Neckpinch singularities in fractional mean curvature flows

Cinti, Eleonora; Sinestrari, Carlo; Valdinoci, Enrico

doi:10.1090/proc/14002

Cited by 29 publications

(24 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In higher dimensions this difference is even more contrasting, as the standard mean curvature of F ε is actually negative at these points. For other barriers of purely nonlocal character see also Proposition 7.3 in [20], Proposition 5 in [13], and Proposition 3.1 and Lemma 8.1 in [11].…”

Section: Proof Of Theoremmentioning

confidence: 97%

On the growth of nonlocal catenoids

Cozzi

Valdinoci

2020

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Proof Of Theoremmentioning

confidence: 97%

On the growth of nonlocal catenoids

Cozzi

Valdinoci

2020

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.

Self Cite

View full text Add to dashboard Cite

show abstract

“…In the case of the plane, it is known that smooth compact level curves never develop an interior, due to a result by Grayson on the evolution of regular compact curves. This result is no more valid for fractional mean curvature flow in the plane, as proved recently in [12]. We recall that examples of fattening of nonregular or noncompact curves in the plane for the mean curvature flow have been given in [3,13,15], where, in particular, the fattening of the evolution starting from the cross is proved.…”

mentioning

confidence: 88%

“…The K-curvature flow has been recently studied from different perspectives, in particular the case fractional mean curvature flow, taking into account geometric features such as conservation of the positivity of the fractional mean curvature, conservation of convexity and formation of neckpinch singularities, see [9,12,18].…”

mentioning

confidence: 99%

Fattening and nonfattening phenomena for planar nonlocal curvature flows

et al. 2018

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…We notice that the formation of neckpinch singularities also in low dimension is a treat shared by other nonlocal geometric flows, see . Nevertheless, the case in is conceptually quite different than that in , since the latter is scaling invariant and the nonlocal aspect of the curvature involves the global geometry of the set (while is not scaling invariant and the calculation of

κ_{r}

only involves a neighborhood of fixed side of a given point).…”

Section: Introductionmentioning

confidence: 96%

On a Minkowski geometric flow in the plane: Evolution of curves with lack of scale invariance

Dipierro

Novaga

Valdinoci

2018

J. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

We consider a planar geometric flow in which the normal velocity is a nonlocal variant of the curvature. The flow is not scaling invariant and in fact has different behaviors at different spatial scales, thus producing phenomena that are different with respect to both the classical mean curvature flow and the fractional mean curvature flow. In particular, we give examples of neckpinch singularity formation, and we discuss convexity properties of the evolution. We also take into account traveling waves for this geometric flow, showing that a new family of C1,1 and convex traveling sets arises in this setting.

show abstract

Neckpinch singularities in fractional mean curvature flows

Cited by 29 publications

References 27 publications

On the growth of nonlocal catenoids

On the growth of nonlocal catenoids

Fattening and nonfattening phenomena for planar nonlocal curvature flows

On a Minkowski geometric flow in the plane: Evolution of curves with lack of scale invariance

Contact Info

Product

Resources

About