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Author: Cozzi, Matteo × Author: Valdinoci, Enrico × End date: 2019 × Start date: 2010 ×

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Now showing 1 - 5 of 5
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    Planelike interfaces in long-range Ising models and connections with nonlocal minimal surfaces
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 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.
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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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    Plane-like minimizers for a non-local Ginzburg-Landau-type energy in a periodic medium
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Cozzi, Matteo; Valdinoci, Enrico
    We 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.
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    Nonlocal phase transitions in homogeneous and periodic media : in honor of Professor Paul Rabinowitz, with great esteem and admiration
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Cozzi, Matteo; Dipierro, Serena; Valdinoci, Enrico
    We discuss some results related to a phase transition model in which the potential energy induced by a double-well function is balanced by a fractional elastic energy. In particular, we present asymptotic results (such as Gamma-convergence, energy bounds and density estimates for level sets), flatness and rigidity results, and the construction of planelike minimizers in periodic media.
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    Gradient bounds and rigidity results for singular, degenerate, anisotropic partial differential equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Cozzi, Matteo; Farina, Alberto; Valdinoci, Enrico
    We consider the Wulff-type energy functional where B is positive, monotone and convex, and H is positive homogeneous of degree 1. The critical points of this functional satisfy a possibly singular or degenerate, quasilinear equation in an anisotropic medium. We prove that the gradient of the solution is bounded at any point by the potential F(u) and we deduce several rigidity and symmetry properties.