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Author: Dipierro, Serena × Subject: fractional Laplacian ×

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Now showing 1 - 3 of 3
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    Dislocation dynamics in crystals: A macroscopic theory in a fractional Laplace setting
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 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.
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    Nonlocal problems with Neumann boundary conditions
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 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.
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    Strongly nonlocal dislocation dynamics in crystals
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Dipierro, Serena; Figalli, Alessio; Valdinoci, Enrico
    We 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.