1,721,022 research outputs found
Regularity Results for Nonlocal Minimal Surfaces
Full text linkIn this note, we present some recent results in the study of nonlocal minimal surfaces. The notion of nonlocal minimal surface was introduced by Caffarelli, Roquejoffre, and Savin, they are boundaries of sets which minimize the nonlocal (or fractional) perimeter. In the last years, much interest has been devoted to the study of their regularity properties. Similarly to the classical local setting, a crucial ingredient in the study of regularity is the classification of minimal cones. In the nonlocal setting, only partial results are available, dealing mainly with the low-dimensional case. We describe the main achievements in the field, focusing in particular on the difference with respect to the classical theory and in the difficulties which arise due to the nonlocal character of the problem
Risultati di regolarità per insiemi isoperimetrici con densità
Full text linkIn this note, we present some recent regularity results for sets which minimize a weighted notion of perimeter under a weighted volume constraint. We focus on the case of two different densities which are merely alpha-Holder continuous, and describe what are the main issues and techniques used in order to establish the optimal regularity C1, alpha/(2-alpha) for the reduced boundary of such sets.In questa nota, presentiamo alcuni recenti risultati di regolarità per insiemi che minimizzano una nozione pesata di perimetro sotto un vincolo di volume pesato. Ci focalizziamo sul caso di due densità diverse, che siano solo Holderiane di ordine alpha e descriviamo quali sono le maggiori difficoltà e le tecniche usate per provare la regolarità ottimale C1, alpha/(2-alpha) per la frontiera ridotta di tali insiemi
Flatness results for nonlocal phase transitions
No full textWe consider a nonlocal version of the Allen-Cahn equation, which models phase transitions problems. In the classical setting, the connection between the Allen-Cahn equation and the classification of entire minimal surfaces is well known and motivates a celebrated conjecture by E. De Giorgi on the one-dimensional symmetry of bounded monotone solutions to the (classical) Allen-Cahn equation up to dimension 8. In this work, we present some recent results in the study of the nonlocal analogue of this phase transition problem. In particular we describe the results obtained in several contributions where the classification of certain entire bounded solutions to the fractional Allen-Cahn equation has been obtained. Moreover we describe the connection between the fractional Allen-Cahn equation and the fractional perimeter functional, and we present also some results in the classifications of nonlocal minimal surfaces
The fractional mean curvature flow
Full text linkIn this note, we present some recent results in the study of the fractional mean curvature flow, that is a geometric evolution of the boundary of a set whose speed is given by the fractional mean curvature. The flow under consideration is of nonlocal type and presents several interesting difference with respect to the classical mean curvature flow. We will describe the main contributions in this field, with particular emphasis on some tipically nonlocal behaviors which are in contrast with the classical local case.In questa nota, presentiamo alcuni risultati recenti riguardanti lo studio del moto per curvatura media frazionaria, che descrive l'evoluzione del bordo di un insieme la cui velocita è data dalla curvatura media frazionaria. Tale flusso ha natura nonlocale e presenta alcune interessanti differenze rispetto al flusso per curvatura media classica. Descriviamo i principali contributi in questo ambito, con particolare enfasi ai comportamente tipicamente nonlocali che sono in contrasto col caso classico
Energy estimates and 1D symmetry for nonlinear equations involving the half-Laplacian
Full text linkWe establish sharp energy estimates for some solutions, such as global minimizers, monotone solutions and saddle-shaped solutions, of the fractional nonlinear equation (−Δ)1/2u=f(u) in Rn. Our energy estimates hold for every nonlinearity f and are sharp since they are optimal for one-dimensional solutions, that is, for solutions depending only on one Euclidian variable. As a consequence, in dimension n=3, we deduce the one-dimensional symmetry of every global minimizer and of every monotone solution. This result is the analog of a conjecture of De Giorgi on one-dimensional symmetry for the classical equation −Δu=f(u) in Rn
Esame ultrasonografico transcutaneo dello sviluppo follicolare durante l’estro nella scrofa.Summa:
No full textCONVEX SETS EVOLVING BY VOLUME-PRESERVING FRACTIONAL MEAN CURVATURE FLOWS
Full text linkWe consider the volume-preserving geometric evolution of the boundary of a set under fractional mean curvature. We show that smooth convex solutions maintain their fractional curvatures bounded for all times, and the long-time asymptotics approach round spheres. The proofs are based on a priori estimates on the inner and outer radii of the solutions
Optimal regularity of isoperimetric sets with Hölder densities
Full text linkWe establish a regularity result for optimal sets of the isoperimetric problem with double density under mild Holder regularity assumptions on the density functions. Our main Theorem improves some previous results and allows to reach in any dimension the regularity class C-1, (a) (2-a) . This class is indeed the optimal one for local minimizers of variational functionals with an integrand that depends a-H & ouml;lder continuous on the minimizer itself, and as such can (the boundary of) the isoperimetric set be locally written (with additional constraint)
A quantitative stability estimate for the fractional Faber-Krahn inequality
No full textWe prove a quantitative version of the Faber-Krahn inequality for the first eigenvalue of the fractional Dirichlet-Laplacian of order s. This is done by using the so-called Caffarelli-Silvestre extension and adapting to the nonlocal setting a trick by Hansen and Nadirashvili. The relevant stability estimate comes with an explicit constant, which is stable as the fractional order of differentiability goes to 1
One-Dimensional Symmetry for the Solutions of a Three-Dimensional Water Wave Problem
Full text linkWe prove a one-dimensional symmetry result for a weighted Dirichlet-to-Neumann problem arising in a model for water waves in dimension 3. More precisely we prove that minimizers and bounded monotone solutions depend on only one Euclidean variable. The analogue of this result for the 2-dimensional case (and without weights) was established in 16. In this paper a crucial ingredient in the proof is given by an energy estimate for minimizers obtained via a comparison argument
- …
