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- ItemStrongly nonlocal dislocation dynamics in crystals(Berlin : Weierstraß-Institut für 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.
- ItemRegularity and Bernstein-type results for nonlocal minimal surfaces(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Figalli, Alessio; Valdinoci, EnricoWe prove that, in every dimension, Lipschitz nonlocal minimal surfaces are smooth. Also, we extend to the nonlocal setting a famous theorem of De Giorgi [5] stating that the validity of Bernsteins theorem in dimension n + 1 is a consequence of the nonexistence of n-dimensional singular minimal cones in IRn.