Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Neckpinch singularities in fractional mean curvature flows


Authors: Eleonora Cinti, Carlo Sinestrari and Enrico Valdinoci
Journal: Proc. Amer. Math. Soc. 146 (2018), 2637-2646
MSC (2010): Primary 53C44, 35R11
DOI: https://doi.org/10.1090/proc/14002
Published electronically: February 21, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider the evolution of boundaries of sets by a fractional mean curvature flow. We show that for any dimension $ n\geqslant 2$, there exist embedded hypersurfaces in $ \mathbb{R}^n$ which develop a singularity without shrinking to a point. Such examples are well known for the classical mean curvature flow for $ n \geqslant 3$. Interestingly, when $ n=2$, our result provides instead a counterexample in the nonlocal framework to the well-known Grayson's Theorem, 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.


References [Enhancements On Off] (What's this?)

  • [1] Roberta Alessandroni and Carlo Sinestrari, Evolution of hypersurfaces by powers of the scalar curvature, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 (2010), no. 3, 541-571. MR 2722655
  • [2] Sigurd B. Angenent, Shrinking doughnuts, Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989) Progr. Nonlinear Differential Equations Appl., vol. 7, Birkhäuser Boston, Boston, MA, 1992, pp. 21-38. MR 1167827
  • [3] Begoña Barrios, Alessio Figalli, and Enrico Valdinoci, Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 13 (2014), no. 3, 609-639. MR 3331523
  • [4] X. Cabré, M. M. Fall, J. Solá-Morales and T. Weth, Curves and surfaces with constant nonlocal mean curvature: meeting Alexandrov and Delaunay, to appear in J. Reine Angew. Math.
  • [5] L. Caffarelli, J.-M. Roquejoffre, and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 (2010), no. 9, 1111-1144. MR 2675483
  • [6] Luis A. Caffarelli and Panagiotis E. Souganidis, Convergence of nonlocal threshold dynamics approximations to front propagation, Arch. Ration. Mech. Anal. 195 (2010), no. 1, 1-23. MR 2564467
  • [7] Luis Caffarelli and Enrico Valdinoci, Regularity properties of nonlocal minimal surfaces via limiting arguments, Adv. Math. 248 (2013), 843-871. MR 3107529
  • [8] Antonin Chambolle, Massimiliano Morini, and Marcello Ponsiglione, A nonlocal mean curvature flow and its semi-implicit time-discrete approximation, SIAM J. Math. Anal. 44 (2012), no. 6, 4048-4077. MR 3023439
  • [9] Antonin Chambolle, Massimiliano Morini, and Marcello Ponsiglione, Nonlocal curvature flows, Arch. Ration. Mech. Anal. 218 (2015), no. 3, 1263-1329. MR 3401008
  • [10] Antonin Chambolle, Matteo Novaga, and Berardo Ruffini, Some results on anisotropic fractional mean curvature flows, Interfaces Free Bound. 19 (2017), no. 3, 393-415. MR 3713894
  • [11] Yun Gang Chen, Yoshikazu Giga, and Shun'ichi Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom. 33 (1991), no. 3, 749-786. MR 1100211
  • [12] E. Cinti, J. Serra, and E. Valdinoci, Quantitative flatness results and $ BV$-estimates for stable nonlocal minimal surfaces, to appear in J. Differential Geom.
  • [13] Matteo Cozzi, On the variation of the fractional mean curvature under the effect of $ C^{1,\alpha}$ perturbations, Discrete Contin. Dyn. Syst. 35 (2015), no. 12, 5769-5786. MR 3393254
  • [14] Klaus Ecker, Regularity theory for mean curvature flow, Progress in Nonlinear Differential Equations and their Applications, vol. 57, Birkhäuser Boston, Inc., Boston, MA, 2004. MR 2024995
  • [15] L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom. 33 (1991), no. 3, 635-681. MR 1100206
  • [16] M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom. 23 (1986), no. 1, 69-96. MR 840401
  • [17] Matthew A. Grayson, A short note on the evolution of a surface by its mean curvature, Duke Math. J. 58 (1989), no. 3, 555-558. MR 1016434
  • [18] Matthew A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom. 26 (1987), no. 2, 285-314. MR 906392
  • [19] Gerhard Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), no. 1, 237-266. MR 772132
  • [20] Gerhard Huisken, A distance comparison principle for evolving curves, Asian J. Math. 2 (1998), no. 1, 127-133. MR 1656553
  • [21] Cyril Imbert, Level set approach for fractional mean curvature flows, Interfaces Free Bound. 11 (2009), no. 1, 153-176. MR 2487027
  • [22] Carlo Mantegazza, Lecture notes on mean curvature flow, Progress in Mathematics, vol. 290, Birkhäuser/Springer Basel AG, Basel, 2011. MR 2815949
  • [23] M. Sáez and E. Valdinoci, On the evolution by fractional mean curvature, to appear in Comm. Anal. Geom. (Available at http://arxiv.org/pdf/1511.06944.pdf.)
  • [24] Ovidiu Savin and Enrico Valdinoci, Regularity of nonlocal minimal cones in dimension 2, Calc. Var. Partial Differential Equations 48 (2013), no. 1-2, 33-39. MR 3090533

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 53C44, 35R11

Retrieve articles in all journals with MSC (2010): 53C44, 35R11


Additional Information

Eleonora Cinti
Affiliation: Weierstraß Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin, Germany
Address at time of publication: Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy
Email: elenora.cinti5@unibo.it

Carlo Sinestrari
Affiliation: Dipartimento di Ingegneria Civile e Ingegneria Informatica, Università di Roma Tor Vergata, Via del Politecnico, 00133 Rome, Italy
Email: sinestra@mat.uniroma2.it

Enrico Valdinoci
Affiliation: School of Mathematics and Statistics, University of Melbourne, 813 Swanston Street, Parkville VIC 3010, Australia – and – Dipartimento di Matematica, Università di Milano, Via Cesare Saldini 50, 20133 Milan, Italy – and – Weierstraß Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin, Germany
Email: enrico.valdinoci@wias-berlin.de

DOI: https://doi.org/10.1090/proc/14002
Keywords: Fractional perimeter, fractional mean curvature flow.
Received by editor(s): July 5, 2017
Published electronically: February 21, 2018
Additional Notes: The first author was supported by grants MTM2011-27739-C04-01 (Spain), 2009SGR345 (Catalunya), and by the ERC Starting Grant “AnOptSetCon” n. 258685.
The first and third authors were supported by the ERC Starting Grant “EPSILON”, Elliptic PDE’s and Symmetry of Interfaces and Layers for Odd Nonlinearities n. 277749.
The second author was supported by the group GNAMPA of INdAM Istituto Nazionale di Alta Matematica.
Communicated by: Lei Ni
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society