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- ItemGradient 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, EnricoWe 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.
- ItemNonlocal phase transitions: Rigidity results and anisotropic geometry(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Dipierro, Serena; Serra, Joaquim; Valdinoci, EnricoFinally, we consider a nonlocal equation with a multiwell potential, motivated by models arising in crystal dislocations, and we construct orbits exhibiting symbolic dynamics, inspired by some classical results by Paul Rabinowitz.
- ItemA nonlinear free boundary problem with a self-driven Bernoulli condition(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Dipierro, Serena; Karakhanyan, Aram; Valdinoci, EnricoWe study a Bernoulli type free boundary problem with two phases and a nonlinear energy superposition. We show that, for this problem, the Bernoulli constant, which determines the gradient jump condition across the free boundary, is of global type and it is indeed determined by the weighted volumes of the phases. In particular, the Bernoulli condition that we obtain can be seen as a pressure prescription in terms of the volume of the two phases of the minimizer itself (and therefore it depends on the minimizer itself and not only on the structural constants of the problem). Another property of this type of problems is that the minimizer in a given domain is not necessarily a minimizer in a smaller subdomain, due to the nonlinear structure of the problem. Due to these features, this problem is highly unstable as opposed to the classical case studied by Alt, Caffarelli and Friedman. It also interpolates the classical case, in the sense that the blow-up limits are minimizers of the Alt-Caffarelli-Friedman functional. Namely, the energy of the problem somehow linearizes in the blow-up limit. We also develop a detailed optimal regularity theory for the minimizers and for their free boundaries.
- ItemThe Dirichlet problem for nonlocal operators with kernels: Convex and nonconvex domains(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Ros-Oton, Xavier; Valdinoci, EnricoWe study the interior regularity of solutions to a Dirichlet problem for anisotropic operators of fractional type. A prototype example is given by the sum of one-dimensional fractional Laplacians in fixed, given directions. We prove here that an interior differentiable regularity theory holds in convex domains. When the spectral measure is a bounded function and the domain is smooth, the same regularity theory applies. In particular, solutions always possess a classical first derivative. The assumptions on the domain are sharp, since if the domain is not convex and the spectral measure is singular, we construct an explicit counterexample.
- ItemGevrey regularity for integro-differential operators(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Albanese, Guglielmo; Fiscella, Alessio; Valdinoci, EnricoWe prove a regularity theory in the Gevrey class for a family of nonlocal differential equations of elliptic type.
- ItemNonlocal minimal surfaces: Interior regularity, quantitative estimates and boundary stickiness(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Dipierro, Serena; Valdinoci, EnricoWe consider surfaces which minimize a nonlocal perimeter functional and we discuss their interior regularity and rigidity properties, in a quantitative and qualitative way, and their (perhaps rather surprising) boundary behavior. We present at least a sketch of the proofs of these results, in a way that aims to be as elementary and self contained as possible, referring to the papers [CRS10, SV13, CV13, BFV14,FV,DSV15,CSV16] for full details.
- ItemOn the evolution by fractional mean curvature(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Sáez, Mariel; Valdinoci, EnricoIn this paper we study smooth solutions to a fractional mean curvature flow equation. We establish a comparison principle and consequences such as uniqueness and finite extinction time for compact solutions. We also establish evolutions equations for fractional geometric quantities that yield preservation of certain quantities (such as positive fractional curvature) and smoothness of graphical evolutions.
- ItemA critical Kirchhoff type problem involving a non-local operator(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Fiscella, Alessio; Valdinoci, EnricoWe show the existence of non-negative solutions for a Kirchhoff type problem driven by a non-local integrodifferential operator.
- ItemCapillarity problems with nonlocal surface tension energies(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Maggi, Francesco; Valdinoci, EnricoA dynamic large deviations principle for a countable reaction network including coagulation-fragmentation models is proved. The rate function is represented as the infimal cost of the reaction fluxes and a minimiser for this variational problem is shown to exist. A weak reversibility condition is used to control the boundary behaviour and to guarantee a representation for the optimal fluxes via a Lagrange multiplier that can be used to construct the changes of measure used in standard tilting arguments. Reflecting the pure jump nature of the approximating processes, their paths are treated as elements of a BV function space.
- ItemA one-dimensional symmetry result for solutions of an integral equation of convolution type(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Hamel, François; Valdinoci, EnricoWe consider an integral equation in the plane, in which the leading operator is of convolution type, and we prove that monotone (or stable) solutions are necessarily one-dimensional.