Abstract
We investigate the stability of the evolution by anysotropic and crystalline curvature starting from an initial surface equal to the Wulff shape. It is well known that the Wulff shape evolves selfsimilarly according to the law V = -k ø n ø .Here the index ø refers to the underlying anisotropy described by the Wulff shape, so that k ø is the relative mean curvature and n ø is the Cahn-Hoffmann conormal vector field. Such selfsimilar evolution is also known to be stable under small perturbations of the initial surface in the isotropic setting (the Wulff shape is a sphere) or in 2D if the underlying anisotropy is symmetric. We show that this evolution is unstable for some specific choices of the Wulff shape both rotationally symmetric and fully crystalline.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
F. Almgren and J.E. Taylor: Flat flow is motion by crystalline curvature for curves with crystalline energies, J. Differential Geom., vol 42, 1995, 1–22
Bellettini and M. Novaga: Approximation and comparison for nonsmooth anisotropic motion by mean curvature in R N, Math. Models Methods Appl. Sci., vol 10, 2000, 1–10
G. Bellettini, M. Novaga, and M. Paolini: Facet-breaking for three dimensional crystals evolving by mean curvature, Interfaces and Free Boundaries, vol 1, 1999, 39–55
M. Paolini: On a crystalline variational problem, part I: first variation and global L ∞ -regularity, Arch. Ration. Mech. Anal., vol 3, 2001, 165–191
M. Paolini: On a crystalline variational problem, part II: BV -regularity and structure of minimizers on facets, Arch. Ration. Mech. Anal., vol 3, 2001, 193–217
M. Paolini: Characterization of facet-breaking for nonsmooth mean curvature flow in the convex case, Interfaces and Free Boundaries, vol 3, 2001, 415–446
G. Bellettini and M. Paolini: Some results on minimal barriers in the sense of De Giorgi applied to driven motion by mean curvature, Rend. Accad. Naz. Sci. XL Mem. Mat. (5), vol 19, 1995, 43–67
M. Paolini: Quasi-optimal error estimates for the mean curvature flow with a forcing term, Differential Integral Equations, vol 8, 1995, 735–752
M. Paolini: Anisotropic motion by mean curvature in the context of Finsler geometry, Hokkaido Math. J., vol 25, 1996, 537–566
Y.G. Chen, Y. Giga, and S. Goto: Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., vol 33, 1991, 749–78
E. De Giorgi: New conjectures on flow by mean curvature, Nonlinear variational problems and partial differential equations (Isola d’Elba, 1990), Longman Sci. Tech., Harlow, 1995, 120–128
P. De Mottoni and M. Schatzman: Geometrical evolution of developed interfaces, Trans. Amer. Math. Soc., vol 347, 1995, 1533–1589
K. Deckelnick and G. Dziuk: Error estimates for a semi-implicit fully discrete finite element scheme for the mean curvature flow of graphs, Interfaces and Free Boundaries, vol 2, 2000, 341–359
L.C. Evans and J. Spruck: Motion of level sets by mean curvature. I, J. Differential Geom., vol 33, 1991, 635–681
F. Fierro and M. Paolini: Numerical evidence of fattening for the mean curvature flow, Math. Models Methods Appl. Sci., vol 6, 1996, 793–813
M.E. Gage: Evolving plane curves by curvature in relative geometries, Duke Math. J., vol 72, 1993, 441–466
M.A. Grayson: The heat equation shrinks embedded plane curves to round points,J. Differential Geom., vol 26, 1987, 285–314
G. Huisken: Local and global behaviour of hypersurfaces moving by mean curvature, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990), Amer. Math. Soc., Providence, RI, 1993
R.H. Nochetto, M. Paolini, and C. Verdi: Quadratic rate of convergence for curvature dependent smooth interfaces: a simple proof, Appl. Math. Lett., vol 7, 1994, 59–63
M. Paolini: A dynamic mesh algorithm for curvature dependent evolving interfaces, J. Comput. Phys., vol 123, 1996, 296–310
R.H. Nochetto and C. Verdi: Combined effect of explicit time-stepping and quadrature for curvature driven flows, Numer. Math., vol 74, 1996, 105–136
M. Paolini: Convergence past singularities for a fully discrete approximation of curvature driven interfaces, SIAM J. Numer. Anal., vol 34, 1997, 490–512
A. Stancu: Asymptotic behavior of solutions to a crystalline flow, Hokkaido Math. J., vol 27, 1998, 303–320
J.E. Taylor: Constructions and conjectures in crystalline nondifferential geometry, Differential Geometry, Pitman Monographs and Surveys in Pure and Applied Math. 52 (B. Lawson and K. Tenenblat, eds.), Longman Scientific and Technical, 1991, 321–336
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer Basel AG
About this paper
Cite this paper
Paolini, M., Pasquarelli, F. (2002). Unstable Crystalline Wulff Shapes in 3D. In: dal Maso, G., Tomarelli, F. (eds) Variational Methods for Discontinuous Structures. Progress in Nonlinear Differential Equations and Their Applications, vol 51. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8193-7_10
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8193-7_10
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9470-8
Online ISBN: 978-3-0348-8193-7
eBook Packages: Springer Book Archive