Abstract
We prove a local existence and uniqueness result of crystalline mean curvature flow starting from a compact convex admissible set in . This theorem can handle the facet breaking/bending phenomena, and can be generalized to any anisotropic mean curvature flow. The method provides also a generalized geometric evolution starting from any compact convex set, existing up to the extinction time, satisfying a comparison principle, and defining a continuous semigroup in time. We prove that, when the initial set is convex, our evolution coincides with the flat φ-curvature flow in the sense of Almgren-Taylor-Wang. As a by-product, it turns out that the flat φ-curvature flow starting from a compact convex set is unique.
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Almgren, F.J., Taylor, J.E.: Flat flow is motion by crystalline curvature for curves with crystalline energies. J. Differential Geom. 42, 1–22 (1995)
Almgren, J.F., Talylor, J.E., Wang, L.-H.: Curvature-driven flows: a variational approach. SIAM J. Control Optim. 31, 387–438 (1993)
Altschuler, S., Angenent, S.B., Giga, Y.: Mean curvature flow through singularities for surfaces of rotation. J. Geom. Anal. 5, 293–358 (1995)
Alvarez, O., Lasry, J.M., Lions, P.L.: Convex viscosity solutions and state constraints. J. Math. Pures Appl. 76, 265–288 (1997)
Amar, M., Bellettini, G.: A notion of total variation depending on a metric with discontinuous coefficients. Ann. Inst. Henri Poincaré 11, 91–133 (1994)
Amborsio, L.: Corso introduttivo alla teoria geometrica della misura ed alle superfici minime. Scuola Normale Superiore di Pisa, 1997
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs, 2000
Angenent, S., Chopp, D.L., Ilmanen, T.: Nonuniqueness of mean curvature flow in R3. Comm. Partial Differential Equations. 20, 1937–1958 (1995)
Anzellotti, G.: Pairings between measures and bounded functions and compensated compactness. Ann. Mat. Pura Appl. 135, 293–318 (1983)
Bellettini, G., Caselles, V., Novaga, M.: The Total Variation Flow in . J. Differential Equations. 184, 475–525 (2002)
Bellettini, G., Novaga, M., Paolini, M.: On a crystalline variational problem I. First variation and global L ∞ regularity. Arch. Ration. Mech. Anal. 157, 165–191 (2001)
Bellettini, G., Novaga, M., Paolini, M.: Facet-breaking for three-dimensional crystals evolving by mean curvature. Interfaces Free Bound. 1, 39–55 (1999)
Bellettini, G., Novaga, M., Paolini, M.: Characterization of facet–breaking for nonsmooth mean curvature flow in the convex case. Interfaces Free Bound. 3, 415–446 (2001)
Bellettini, G., Novaga, M.: Approximation and comparison for non-smooth anisotropic motion by mean curvature in . Math. Mod. Meth. Appl. Sc. 10, 1–10 (2000)
Bence, J., Merriman, B., Osher, S.: Diffusion Generated Motion by Mean Curvature. In: AMS Selected Lectures in Math., The Comput. Crystal Grower's Workshop, edited by J. Taylor. Am. Math Soc., Providence, RI, 1993, p. 73
Brezis, H.: Operateurs Maximaux Monotones et Semi-Groups de Contractions dans les Espaces de Hilbert. North Holland, 1973
Cahn, J.W., Handwerker, C.A., Taylor, J.E.: Geometric models of crystal growth. Acta Metall. Mater. 40, 1443–1474 (1992)
Cahn, J.W., Hoffman, D.W.: A vector thermodynamics for anisotropic interfaces. 1. Fundamentals and applications to plane surface junctions. Surface Sci. 31, 368–388 (1972)
Caselles, V., Chambolle, A.: Anisotropic curvature-driven flow of convex sets. Preprint R.I. 528, April 2004, Ecole Polytechnique, Centre de Mathématiques Appliqués.
Chambolle, A.: An algorithm for mean curvature motion. Interfaces Free Bound. 6, 195–218 (2004)
Delfour, M.C., Zolesio, J.P.: Shapes analysis via oriented distance functions. J. Funct. Anal. 123, 129–201 (1994)
Delfour, M.C., Zolesio, J.P.: Shapes and Geometries. Analysis, differential calculus, and optimization. Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2001
Ecker, K., Huisken, G.: Mean curvature evolution of entire graphs. Ann. of Math. (2), 130, 453–471 (1989)
Evans, L.C.: Convergence of an algorithm for mean curvature motion. Indiana Univ. Math. J. 42, 533–557 (1993)
Evans, L.C., Spruck, J.: Motion of level sets by mean curvature. I. J. Differential Geom. 33, 635–681 (1991)
Evans, L.C., Spruck, J.: Motion of level sets by mean curvature. II. Trans. Amer. Math. Soc. 330, 321–332 (1992)
Evans, L.C., Spruck, J.: Motion of level sets by mean curvature. III. J. Geom. Anal. 3, 121–150 (1992)
Gage, M.E., Hamilton, R.S.: The heat equation shrinking convex plane curves. J. Differential Geom. 23, 69–96 (1986)
Giga, Y.: Singular diffusivity-facets, shocks and more. Hokkaido University Preprint Series in Mathematics, Series 604, September 2003
Giga, Y., Goto, S., Ishii, H., Sato, M.H.: Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains. Indiana Univ. Math. J. 40, 443–470 (1991)
Giga, Y., Gurtin, M.E.: A comparison theorem for crystalline evolutions in the plane. Quart. Appl. Math. LIV, 727–737 (1996)
Gurtin, M.: Thermomechanics of Evolving Phase Boundaries in the Plane. Clarendon Press, Oxford, 1993
Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differential Geom. 20, 237–266 (1984)
Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Differential Geom. 31, 285–299 (1990)
Huisken, G., Sinestrari, C.: Convexity estimates for mean curvature flow and singularities of mean convex surfaces. Acta Math. 183, 45–70 (1999)
Luckhaus, S., Sturzenhecker, T.: Implicit time discretization for the mean curvature flow equation. Calc. Var. Partial Differential Equations 3, 253–271 (1995)
Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory. In: Encyclopedia of Mathematics and its Applications 44. Cambridge University Press, 1993
Taylor, J.E.: Crystalline variational problems. Bull. Amer. Math. Soc. (N.S.) 84, 568–588 (1978)
Taylor, J.E.: II-Mean curvature and weighted mean curvature. Acta Metall. Mater. 40, 1475–1485 (1992)
White, B.: The nature of singularities in mean curvature flow of mean-convex sets. J. Amer. Math. Soc. 16, 123–138 (2003)
Yunger, J.: Facet stepping and motion by crystalline curvature. PhD. Thesis, Rutgers University 1998
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Bellettini, G., Caselles, V., Chambolle, A. et al. Crystalline Mean Curvature Flow of Convex Sets. Arch. Rational Mech. Anal. 179, 109–152 (2006). https://doi.org/10.1007/s00205-005-0387-0
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DOI: https://doi.org/10.1007/s00205-005-0387-0