reaction to the external stress.
- ItemOverdetermined problems for the fractional Laplacian in exterior and annular sets(Berlin : WeierstraĆ-Institut fĆ¼r Angewandte Analysis und Stochastik, 2014) Soave, Nicola; Valdinoci, EnricoWe consider a fractional elliptic equation in an unbounded set with both Dirichlet and fractional normal derivative datum prescribed. We prove that the domain and the solution are necessarily radially symmetric. The extension of the result in bounded non-convex regions is also studied, as well as the radial symmetry of the solution when the set is a priori supposed to be rotationally symmetric.
- ItemA Widder's type theorem for the heat equation with nonlocal diffusion(Berlin : WeierstraĆ-Institut fĆ¼r Angewandte Analysis und Stochastik, 2013) Barrios, BegoƱa; Peral, Ireneo; Soria, Fernando; Valdinoci, EnricoI
- ItemPlane-like minimizers for a non-local Ginzburg-Landau-type energy in a periodic medium(Berlin : WeierstraĆ-Institut fĆ¼r Angewandte Analysis und Stochastik, 2015) Cozzi, Matteo; Valdinoci, EnricoWe consider a non-local phase transition equation set in a periodic medium and we construct solutions whose interface stays in a slab of prescribed direction and universal width. The solutions constructed also enjoy a local minimality property with respect to a suitable non-local energy functional.
- ItemDislocation dynamics in crystals: A macroscopic theory in a fractional Laplace setting(Berlin : WeierstraĆ-Institut fĆ¼r Angewandte Analysis und Stochastik, 2013) Dipierro, Serena; Palatucci, Giampiero; Valdinoci, EnricoWe 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.
- ItemNonlocal problems with Neumann boundary conditions(Berlin : WeierstraĆ-Institut fĆ¼r Angewandte Analysis und Stochastik, 2014) Dipierro, Serena; Ros-Oton, Xavier; Valdinoci, EnricoWe 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.
- ItemAsymptotically linear problems driven by fractional Laplacian operators(Berlin : WeierstraĆ-Institut fĆ¼r Angewandte Analysis und Stochastik, 2014) Fiscella, Alessio; Servadei, Raffaella; Valdinoci, EnricoIn this paper we study a non-local fractional Laplace equation, depending on a parameter, with asymptotically linear right-hand side. Our main result concerns the existence of weak solutions for this equation and it is obtained using variational and topological methods. We treat both the nonresonant case and the resonant one.