Abstract
Solids can exist in polygonal shapes with boundaries unions of flat pieces called facets. Analyzing the growth of such crystalline shapes is an important problem in materials science. In this paper we derive equations that govern the evolution of such shapes; we formulate the corresponding initial-value problem variationally; and we use this formulation to establish a comparison principle for crystalline evolutions. This principle asserts that two evolving crystals one initially inside the other will remain in that configuration for all time.
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References
H. Attouch and A. Damlamian, Application des methods de convexité et monotonie a l’etude de certaines equations quasi lineares. Proc. Roy. Soc. Edinburgh,79A (1977), 107–129.
S. Angenent and M.E. Gurtin, Multiphase thermomechanics with interfacial structure. 2. Evolution of an isothermal interface. Arch. Rational Mech. Anal.,108 (1989), 323–391.
F. Almgren and J. Taylor, Flat flow is motion by crystalline curvature for curves with crystalline energies. J. Differential Geometry,42 (1995), 1–22.
F. Almgren, J. Taylor and L. Wang, Curvature-driven flows: a variational approach, SIAM J. Control Optim.,31 (1993), 387–437.
V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Space. Noordhoff, Groningen, 1976.
H. Brezis, Monotonicity methods in Hilbert spaces and some applications to nonlinear differential equations. Contributions to Nonlinear Functional Analysis (ed. E. Zarantonello), Academic Press, New York, 1971, 101–156.
K.A. Brakke, The motion of a Surface by its Mean Curvature. Princeton Univ. Press, 1978.
Y.-G. Chen, Y. Giga and S. Goto, Uniqueness and Existence of viscosity solutions of generalized mean curvature flow equations. J. Differential Geometry,33 (1991), 749–786.
J.W. Cahn and D.W. Hoffman, A vector thermodynamics for anisotropic surfaces — 2. curved and faceted surfaces. Act. Metall.,22 (1974), 1205–1214.
J.W. Cahn and J. Taylor, A contribution to the theory of surface energy minimizing shapes. Scripta Metall.,18 (1984), 1117–1120.
C.M. Elliott, A.R. Gardiner and R. Schatzle, Crystalline curvature flow of a graph in a variational setting. Univ. of Sussex at Brighton, Research Report N 95/06.
L.C. Evans and J. Spruck, Motion of level sets by mean curvature, I. J. Differential Geometry,33 (1991), 635–681.
T. Fukui and Y. Giga, Motion of a graph by nonsmooth weighted curvature. Proceedings of the First World Congress of Nonlinear Analysts,1 (ed. V. Lakshmikantham), Walter de Gruyter, Hawthorne, Berlin, 1995, 47–56.
M.-H. Giga and Y. Giga, Geometric evolution by nonsmooth interfacial energy. Proc. of Banach Center Minisemester, “Nonlinear Analysis and Applications” (eds. N. Kenmochi, M. Niezgódka and P. Strzelecki), Gakuto, 1995, 125–140.
M.-H. Giga and Y. Giga, Consistency in evolutions by crystalline curvature. Proceedings of Free Boundary Problems 95 (eds. M. Niezgódka and P. Strzelecki) (Zakopane, Poland, 1995), 186–202.
Y. Giga and M.E. Gurtin, A comparison theorem for crystalline evolution in the plane. Q. Appl. Math.,54 (1996), 727–737.
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), 1443–1474.
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.
P.M. Girão and R.V. Kohn, The crystalline algorithm for computing motion by curvature. Variational Methods for Discontinuous Structure (eds. R. Serapioni and F. Tomarelli), Birkhauser, 1996, 7–18.
M.E. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane. Oxford Univ. Press, 1993.
M.E. Gurtin and A. Struthers, Multiphase thermomechanics with interfacial structure. 3. Evolving phase boundaries in the presence of bulk deformation. Arch. Rational Mech. Anal.,112 (1990), 97–160.
D.W. Hoffman and J.W. Cahn, A vector thermodynamics for anisotropic surfaces — 1. Fundamentals and applications to plane surface junctions. Surface Sci.,31 (1972), 368–388.
H.M. Soner, Motion of a set by the curvature of its boundary. J. Differential Equations,101 (1993), 313–372.
A. Stancu, Uniqueness of self-similar solutions for a crystalline flow. Indiana Univ. Math. J.,45 (1996), 1159–1160.
J.E. Taylor, Constructions and conjectures in crystalline nondifferential geometry. Monographs in Pure and Applied Mathematics,55 (eds. B. Lawson and K. Tanenblat), Pitman, London, 1991, 321–336.
J.E. Taylor, Mean curvature and weighted mean curvature. Act. Metall.,40 (1992), 1475–1485.
R. Teman, Navier-Stokes Equations. North Holland, Amsterdam, 1977.
J.E. Taylor and J.W. Cahn, Catalog of saddle shaped surfaces in crystals. Act. Metall.,34 (1986), 1–12.
J.E. Taylor, J.W. Cahn and C. Handwerker, Geometric models of crystal growth. Act. Metall.,40 (1992), 1443–1474.
J. Watanabe, Approximation of nonlinear problems of a certain type. Lecture Notes in Numer. Appl.,1, North Holland, Amsterdam, 1979, 147–163.
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Dedicated toProfessor Kôji Kubota on his sixtieth birthday.
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Giga, Y., Gurtin, M.E. & Matias, J. On the dynamics of crystalline motions. Japan J. Indust. Appl. Math. 15, 7–50 (1998). https://doi.org/10.1007/BF03167395
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DOI: https://doi.org/10.1007/BF03167395