Uniform estimates and limiting arguments for nonlocal minimal surfaces

By Luis Caffarelli, Enrico Valdinoci. 2011

241 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/9ab4d97a0004083da4c43b8839334e4d07c3809d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional p-Laplacian in RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setl (2015, background): “This type of operators arises in a quite natural way in many different applications, such as, continuum mechanics, phase transition phenomena, population dynamics, minimal surfaces and game theory, as they are the typical outcome of stochastically stabilization of Lévy processes, see for example [4,”
On an Isoperimetric Problem with a Competing Nonlocal Term II: The General Case (2012, background): “In fact, when ̨ ! n C 1 from below, the latter -converges, after a suitable rescaling, to the usual perimeter [3] (see also [9]).”
A Notion of Nonlocal Curvature (2014, background): “For instance, it is proved in [1, 7] that, as s ↗ 12 , the s -perimeter reduces to the classical perimeter, namely, lim s↗ (1)2 (1 − 2s)Pers(E ,Br ) = Per(E ,Br ) for a.”
Isoperimetry and Stability Properties of Balls with Respect to Nonlocal Energies (2014, background): “A regularity theory for such nonlocal minimal surfaces has been developed by several authors [10, 4, 18, 35, 12], while the relation of nonlocal perimeters with their local counterpart has been investigated in [8, 3].”
Graph properties for nonlocal minimal surfaces (2015, background): “Several analytic and geometric properties of s-minimal sets have been recently investigated, in terms, for instance, of asymptotics [18, 3, 9, 1, 13], regularity [10, 21, 15] and classification [4, 11].”
Curves and surfaces with constant nonlocal mean curvature: Meeting Alexandrov and Delaunay (2015, background): “Regarding nonlocal minimal curves in the plane, let us mention here that Savin and Valdinoci [22] have proved that the only minimizing nonlocal minimal curves in the plane are the straight lines.”
Weyl and Marchaud Derivatives: A Forgotten History (2017, background): “Other applications of the fractional calculus to the geometric measure theory can be found, for instance, in [16], [4], [15] and also coming to a very recently result [17] and [26], where the definition of nonlocal (fractional) perimeters is discussed.”
Gamma-convergence of nonlocal perimeter functionals (2010, methodology): “Motivated by this, Caffarelli and Valdinoci [5] studied the limiting properties of minimal sets for the s-perimeter as s→ 1−.”
Anisotropic fractional perimeters (2013, unknown): “Fractional s-perimeters have attracted increased attention in recent years (see [3, 6–8,11,14,15,36] and the references therein).”
Nonlocal Diffusion and Applications (2015, background): “When s Ñ 12 , the fractional perimeter Pers approaches the classical perimeter, as proved in [23].”
Concentration phenomenon for fractional nonlinear Schr\"{o}dinger equations (2013, background): “[32], finance [19], phase transition [2, 11, 65, 33, 40], stratified materials [61], anomalous diffusion [56], crystal dislocation [67, 41], soft thin films [47], semipermeable membranes and flame propagation [12], conservation laws [9], ultra-relativistic limits of quantum mechanics [34], quasi-geo”