On the evolution by fractional mean curvature

Sáez, Mariel; Valdinoci, Enrico

doi:10.4310/cag.2019.v27.n1.a6

Cited by 31 publications

(34 citation statements)

References 31 publications

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“…Thus, we have shown that −(P 1 ∩ S) ⊆ P 3 ∩ S, up to negligible sets. Combining this with (22), we conclude the proof of (21). Also, we notice that H ∩ P 1 ∪ P 2 = ∅ and K ⊆ P 1 .…”

Section: The Nonlocal Curvature Of Perturbed Stripssupporting

confidence: 67%

“…Other recent contributions in the study of the fractional (and more general nonlocal) mean curvature flows are the articles [6,10,22]. In [6], the convergence of a class of threshold dynamics approximations to moving fronts was established.…”

Section: Introductionmentioning

confidence: 99%

“…Lemma 8 (Lemma 2 and Corollary 3 in[22]). The fractional mean curvature of the ball of radius R is equal, up to dimensional constants, to R −s .More precisely, for any x ∈ ∂B 1 H s B 1 (x) =ω, for someω > 0, and for any x ∈ ∂B R (0),…”

mentioning

confidence: 98%

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Neckpinch singularities in fractional mean curvature flows

Cinti

Sinestrari

Valdinoci

2018

Proc. Amer. Math. Soc.

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In this paper we consider the evolution of boundaries of sets by a fractional mean curvature flow. We show that, for any dimension n 2, there exist embedded hypersurfaces in R n which develop a singularity without shrinking to a point. Such examples are well known for the classical mean curvature flow for n 3. Interestingly, when n = 2, our result provides instead a counterexample in the nonlocal framework to the well known Grayson's Theorem [17], which states that any smooth embedded curve in the plane evolving by (classical) MCF shrinks to a point. The essential step in our construction is an estimate which ensures that a suitably small perturbation of a thin strip has positive fractional curvature at every boundary point.2010 Mathematics Subject Classification. 53C44, 35R11.

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Section: The Nonlocal Curvature Of Perturbed Stripssupporting

confidence: 67%

Section: Introductionmentioning

confidence: 99%

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Neckpinch singularities in fractional mean curvature flows

Cinti

Sinestrari

Valdinoci

2018

Proc. Amer. Math. Soc.

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“…is finite for every R > 0 and also that lim R→+∞ c(R) = 0, see [8,16]. For the computation in the case of fractional kernels, see [18].…”

Section: Appendix a Viscosity Solutions And Geometric Barriersmentioning

confidence: 99%

Fattening and nonfattening phenomena for planar nonlocal curvature flows

et al. 2018

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We discuss fattening phenomenon for the evolution of sets according to their nonlocal curvature. More precisely, we consider a class of generalized curvatures which correspond to the first variation of suitable nonlocal perimeter functionals, defined in terms of an interaction kernel K, which is symmetric, nonnegative, possibly singular at the origin, and satisfies appropriate integrability conditions.We prove a general result about uniqueness of the geometric evolutions starting from regular sets with positive K-curvature in R n and we discuss the fattening phenomenon in R 2 for the evolution starting from the cross, showing that this phenomenon is very sensitive to the strength of the interactions. As a matter of fact, we show that the fattening of the cross occurs for kernels with sufficiently large mass near the origin, while for kernels that are sufficiently weak near the origin such a fattening phenomenon does not occur.We also provide some further results in the case of the fractional mean curvature flow, showing that strictly starshaped sets in R n have a unique geometric evolution.Moreover, we exhibit two illustrative examples in R 2 of closed nonregular curves, the first with a Lipschitztype singularity and the second with a cusp-type singularity, given by two tangent circles of equal radius, whose evolution develops fattening in the first case, and is uniquely defined in the second, thus remarking the high sensitivity of the fattening phenomenon in terms of the regularity of the initial datum. The latter example is in striking contrast to the classical case of the (local) curvature flow, where two tangent circles always develop fattening.As a byproduct of our analysis, we provide also a simple proof of the fact that the cross in R 2 is not a K-minimal set for the nonlocal perimeter functional associated to K.

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“…We denote by B r the ball of center 0 and radius r > 0, and we let B r (x) = x + B r . Moreover, we recall that, by the scale invariance of fractional mean curvature, for all x r ∈ ∂B r there holds (see [19, Lemma 2])…”

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confidence: 99%

Symmetric Self-Shrinkers for the Fractional Mean Curvature Flow

Cesaroni

Novaga

2019

J Geom Anal

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We show existence of homothetically shrinking solutions of the fractional mean curvature flow, whose boundary consists in a prescribed number of concentric spheres. We prove that all these solutions, except from the ball, are dynamically unstable. Existence of symmetric self-shrinkersWe start with a technical result which will be useful in the sequel. We denote by B r the ball of center 0 and radius r > 0, and we let B r (x) = x + B r . Moreover, we recall that, by the scale invariance of fractional mean curvature, for all x r ∈ ∂B r there holds (see [19, Lemma 2])

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On the evolution by fractional mean curvature

Cited by 31 publications

References 31 publications

Neckpinch singularities in fractional mean curvature flows

Neckpinch singularities in fractional mean curvature flows

Fattening and nonfattening phenomena for planar nonlocal curvature flows

Symmetric Self-Shrinkers for the Fractional Mean Curvature Flow

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