Regularity properties of nonlocal minimal surfaces via limiting arguments

By Luis Caffarelli, Enrico Valdinoci. 2011

142 citations recorded by the source corpus.

Abstract We prove an improvement of flatness result for nonlocal minimal surfaces which is independent of the fractional parameter s when s → 1 − . As a consequence, we obtain that all the nonlocal minimal cones are flat and that all the nonlocal minimal surfaces are smooth when the dimension of the ambient space is less or equal than 7 and s is close to 1.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f1839d78ddf81a07a7f08ab26b34d16154859c21

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Nonlocal Diffusion and Applications (2015, background): “Further regularity results of the s minimal surfaces can be found in [24].”
A Notion of Nonlocal Curvature (2014, background): “2We refer here to sets as in footnote 1 in [6].”
Gamma-convergence of nonlocal perimeter functionals (2010, background): “The proof of the lim sup inequality, which is typically constructive and by density, is slightly different from the analogous results in [5], since we approximate with polyhedra, rather than C1,α sets.”
Regularity of nonlocal minimal cones in dimension 2 (2012, background): “In [4], the authors used these results to show that if s is sufficiently close to 1 the same holds for s-minimal surfaces i.”
Nonlocal quantitative isoperimetric inequalities (2015, background): “The notion of fractional perimeter has been introduced in [36, 9] and it has been extensively studied in several recent papers (see for instance [24, 33, 34, 11, 18, 15] and references therein).”
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].”
Fractional Laplacian in conformal geometry (2010, background): “In particular, we pose the question of finding the relationship between the fractional order mean curvature very recently defined (see Caffarelli-Roquejoffre-Savin [4], Caffarelli-Valdinoci [8]), and our notion of Qγ curvature coming from the fractional Paneitz operator.”