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Author: Caffarelli, Luis × Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik ×

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Now showing 1 - 3 of 3
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    On a long range segregation model
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Caffarelli, Luis; Patrizi, Stefania; Quitalo, Veronica
    Segregation phenomena occurs in many areas of mathematics and science: from equipartition problems in geometry, to social and biological processes (cells, bacteria, ants, mammals) to finance (sellers and buyers). There is a large body of literature studying segregation models where the interaction between species is punctual. There are many processes though, where the growth of a population at a point is inhibited by the populations in a full area surrounding that point. This work is a first attempt to study the properties of such a segregation process.
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    A logistic equation with nonlocal interactions
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Caffarelli, Luis; Dipierro, Serena; Outrata, Jir̆í
    We consider here a logistic equation, modeling processes of nonlocal character both in the diffusion and proliferation terms. More precisely, for populations that propagate according to a Levy process and can reach resources in a neighborhood of their position, we compare (and find explicit threshold for survival) the local and nonlocal case. As ambient space, we can consider: bounded domains, periodic environments, transition problems, where the environment consists of a block of infinitesimal diffusion and an adjacent nonlocal one. In each of these cases, we analyze the existence/nonexistence of solutions in terms of the spectral properties of the domain. In particular, we give a detailed description of the fact that nonlocal populations may better adapt to sparse resources and small environments.
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    Minimization of a fractional perimeter-Dirichlet integral functional
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Caffarelli, Luis; Savin, Ovidiu; Valdinoci, Enrico
    We consider a minimization problem that combines the Dirichlet energy with the nonlocal perimeter of a level set. We obtain regularity results for the minimizers and for their free boundaries using blow-up analysis, density estimates, monotonicity formulas, Euler-Lagrange equations and extension problems.