Browsing by Author "Valdinoci, Enrico"
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- Item1D symmetry for semilinear pdes from the limit interface of the solution(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Farina, Alberto; Valdinoci, EnricoWe 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.
- ItemAll functions are locally s-harmonic up to a small error(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Dipierro, Serena; Savin, Ovidiu; Valdinoci, EnricoWe 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.
- ItemAsymptotic expansions of the contact angle in nonlocal capillarity problems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Dipierro, Serena; Maggi, Francesco; Valdinoci, EnricoWe consider a family of nonlocal capillarity models, where surface tension is modeled by exploiting a family of fractional interaction kernels The fractional Young's law (contact angle condition) predicted by these models coincides, in the limit, with the classical Young's law determined by the Gauss free energy. Here we refine this asymptotics by showing that, for s close to 1, the fractional contact angle is always smaller than its classical counterpart when the relative adhesion coefficient is negative, and larger if it is positive. In addition, we address the asymptotics of the fractional Young's law in the limit case s close to 0 of interaction kernels with heavy tails. Interestingly, forsmall s, the dependence of the contact angle from the relative adhesion coefficient becomes linear.
- ItemAsymptotically linear problems driven by fractional Laplacian operators(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Fiscella, Alessio; Servadei, Raffaella; Valdinoci, EnricoIn 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.
- 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.
- ItemBoundary behavior of nonlocal minimal surfaces(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Dipierro, Verena; Savin, Ovidiu; Valdinoci, EnricoWe consider the behavior of the nonlocal minimal surfaces in the vicinity of the boundary. By a series of detailed examples, we show that nonlocal minimal surfaces may stick at the boundary of the domain, even when the domain is smooth and convex. This is a purely nonlocal phenomenon, and it is in sharp contrast with the boundary properties of the classical minimal surfaces. In particular, we show stickiness phenomena to half-balls when the datum outside the ball is a small half-ring and to the side of a two-dimensional box when the oscillation between the datum on the right and on the left is large enough. When the fractional parameter is small, the sticking effects may become more and more evident. Moreover, we show that lines in the plane are unstable at the boundary: namely, small compactly supported perturbations of lines cause the minimizers in a slab to stick at the boundary, by a quantity that is proportional to a power of the perturbation. In all the examples, we present concrete estimates on the stickiness phenomena. Also, we construct a family of compactly supported barriers which can have independent interest.
- 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.
- ItemChaotic orbits for systems of nonlocal equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Dipierro, Serena; Patrizi, Stefania; Valdinoci, EnricoWe consider a system of nonlocal equations driven by a perturbed periodic potential. We construct multibump solutions that connect one integer point to another one in a prescribed way. In particular, heteroclinc, homoclinic and chaotic trajectories are constructed. This is the first attempt to consider a nonlocal version of this type of dynamical systems in a variational setting and the first result regarding symbolic dynamics in a fractional framework.
- ItemA class of unstable free boundary problems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Dipierro, Serena; Karakhanyan, Aram; Valdinoci, EnricoWe consider the free boundary problem arising from an energy functional which is the sum of a Dirichlet energy and a nonlinear function of either the classical or the fractional perimeter. The main difference with the existing literature is that the total energy is here a nonlinear superposition of the either local or nonlocal surface tension effect with the elastic energy. In sharp contrast with the linear case, the problem considered in this paper is unstable, namely a minimizer in a given domain is not necessarily a minimizer in a smaller domain. We provide an explicit example for this instability. We also give a free boundary condition, which emphasizes the role played by the domain in the geometry of the free boundary. In addition, we provide density estimates for the free boundary and regularity results for the minimal solution. As far as we know, this is the first case in which a nonlinear function of the perimeter is studied in this type of problems. Also, the results obtained in this nonlinear setting are new even in the case of the local perimeter, and indeed the instability feature is not a consequence of the possibly nonlocality of the problem, but it is due to the nonlinear character of the energy functional.
- 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.
- ItemContinuity and density results for a one-phase nonlocal free boundary problem(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Dipierro, Serena; Valdinoci, EnricoWe consider a one-phase nonlocal free boundary problem obtained by the superposition of a fractional Dirichlet energy plus a nonlocal perimeter functional. We prove that the minimizers are Hölder continuous and the free boundary has positive density from both sides. For this, we also introduce a new notion of fractional harmonic replacement in the extended variables and we study its basic properties.
- 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.
- ItemCrystal dislocations with different orientations and collisions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Patrizi, Stefania; Valdinoci, EnricoWe 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.
- ItemDefinition of fractional Laplacian for functions with polynomial growth(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Dipierro, Serena; Savin, Ovidiu; Valdinoci, EnricoWe introduce a notion of fractional Laplacian for functions which grow more than linearly at infinity. In such case, the operator is not defined in the classical sense: nevertheless, we can give an ad-hoc definition which can be useful for applications in various fields, such as blowup and free boundary problems. In this setting, when the solution has a polynomial growth at infinity, the right hand side of the equation is not just a function, but an equivalence class of functions modulo polynomials of a fixed order. We also give a sharp version of the Schauder estimates in this framework, in which the full smooth Hölder norm of the solution is controlled in terms of the seminorm of the nonlinearity. Though the method presented is very general and potentially works for general nonlocal operators, for clarity and concreteness we focus here on the case of the fractional Laplacian.
- ItemA density property for fractional weighted Sobolev spaces(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Dipierro, Serena; Valdinoci, EnricoIn this paper we show a density property for fractional weighted Sobolev spaces. That is, we prove that any function in a fractional weighted Sobolev space can be approximated by a smooth function with compact support. The additional difficulty in this nonlocal setting is caused by the fact that the weights are not necessarily translation invariant.
- 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.
- ItemDislocation 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, EnricoWe 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.
- ItemThe effect on Fisher-KPP propagation in a cylinder with fast diffusion on the boundary(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Rossi, Luca; Tellini, Andrea; Valdinoci, EnricoIn this paper we consider a reaction-diffusion equation of Fisher-KPP type inside an infinite cylindrical domain in RN+1, coupled with a reaction-diffusion equation on the boundary of the domain, where potentially fast diffusion is allowed. We will study the existence of an asymptotic speed of propagation for solutions of the Cauchy problem associated with such system, as well as the dependence of this speed on the diffusivity at the boundary and the amplitude of the cylinder. When N = 1 the domain reduces to a strip between two straight lines. This models the effect of two roads with fast diffusion on a strip-shaped field bounded by them.
- ItemFractional elliptic problems with critical growth in the whole of Rn(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Dipierro, Serena; Medina, María; Valdinoci, EnricoWe study a nonlinear and nonlocal elliptic equation. The problem has a variational structure, and this allows us to find a positive solution by looking at critical points of a suitable energy functional. In particular, in this paper, we find a local minimum and a mountain pass solution of this functional. One of the crucial ingredient is a Concentration-Compactness principle. Some difficulties arise from the nonlocal structure of the problem and from the fact that we deal with an equation in the whole of the space (and this causes lack of compactness of some embeddings). We overcome these difficulties by looking at an equivalent extended problem.
- 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.