Regularity of nonlocal minimal cones in dimension 2

By Ovidiu Savin, Enrico Valdinoci. 2012

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We show that the only nonlocal s-minimal cones in $${\mathbb{R}^2}$$ are the trivial ones for all $${s \in (0, 1).}$$ As a consequence we obtain that the singular set of a nonlocal minimal surface has at most n − 3 Hausdorff dimension.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/3e0d393f864d4e91c5e4c4169fc84cf13eaf49ee

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

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].”
Curves and surfaces with constant nonlocal mean curvature: Meeting Alexandrov and Delaunay (2015, background): “For N D 2 and any ˛, Savin and Valdinoci [22] established that they are C1.”
Regularity properties of nonlocal minimal surfaces via limiting arguments (2011, background): “For other recent regularity results for nonlocal minimal surfaces see [3,13].”
Graph properties for nonlocal minimal surfaces (2015, background): “As customary in the calculus of variation literature, one says that E is s-minimal in a bounded open set Ω if Pers(E,Ω) +∞ and Pers(E,Ω) 6 Pers(F,Ω) among all the sets F which coincide with E outside Ω (with a slight abuse of language, when Ω is unbounded, we say that E is s-minimal in Ω if it is…”
Nonlocal Diffusion and Applications (2015, methodology): “m6.2.1will be completely diﬀerent than the one of Proposition6.2.3anditwillrelyonanappropriatedomainperturbationargument. TheproofofTheorem6.2.1thatwepresenthereisactuallydiﬀerentthanthe originalonein[99]. Indeed, in[99], theresultwasprovedbyusingtheharmonic extension for the fractional Laplacian. H”
Rigidity and sharp stability estimates for hypersurfaces with constant and almost-constant nonlocal mean curvature (2015, unknown): “Ωc dx dy |x− y|n+2 s , Ω = R \ Ω , corresponding to some value of s ∈ (0, 1/2) (see for instance [18, 2, 25, 6, 22, 20, 21, 19]).”
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).”