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April  2022, 42(4): 2027-2051. doi: 10.3934/dcds.2021182

Crystalline flow starting from a general polygon

1. 

Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan

2. 

Department of Indian Philosophy and Buddhist Studies, Humanities, Faculty of Letters, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

3. 

Airitech Co., Ltd., Masonic 39MT Building, 2-4-5 Azabudai, Minato-ku, Tokyo 106-0041, Japan

* Corresponding author: Yoshikazu Giga

Received  March 2021 Revised  September 2021 Published  April 2022 Early access  November 2021

Fund Project: The work of the second author was partly supported by the Japan Society for the Promotion of Science (JSPS) through the grants KAKENHI No. 19H00639, No. 18H05323, No. 17H01091 and by Arithmer Inc. through collaborative grant

This paper solves a singular initial value problem for a system of ordinary differential equations describing a polygonal flow called a crystalline flow. Such a problem corresponds to a crystalline flow starting from a general polygon not necessarily admissible in the sense that the corresponding initial value problem is singular. To solve the problem, a self-similar expanding solution constructed by the first two authors with H. Hontani (2006) is effectively used.

Citation: Mi-Ho Giga, Yoshikazu Giga, Ryo Kuroda, Yusuke Ochiai. Crystalline flow starting from a general polygon. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 2027-2051. doi: 10.3934/dcds.2021182
References:
[1]

B. Andrews, Singularities in crystalline curvature flows, Asian J. Math., 6 (2002), 101-121.  doi: 10.4310/AJM.2002.v6.n1.a6.

[2]

S. Angenent and M. E. Gurtin, Multiphase thermomechanics with interfacial structure. II. Evolution of an isothermal interface, Arch. Rational Mech. Anal., 108 (1989), 323-391.  doi: 10.1007/BF01041068.

[3]

G. BellettiniM. Novaga and M. Paolini, Facet-breaking for three-dimensional crystals evolving by mean curvature, Interfaces Free Bound., 1 (1999), 39-55.  doi: 10.4171/IFB/3.

[4]

G. BellettiniM. Novaga and M. Paolini, Characterization of facet breaking for nonsmooth mean curvature flow in the convex case, Interfaces Free Bound., 3 (2001), 415-446.  doi: 10.4171/IFB/47.

[5]

D. Campbell, A First Glance at Crystal Motion, Master's thesis, Rutgers University, New Brunswick, NJ, 2002.

[6]

A. ChambolleM. MoriniM. Novaga and M. Ponsiglione, Existence and uniqueness for anisotropic and crystalline mean curvature flows, J. Amer. Math. Soc., 32 (2019), 779-824.  doi: 10.1090/jams/919.

[7]

A. ChambolleM. Morini and M. Ponsiglione, Existence and uniqueness for a crystalline mean curvature flow, Comm. Pure Appl. Math., 70 (2017), 1084-1114.  doi: 10.1002/cpa.21668.

[8]

Y. G. ChenY. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., 33 (1991), 749-786. 

[9]

C. Dohmen and Y. Giga, Selfsimilar shrinking curves for anisotropic curvature flow equations, Proc. Japan Acad. Ser. A Math. Sci., 70 (1994), 252-255. 

[10]

C. DohmenY. Giga and N. Mizoguchi, Existence of selfsimilar shrinking curves for anisotropic curvature flow equations, Calc. Var. Partial Differential Equations, 4 (1996), 103-119.  doi: 10.1007/BF01189949.

[11]

C. M. ElliottA. R. Gardiner and R. Schätzle, Crystalline curvature flow of a graph in a variational setting, Adv. Math. Sci. Appl., 8 (1998), 425-460. 

[12]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom., 33 (1991), 635-681. 

[13]

T. Fukui and Y. Giga, Motion of a graph by nonsmooth weighted curvature, In World Congress of Nonlinear Analysts '92, Vol. I–IV (Tampa, FL, 1992), de Gruyter, Berlin, (1996), 47–56.

[14]

M. E. Gage, Evolving plane curves by curvature in relative geometries, Duke Math. J., 72 (1993), 441-466.  doi: 10.1215/S0012-7094-93-07216-X.

[15]

M. E. Gage and Y. Li, Evolving plane curves by curvature in relative geometries. II, Duke Math. J., 75 (1994), 79-98.  doi: 10.1215/S0012-7094-94-07503-0.

[16]

R. Gérard and H. Tahara, Singular Nonlinear Partial Differential Equations, Aspects of Mathematics Friedr. Vieweg & Sohn, Braunschweig, 1996. doi: 10.1007/978-3-322-80284-2.

[17]

M.-H. Giga and Y. Giga, Consistency in evolutions by crystalline curvature, Free Boundary Problems, Theory and Applications (Zakopane, 1995), Pitman Res. Notes Math. Ser., Longman, Harlow, 363 (1996), 186–202.

[18]

M.-H. Giga and Y. Giga, Geometric evolution by nonsmooth interfacial energy, Nonlinear Analysis and Applications (Warsaw, 1994), GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho, Tokyo, 7 (1996), 125–140.

[19]

M.-H. Giga and Y. Giga, A subdifferential interpretation of crystalline motion under nonuniform driving force, Discrete Contin. Dynam. Systems, 1 (1998), 276-287. 

[20]

M.-H. Giga and Y. Giga, Evolving graphs by singular weighted curvature, Arch. Rational Mech. Anal., 141 (1998), 117-198.  doi: 10.1007/s002050050075.

[21]

M.-H. Giga and Y. Giga, Stability for evolving graphs by nonlocal weighted curvature, Comm. Partial Differential Equations, 24 (1999), 109-184.  doi: 10.1080/03605309908821419.

[22]

M.-H. Giga and Y. Giga, Crystalline and level set flow – convergence of a crystalline algorithm for a general anisotropic curvature flow in the plane, Free Boundary Problems: Theory and Applications, I (Chiba, 1999), GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho, Tokyo, 13 (2000), 64–79.

[23]

M.-H. Giga and Y. Giga, Generalized motion by nonlocal curvature in the plane, Arch. Ration. Mech. Anal., 159 (2001), 295-333.  doi: 10.1007/s002050100154.

[24]

M.-H. Giga and Y. Giga, On the role of kinetic and interfacial anisotropy in the crystal growth theory, Interfaces Free Bound., 15 (2013), 429-450.  doi: 10.4171/IFB/309.

[25]

M.-H. GigaY. Giga and H. Hontani, Self-similar expanding solutions in a sector for a crystalline flow, SIAM J. Math. Anal., 37 (2005), 1207-1226.  doi: 10.1137/040614372.

[26]

Y. Giga, Motion of a graph by convexified energy, Hokkaido Math. J., 23 (1994), 185-212.  doi: 10.14492/hokmj/1381412492.

[27]

Y. Giga, Surface Evolution Equations. A Level Set Approach, Monographs in Mathematics, 99, Birkhäuser Verlag, Basel, 2006. doi: 10.1007/3-7643-7391-1.

[28]

Y. Giga and M. E. Gurtin, A comparison theorem for crystalline evolution in the plane, Quart. Appl. Math., 54 (1996), 727-737.  doi: 10.1090/qam/1417236.

[29]

Y. Giga and N. Požár, A level set crystalline mean curvature flow of surfaces, Adv. Differential Equations, 21 (2016), 631-698. 

[30]

Y. Giga and N. Požár, Approximation of general facets by regular facets with respect to anisotropic total variation energies and its application to crystalline mean curvature flow, Comm. Pure Appl. Math., 71 (2018), 1461-1491.  doi: 10.1002/cpa.21752.

[31]

Y. Giga and N. Požár, Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term, SN Partial Differ. Equ. Appl., 1 (2020), Article number: 39. doi: 10.1007/s42985-020-00040-0.

[32]

P. M. Girão, Convergence of a crystalline algorithm for the motion of a simple closed convex curve by weighted curvature, SIAM J. Numer. Anal., 32 (1995), 886-899.  doi: 10.1137/0732041.

[33]

P. M. Girão and R. V. Kohn, Convergence of a crystalline algorithm for the heat equation in one dimension and for the motion of a graph by weighted curvature, Numer. Math., 67 (1994), 41-70.  doi: 10.1007/s002110050017.

[34]

M. A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom., 26 (1987), 285-314. 

[35] M. E. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993. 
[36]

M. E. GurtinH. M. Soner and P. E. Souganidis, Anisotropic motion of an interface relaxed by the formation of infinitesimal wrinkles, J. Differential Equations, 119 (1995), 54-108.  doi: 10.1006/jdeq.1995.1084.

[37]

K. Ishii and H. M. Soner, Regularity and convergence of crystalline motion, SIAM J. Math. Anal., 30 (1999), 19-37.  doi: 10.1137/S0036141097317347.

[38]

T. Ishiwata, Crystalline Undou ni Tsuite: Heimenjou No Takakukei No Undou No Kaiseki, (Japanese) [On crystalline motion: Analysis on motion of a polygon in the plane], Lecture Series in Mathematics GP-TML06, Graduate School of Science, Tohoku University, 2008.

[39]

T. Ishiwata, Motion of non-convex polygons by crystalline curvature and almost convexity phenomena, Japan J. Indust. Appl. Math., 25 (2008), 233-253.  doi: 10.1007/BF03167521.

[40]

T. Ishiwata, Motion of polygonal curved fronts by crystalline motion: V-shaped solutions and eventual monotonicity, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications. 8$^{th}$ AIMS Conference. 1 (2011), 717–726.

[41]

T. Ishiwata, On the motion of polygonal curves with asymptotic lines by crystalline curvature flow with bulk effect, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 865-873.  doi: 10.3934/dcdss.2011.4.865.

[42]

T. Ishiwata, Crystalline motion of spiral-shaped polygonal curves with a tip motion, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 53-62.  doi: 10.3934/dcdss.2014.7.53.

[43]

T. Ishiwata and T. Ohtsuka, Evolution of a spiral-shaped polygonal curve by the crystalline curvature flow with a pinned tip, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5261-5295.  doi: 10.3934/dcdsb.2019058.

[44]

T. Ishiwata and T. Ohtsuka, Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 893-907.  doi: 10.3934/dcdss.2020390.

[45]

T. IshiwataT. K. UshijimaH. Yagisita and S. Yazaki, Two examples of nonconvex self-similar solution curves for a crystalline curvature flow, Proc. Japan Acad. Ser. A Math. Sci., 80 (2004), 151-154. 

[46]

R. Kuroda, Facet-Creation Between Two Facets Moved by Crystalline Flow or Similar Equations, Bachelor's thesis, The University of Tokyo, Tokyo, 2019.

[47]

P. B. Mucha, Regular solutions to a monodimensional model with discontinuous elliptic operator, Interfaces Free Bound., 14 (2012), 145-152.  doi: 10.4171/IFB/276.

[48]

P. B. Mucha and P. Rybka, A note on a model system with sudden directional diffusion, J. Stat. Phys., 146 (2012), 975-988.  doi: 10.1007/s10955-012-0446-5.

[49]

P. B. Mucha and P. Rybka, Well posedness of sudden directional diffusion equations, Math. Methods Appl. Sci., 36 (2013), 2359-2370.  doi: 10.1002/mma.2759.

[50]

A. ObermanS. OsherR. Takei and R. Tsai, Numerical methods for anisotropic mean curvature flow based on a discrete time variational formulation, Commun. Math. Sci., 9 (2011), 637-662.  doi: 10.4310/CMS.2011.v9.n3.a1.

[51]

Y. Ochiai, Facet-Creation Between Two Facets Moved by Crystalline Curvature, Master's thesis, The University of Tokyo, Tokyo, 2009.

[52]

A. Stancu, Uniqueness of self-similar solutions for a crystalline flow, Indiana Univ. Math. J., 45 (1996), 1157-1174.  doi: 10.1512/iumj.1996.45.1159.

[53]

A. Stancu, Asymptotic behavior of solutions to a crystalline flow, Hokkaido Math. J., 27 (1998), 303-320.  doi: 10.14492/hokmj/1351001287.

[54]

J. E. Taylor, Constructions and conjectures in crystalline nondifferential geometry, Differential Geometry, Pitman Monogr. Surveys Pure Appl. Math. Longman Sci. Tech. Harlow, 52 (1991), 321–336.

[55]

J. E. Taylor, Mean curvature and weighted mean curvature, Acta Metall. Mater., 40 (1992), 1475-1485.  doi: 10.1016/0956-7151(92)90091-R.

[56]

J. E. Taylor, Motion of curves by crystalline curvature, including triple junctions and boundary points, Differential Geometry: Partial Differential Equations on Manifolds (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., Part 1, Amer. Math. Soc., Providence, RI, 54 (1993), 417–438. doi: 10.1090/pspum/054.1/1216599.

[57]

S. Yazaki, On an area-preserving crystalline motion, Calc. Var. Partial Differential Equations, 14 (2002), 85-105.  doi: 10.1007/s005260100094.

[58]

S. Yazaki, Motion of nonadmissible convex polygons by crystalline curvature, Publ. Res. Inst. Math. Sci., 43 (2007), 155-170.  doi: 10.2977/prims/1199403812.

show all references

References:
[1]

B. Andrews, Singularities in crystalline curvature flows, Asian J. Math., 6 (2002), 101-121.  doi: 10.4310/AJM.2002.v6.n1.a6.

[2]

S. Angenent and M. E. Gurtin, Multiphase thermomechanics with interfacial structure. II. Evolution of an isothermal interface, Arch. Rational Mech. Anal., 108 (1989), 323-391.  doi: 10.1007/BF01041068.

[3]

G. BellettiniM. Novaga and M. Paolini, Facet-breaking for three-dimensional crystals evolving by mean curvature, Interfaces Free Bound., 1 (1999), 39-55.  doi: 10.4171/IFB/3.

[4]

G. BellettiniM. Novaga and M. Paolini, Characterization of facet breaking for nonsmooth mean curvature flow in the convex case, Interfaces Free Bound., 3 (2001), 415-446.  doi: 10.4171/IFB/47.

[5]

D. Campbell, A First Glance at Crystal Motion, Master's thesis, Rutgers University, New Brunswick, NJ, 2002.

[6]

A. ChambolleM. MoriniM. Novaga and M. Ponsiglione, Existence and uniqueness for anisotropic and crystalline mean curvature flows, J. Amer. Math. Soc., 32 (2019), 779-824.  doi: 10.1090/jams/919.

[7]

A. ChambolleM. Morini and M. Ponsiglione, Existence and uniqueness for a crystalline mean curvature flow, Comm. Pure Appl. Math., 70 (2017), 1084-1114.  doi: 10.1002/cpa.21668.

[8]

Y. G. ChenY. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., 33 (1991), 749-786. 

[9]

C. Dohmen and Y. Giga, Selfsimilar shrinking curves for anisotropic curvature flow equations, Proc. Japan Acad. Ser. A Math. Sci., 70 (1994), 252-255. 

[10]

C. DohmenY. Giga and N. Mizoguchi, Existence of selfsimilar shrinking curves for anisotropic curvature flow equations, Calc. Var. Partial Differential Equations, 4 (1996), 103-119.  doi: 10.1007/BF01189949.

[11]

C. M. ElliottA. R. Gardiner and R. Schätzle, Crystalline curvature flow of a graph in a variational setting, Adv. Math. Sci. Appl., 8 (1998), 425-460. 

[12]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom., 33 (1991), 635-681. 

[13]

T. Fukui and Y. Giga, Motion of a graph by nonsmooth weighted curvature, In World Congress of Nonlinear Analysts '92, Vol. I–IV (Tampa, FL, 1992), de Gruyter, Berlin, (1996), 47–56.

[14]

M. E. Gage, Evolving plane curves by curvature in relative geometries, Duke Math. J., 72 (1993), 441-466.  doi: 10.1215/S0012-7094-93-07216-X.

[15]

M. E. Gage and Y. Li, Evolving plane curves by curvature in relative geometries. II, Duke Math. J., 75 (1994), 79-98.  doi: 10.1215/S0012-7094-94-07503-0.

[16]

R. Gérard and H. Tahara, Singular Nonlinear Partial Differential Equations, Aspects of Mathematics Friedr. Vieweg & Sohn, Braunschweig, 1996. doi: 10.1007/978-3-322-80284-2.

[17]

M.-H. Giga and Y. Giga, Consistency in evolutions by crystalline curvature, Free Boundary Problems, Theory and Applications (Zakopane, 1995), Pitman Res. Notes Math. Ser., Longman, Harlow, 363 (1996), 186–202.

[18]

M.-H. Giga and Y. Giga, Geometric evolution by nonsmooth interfacial energy, Nonlinear Analysis and Applications (Warsaw, 1994), GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho, Tokyo, 7 (1996), 125–140.

[19]

M.-H. Giga and Y. Giga, A subdifferential interpretation of crystalline motion under nonuniform driving force, Discrete Contin. Dynam. Systems, 1 (1998), 276-287. 

[20]

M.-H. Giga and Y. Giga, Evolving graphs by singular weighted curvature, Arch. Rational Mech. Anal., 141 (1998), 117-198.  doi: 10.1007/s002050050075.

[21]

M.-H. Giga and Y. Giga, Stability for evolving graphs by nonlocal weighted curvature, Comm. Partial Differential Equations, 24 (1999), 109-184.  doi: 10.1080/03605309908821419.

[22]

M.-H. Giga and Y. Giga, Crystalline and level set flow – convergence of a crystalline algorithm for a general anisotropic curvature flow in the plane, Free Boundary Problems: Theory and Applications, I (Chiba, 1999), GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho, Tokyo, 13 (2000), 64–79.

[23]

M.-H. Giga and Y. Giga, Generalized motion by nonlocal curvature in the plane, Arch. Ration. Mech. Anal., 159 (2001), 295-333.  doi: 10.1007/s002050100154.

[24]

M.-H. Giga and Y. Giga, On the role of kinetic and interfacial anisotropy in the crystal growth theory, Interfaces Free Bound., 15 (2013), 429-450.  doi: 10.4171/IFB/309.

[25]

M.-H. GigaY. Giga and H. Hontani, Self-similar expanding solutions in a sector for a crystalline flow, SIAM J. Math. Anal., 37 (2005), 1207-1226.  doi: 10.1137/040614372.

[26]

Y. Giga, Motion of a graph by convexified energy, Hokkaido Math. J., 23 (1994), 185-212.  doi: 10.14492/hokmj/1381412492.

[27]

Y. Giga, Surface Evolution Equations. A Level Set Approach, Monographs in Mathematics, 99, Birkhäuser Verlag, Basel, 2006. doi: 10.1007/3-7643-7391-1.

[28]

Y. Giga and M. E. Gurtin, A comparison theorem for crystalline evolution in the plane, Quart. Appl. Math., 54 (1996), 727-737.  doi: 10.1090/qam/1417236.

[29]

Y. Giga and N. Požár, A level set crystalline mean curvature flow of surfaces, Adv. Differential Equations, 21 (2016), 631-698. 

[30]

Y. Giga and N. Požár, Approximation of general facets by regular facets with respect to anisotropic total variation energies and its application to crystalline mean curvature flow, Comm. Pure Appl. Math., 71 (2018), 1461-1491.  doi: 10.1002/cpa.21752.

[31]

Y. Giga and N. Požár, Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term, SN Partial Differ. Equ. Appl., 1 (2020), Article number: 39. doi: 10.1007/s42985-020-00040-0.

[32]

P. M. Girão, Convergence of a crystalline algorithm for the motion of a simple closed convex curve by weighted curvature, SIAM J. Numer. Anal., 32 (1995), 886-899.  doi: 10.1137/0732041.

[33]

P. M. Girão and R. V. Kohn, Convergence of a crystalline algorithm for the heat equation in one dimension and for the motion of a graph by weighted curvature, Numer. Math., 67 (1994), 41-70.  doi: 10.1007/s002110050017.

[34]

M. A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom., 26 (1987), 285-314. 

[35] M. E. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993. 
[36]

M. E. GurtinH. M. Soner and P. E. Souganidis, Anisotropic motion of an interface relaxed by the formation of infinitesimal wrinkles, J. Differential Equations, 119 (1995), 54-108.  doi: 10.1006/jdeq.1995.1084.

[37]

K. Ishii and H. M. Soner, Regularity and convergence of crystalline motion, SIAM J. Math. Anal., 30 (1999), 19-37.  doi: 10.1137/S0036141097317347.

[38]

T. Ishiwata, Crystalline Undou ni Tsuite: Heimenjou No Takakukei No Undou No Kaiseki, (Japanese) [On crystalline motion: Analysis on motion of a polygon in the plane], Lecture Series in Mathematics GP-TML06, Graduate School of Science, Tohoku University, 2008.

[39]

T. Ishiwata, Motion of non-convex polygons by crystalline curvature and almost convexity phenomena, Japan J. Indust. Appl. Math., 25 (2008), 233-253.  doi: 10.1007/BF03167521.

[40]

T. Ishiwata, Motion of polygonal curved fronts by crystalline motion: V-shaped solutions and eventual monotonicity, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications. 8$^{th}$ AIMS Conference. 1 (2011), 717–726.

[41]

T. Ishiwata, On the motion of polygonal curves with asymptotic lines by crystalline curvature flow with bulk effect, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 865-873.  doi: 10.3934/dcdss.2011.4.865.

[42]

T. Ishiwata, Crystalline motion of spiral-shaped polygonal curves with a tip motion, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 53-62.  doi: 10.3934/dcdss.2014.7.53.

[43]

T. Ishiwata and T. Ohtsuka, Evolution of a spiral-shaped polygonal curve by the crystalline curvature flow with a pinned tip, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5261-5295.  doi: 10.3934/dcdsb.2019058.

[44]

T. Ishiwata and T. Ohtsuka, Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 893-907.  doi: 10.3934/dcdss.2020390.

[45]

T. IshiwataT. K. UshijimaH. Yagisita and S. Yazaki, Two examples of nonconvex self-similar solution curves for a crystalline curvature flow, Proc. Japan Acad. Ser. A Math. Sci., 80 (2004), 151-154. 

[46]

R. Kuroda, Facet-Creation Between Two Facets Moved by Crystalline Flow or Similar Equations, Bachelor's thesis, The University of Tokyo, Tokyo, 2019.

[47]

P. B. Mucha, Regular solutions to a monodimensional model with discontinuous elliptic operator, Interfaces Free Bound., 14 (2012), 145-152.  doi: 10.4171/IFB/276.

[48]

P. B. Mucha and P. Rybka, A note on a model system with sudden directional diffusion, J. Stat. Phys., 146 (2012), 975-988.  doi: 10.1007/s10955-012-0446-5.

[49]

P. B. Mucha and P. Rybka, Well posedness of sudden directional diffusion equations, Math. Methods Appl. Sci., 36 (2013), 2359-2370.  doi: 10.1002/mma.2759.

[50]

A. ObermanS. OsherR. Takei and R. Tsai, Numerical methods for anisotropic mean curvature flow based on a discrete time variational formulation, Commun. Math. Sci., 9 (2011), 637-662.  doi: 10.4310/CMS.2011.v9.n3.a1.

[51]

Y. Ochiai, Facet-Creation Between Two Facets Moved by Crystalline Curvature, Master's thesis, The University of Tokyo, Tokyo, 2009.

[52]

A. Stancu, Uniqueness of self-similar solutions for a crystalline flow, Indiana Univ. Math. J., 45 (1996), 1157-1174.  doi: 10.1512/iumj.1996.45.1159.

[53]

A. Stancu, Asymptotic behavior of solutions to a crystalline flow, Hokkaido Math. J., 27 (1998), 303-320.  doi: 10.14492/hokmj/1351001287.

[54]

J. E. Taylor, Constructions and conjectures in crystalline nondifferential geometry, Differential Geometry, Pitman Monogr. Surveys Pure Appl. Math. Longman Sci. Tech. Harlow, 52 (1991), 321–336.

[55]

J. E. Taylor, Mean curvature and weighted mean curvature, Acta Metall. Mater., 40 (1992), 1475-1485.  doi: 10.1016/0956-7151(92)90091-R.

[56]

J. E. Taylor, Motion of curves by crystalline curvature, including triple junctions and boundary points, Differential Geometry: Partial Differential Equations on Manifolds (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., Part 1, Amer. Math. Soc., Providence, RI, 54 (1993), 417–438. doi: 10.1090/pspum/054.1/1216599.

[57]

S. Yazaki, On an area-preserving crystalline motion, Calc. Var. Partial Differential Equations, 14 (2002), 85-105.  doi: 10.1007/s005260100094.

[58]

S. Yazaki, Motion of nonadmissible convex polygons by crystalline curvature, Publ. Res. Inst. Math. Sci., 43 (2007), 155-170.  doi: 10.2977/prims/1199403812.

Figure 1.  Each arrow indicates the positive direction
Figure 2.  A given Wulff shape and initial condition
Figure 3.  Self-similar solution
Figure 4.  Wulff shape and newly created facets
Figure 5.  $ i $-th, $ (i-1) $-th and $ (i+1) $-th facet
Figure 6.  Distance function $ d_i^s(t) $
Figure 7.  New self-similar solution
Figure 8.  Wulff shape
Figure 9.  Compare with a self-similar solution
Figure 10.  An example of the numerical calculation
Figure 11.  Another example of the numerical calculation (enlarged)
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