On volume-preserving crystalline mean curvature flow

In this work we consider the global existence of volume-preserving crystalline curvature flow in a non-convex setting. We show that a natural geometric property, associated with reflection symmetries of the Wulff shape, is preserved with the flow. Using this geometric property, we address global existence and regularity of the flow for smooth anisotropies. For the non-smooth case we establish global existence results for the types of anisotropies known to be globally well-posed.

[1]  L. Evans,et al.  Motion of level sets by mean curvature. II , 1992 .

[2]  A. Chambolle.  An algorithm for Mean Curvature Motion , 2004 .

[3]  G. Barles,et al.  Front propagation and phase field theory , 1993 .

[4]  Inwon C. Kim,et al.  On mean curvature flow with forcing , 2015, 1512.07549.

[5]  Sabrina Hirsch,et al.  Reflection Groups And Coxeter Groups , 2016 .

[6]  S. Luckhaus,et al.  Implicit time discretization for the mean curvature flow equation , 1995 .

[7]  Jennifer Cahn,et al.  A vector thermodlnamics for anisotropic surfaces—II. Curved and faceted surfaces , 1974 .

[8]  Y. Giga,et al.  A level set crystalline mean curvature flow of surfaces , 2016, Advances in Differential Equations.

[9]  Tim Laux,et al.  Convergence of thresholding schemes incorporating bulk effects , 2016, 1601.02467.

[10]  A. Chambolle,et al.  The volume preserving crystalline mean curvature flow of convex sets in R^N , 2009 .

[11]  W. M. Feldman,et al.  Dynamic Stability of Equilibrium Capillary Drops , 2012, 1212.3625.

[12]  M. Novaga,et al.  Facet-breaking for three-dimensional crystals evolving by mean curvature , 1999 .

[13]  Yun-Gang Chen,et al.  Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations , 1989 .

[14]  H. Spohn.  Interface motion in models with stochastic dynamics , 1993 .

[15]  A. Chambolle,et al.  Existence and Uniqueness for a Crystalline Mean Curvature Flow , 2015, 1508.03598.

[16]  J. Humphreys.  Reflection groups and coxeter groups , 1990 .

[17]  A. Chambolle,et al.  Generalized crystalline evolutions as limits of flows with smooth anisotropies , 2017, Analysis & PDE.

[18]  H. Soner.  MOTION OF A SET BY THE CURVATURE OF ITS BOUNDARY , 1993 .

[19]  Y. Giga,et al.  Generalized Motion¶by Nonlocal Curvature in the Plane , 2000 .

[20]  J. Taylor,et al.  Crystalline variational problems , 1978 .

[21]  Sigurd B. Angenent,et al.  Multiphase thermomechanics with interfacial structure 2. Evolution of an isothermal interface , 1989 .

[22]  A. Chambolle,et al.  Implicit time discretization of the mean curvature flow with a discontinuous forcing term , 2008 .

[23]  U. Dini,et al.  Facet{breaking for Three{dimensional Crystals Evolving by Mean Curvature , 1998 .

[24]  G. Huisken.  The volume preserving mean curvature flow. , 1987 .

[25]  Inwon C. Kim,et al.  Volume preserving mean curvature flow for star-shaped sets , 2018, Calculus of Variations and Partial Differential Equations.

[26]  B. Andrews,et al.  Volume-Preserving Anisotropic Mean Curvature Flow , 2001 .

[27]  Luca Mugnai,et al.  Global solutions to the volume-preserving mean-curvature flow , 2015, 1502.07232.

[28]  Y. Giga,et al.  Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term , 2020, SN Partial Differential Equations and Applications.

[29]  G. Bellettini.  An introduction to anisotropic and crystalline mean curvature flow , 2010 .

[30]  K. Takasao.  Existence of weak solution for volume preserving mean curvature flow via phase field method , 2015, 1511.01687.

[31]  L. Caffarelli,et al.  A Geometric Approach to Free Boundary Problems , 2005 .

[32]  D. Kraft,et al.  Measure-Theoretic Properties of Level Sets of Distance Functions , 2016 .

[33]  F. Almgren,et al.  Curvature-driven flows: a variational approach , 1993 .

[34]  Y. Giga,et al.  Evolving Graphs by Singular Weighted Curvature , 1998 .

[35]  Y. Giga,et al.  Approximation of General Facets by Regular Facets with Respect to Anisotropic Total Variation Energies and Its Application to Crystalline Mean Curvature Flow , 2017, 1702.05220.

[36]  Flat flow is motion by crystalline curvature for curves with crystalline energies , 1995 .

[37]  A. Chambolle,et al.  Existence and uniqueness for anisotropic and crystalline mean curvature flows , 2017, Journal of the American Mathematical Society.