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Author: Valdinoci, Enrico × End date: 2020 × Start date: 2019 × End date: 2014 × Start date: 2013 ×

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Now showing 1 - 10 of 23
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    All functions are locally s-harmonic up to a small error
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 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.
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    Asymptotically linear problems driven by fractional Laplacian operators
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Fiscella, Alessio; Servadei, Raffaella; Valdinoci, Enrico
    In 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.
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    Regularity and Bernstein-type results for nonlocal minimal surfaces
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Figalli, Alessio; Valdinoci, Enrico
    We 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.
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    Monotonicity formulae and classifcation results for singular, degenerate, anisotropic PDEs
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Cozzi, Matteo; Farina, Alberto; Valdinoci, Enrico
    We consider possibly degenerate and singular elliptic equations in a possibly anisotropic medium. We obtain monotonicity results for the energy density, rigidity results for the solutions and classi?cation results for the singularity/degeneracy/anisotropy allowed. As far as we know, these results are new even in the case of non-singular and non- degenerate anisotropic equations.
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    A critical Kirchhoff type problem involving a non-local operator
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Fiscella, Alessio; Valdinoci, Enrico
    We show the existence of non-negative solutions for a Kirchhoff type problem driven by a non-local integrodifferential operator.
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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.
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    Crystal dislocations with different orientations and collisions
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Patrizi, Stefania; Valdinoci, Enrico
    We study a parabolic differential equation whose solution represents the atom dislocation in a crystal for a general type of Peierls-Nabarro model with possibly long range interactions and an external stress. Differently from the previous literature, we treat here the case in which such dislocation is not the superpositions of transitions all occurring with the same orientations (i.e. opposite orientations are allowed as well). We show that, at a long time scale, and at a macroscopic space scale, the dislocations have the tendency to concentrate as pure jumps at points which evolve in time, driven by the external stress and by a singular potential. Due to differences in the dislocation orientations, these points may collide in finite time.
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    1D symmetry for semilinear pdes from the limit interface of the solution
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Farina, Alberto; Valdinoci, Enrico
    We study bounded, entire, monotone solutions of the Allen-Cahn equation. We prove that under suitable assumptions on the limit interface and on the energy growth, the solution is 1D. In particular, differently from the previous literature, the solution is not assumed to have minimal properties. We think that this approach could be fruitful in concrete situations, where one can observe the phase separation at a large scale and whishes to deduce the values of the state parameter in the vicinity of the interface. As a simple example of the results obtained with this point of view, we mention that monotone solutions with energy bounds, whose limit interface does not contain a vertical line through the origin, are 1D, at least up to dimension 4.