Nonlocal minimal surfaces

By Luis Caffarelli, Jean-Michel Roquejoffre, Ovidiu Savin. 2009

483 citations recorded by the source corpus.

The de Giorgi theory for minimal surfaces consists in studying sets whose indicator function is a (local) minimum of the BV norm. In this paper we replace the BV norm by the $H^\sigma$ norm, with $\sigma<1/2$, and try to understand what the minimisers look like. Parallel to the de Giorgi theory we prove that, if the boundary of a minimiser is sufficiently flat in the unit ball, then it is a smooth piece of hypersurface.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d1139c2682647b15977d9658a6ed9b47de92348e

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Regularity properties of nonlocal minimal surfaces via limiting arguments (2011, background): “1) improves (while the classical minimal surfaces flatness εo is a fixed constant), we can apply the decrease of oscillation more and more times, so that in the vertical blow-up limit we get a Hölder graph, that is harmonic in viscosity sense (see [6]).”
Isoperimetry and Stability Properties of Balls with Respect to Nonlocal Energies (2014, methodology): “The parts of the regularity theory for nonlocal perimeter minimizers that are relevant to us have been developed in [7, 10] with the parameter s fixed.”
On doubly nonlocal fractional elliptic equations (2015, background): “Very recently, the nonlocal fractional counterpart of Kirchhoff-type problems has been considered (see [12] and [17, 22]).”
Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: A unified approach via fractional De Giorgi classes (2016, background): “ of sets with smooth boundaries are in Ws,p, if sp < 1. In fact, they play a central role in the theory of nonlocal perimeters recently developed by L. Caffarelli, J.-M. Roquejoﬀre and O. Savin in [17]. Therefore, the hopes of ﬁnding an appropriate generalization of (1.7) to fractional Sobolev sp”
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).”
Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces (2012, background): “Introduction Motivated by the structure of interphases arising in phase transition models with long range interactions, in [4] the authors introduced a nonlocal version of minimal surfaces.”
A Notion of Nonlocal Curvature (2014, background): “Of course this equation makes sense if E is smooth enough near p, so in general [4] has to deal with (13) in a suitable weak (and in fact viscosity) sense.”
Nonlocal minimal surfaces: Interior regularity, quantitative estimates and boundary stickiness (2016, methodology): “g this term may turn out to be important from the mathematical point of view, since this term may provide an innite contribution to the energy): for this reason, the nonlocal perimeter considered in [CRS10] is given by the sum of the rst three terms in (1.10), namely one denes Per s(E; ) := I(E\ c;E”
Multiplicity results for the non-homogeneous fractional p-Kirchhoff equations with concave–convex nonlinearities (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 [5–9] and the re”
All functions are locally $s$-harmonic up to a small error (2014, background): “rfaces. Indeed, the uniform density estimates satisﬁed by the nonlocal minimal surfaces prescribe a severe geometric restriction that prevent the formation of sharp edges and thin spikes. We refer to [2] for the deﬁnition of nonlocal minimal surfaces and for their density properties: as a matter off”
On an Isoperimetric Problem with a Competing Nonlocal Term II: The General Case (2012, background): “n; n C 1/ leads to a nonlocal isoperimetric problem in which P ̨ gives a generalized notion of the perimeter of [8].”
Nonlocal Diffusion and Applications (2015, background): “2 in [17] for the proof of this statement).”