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- ItemConcentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Dávila, Juan; del Pino, Manuael; Dipierro, Serena; Valdinoci, EnricoFor a smooth, bounded Euclidean domain, we consider a nonlocal Schrödinger equation with zero Dirichlet datum. We construct a family of solutions that concentrate at an interior point of the domain in the form of a scaling of the ground state in entire space. Unlike the classical case, the leading order of the associated reduced energy functional in a variational reduction procedure is of polynomial instead of exponential order on the distance from the boundary, due to the nonlocal effect. Delicate analysis is needed to overcome the lack of localization, in particular establishing the rather unexpected asymptotics for the Green function in the expanding domain.
- ItemRegularity and rigidity theorems for a class of anisotropic nonlocal operators(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Farina, Alberto; Valdinoci, EnricoWe consider here operators which are sum of (possibly) fractional derivatives, with (possibly different) order. The main constructive assumption is that the operator is of order 2 in one variable. By constructing an explicit barrier, we prove a Lipschitz estimate which controls the oscillation of the solutions in such direction with respect to the oscillation of the nonlinearity in the same direction. As a consequence, we obtain a rigidity result that, roughly speaking, states that if the nonlinearity is independent of a coordinate direction, then so is any global solution (provided that the solution does not grow too much at infinity). A Liouville type result then follows as a byproduct.
- 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.
- ItemRigidity of critical points for a nonlocal Ohta-Kawasaki energy(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Dipierro, Serena; Novaga, Matteo; Valdinoci, EnricoWe investigate the shape of critical points for a free energy consisting of a nonlocal perimeter plus a nonlocal repulsive term. In particular, we prove that a volume-constrained critical point is necessarily a ball if its volume is sufficiently small with respect to its isodiametric ratio, thus extending a result previously known only for global minimizers. We also show that, at least in one-dimension, there exist critical points with arbitrarily small volume and large isodiametric ratio. This example shows that a constraint on the diameter is, in general, necessary to establish the radial symmetry of the critical points.
- ItemBifurcation results for a fractional elliptic equation with critical exponent in Rn(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Dipierro, Serena; Medina, Maria; Peral, Ireneo; Valdinoci, EnricoIn this paper we study some nonlinear elliptic equations obtained as a perturbation of the problem with the fractional critical Sobolev exponent. To construct solutions to this equation, we use the Lyapunov-Schmidt reduction, that takes advantage of the variational structure of the problem. Some cases of the parameter range are particularly difficult, due to the lack of regularity of the associated energy functional, and we need to introduce a new functional setting and develop an appropriate fractional elliptic regularity theory.
- ItemA nonlocal concave-convex problem with nonlocal mixed boundary data(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Abdellaoui, Boumediene; Dieb, Abdelrazek; Valdinoci, EnricoThe aim of this paper is to study a nonlocal equation with mixed Neumann and Dirichlet external conditions. We prove existence, nonexistence and multiplicity of positive energy solutions and analyze the interaction between the concave-convex nonlinearity and the Dirichlet-Neumann data.
- ItemRelaxation times for atom dislocations in crystals(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Patrizi, Stefania; Valdinoci, EnricoWe study the relaxation times for a parabolic differential equation whose solution represents the atom dislocation in a crystal. The equation that we consider comprises the classical Peierls-Nabarro model as a particular case, and it allows also long range interactions. It is known that the dislocation function of such a model has the tendency to concentrate at single points, which evolve in time according to the external stress and a singular, long range potential. Depending on the orientation of the dislocation function at these points, the potential may be either attractive or repulsive, hence collisions may occur in the latter case and, at the collision time, the dislocation function does not disappear. The goal of this paper is to provide accurate estimates on the relaxation times of the system after collision. More precisely, we take into account the case of two and three colliding points, and we show that, after a small transition time subsequent to the collision, the dislocation function relaxes exponentially fast to a steady state. We stress that the exponential decay is somehow exceptional in nonlocal problems (for instance, the spatial decay in this case is polynomial). The exponential time decay is due to the coupling (in a suitable space/time scale) between the evolution term and the potential induced by the periodicity of the crystal.
- ItemOn stable solutions of boundary reaction-diffusion equations and applications to nonlocal problems with Neumann data(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Dipierro, Serena; Soave, Nicola; Valdinoci, EnricoWe study reaction-diffusion equations in cylinders with possibly nonlinear diffusion and possibly nonlinear Neumann boundary conditions. We provide a geometric Poincare-type inequality and classification results for stable solutions, and we apply them to the study of an associated nonlocal problem. We also establish a counterexample in the corresponding framework for the fractional Laplacian.
- ItemGround states and concentration phenomena for the fractional Schrödinger equation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Fall, Mouhamed Moustapha; Mahmoudi, Fethi; Valdinoci, EnricoWe consider here solutions of the nonlinear fractional Schrödinger equation. We show that concentration points must be critical points for the potential. We also prove that, if the potential is coercive and has a unique global minimum, then ground states concentrate suitably at such minimal point. In addition, if the potential is radial, then the minimizer is unique.
- 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.