Computation of geometric partial differential equations and mean curvature flow

Acta Numerica

Volumn 14, Issue , 2005, Pages 139-232

  Deckelnick, Klaus ^a   Dziuk, Gerhard ^b   Elliott, Charles M ^c  

^a OTTO VON GUERICKE UNIVERSITY   (Germany)

^b UNIVERSITY OF FREIBURG   (Germany)

^c UNIVERSITY OF SUSSEX   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 19844369196     PISSN: 09624929     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0962492904000224     Document Type: Review

References (148)

1. S. Allen and J. Cahn (1979), 'A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening', Acta Metall. 27, 1084-1095.
2. R. Almgren (1993), 'Variational algorithms and pattern formation in dendritic solidification', J. Comput. Phys. 106, 337-354.
3. S. J. Altschuler and M. A. Grayson (1992), 'Shortening space curves and flow through singularities', J. Differential Geom. 35, 283-298.
4. S. Angenent (1991), 'Parabolic equations for curves on surfaces, Part II: Intersections, blow-up and generalized solutions', Ann. of Math. 133, 171-215.
5. S. Angenent and M. Gurtin (1989), 'Multiphase thermomechanics with interfacial structure, 2: Evolution of an isothermal interface', Arch. Rat. Mech. Anal. 108, 323-391.
6. G. Aubert and P. Kornprobst (2002), Mathematical Problems in Image Processing, Vol. 147 of Applied Mathematical Sciences, Springer.
7. E. Bänsch and A. Schmidt (2000), 'Simulation of dendritic crystal growth with thermal convection', Interfaces Free Bound. 2, 95-115.
8. E. Bänsch, P. Morin and R. H. Nochetto (2004), 'Surface diffusion of graphs: Variational formulation, error analysis and simulation', SIAM J. Numer. Anal. 42, 773-799.
9. J. W. Barrett, R. Nürnberg and V. Styles (2004), 'Finite element approximation of a phase field model for void electromigration', SIAM J. Numer. Anal. 42, 738-772.
10. G. Bellettini and M. Paolini (1995), 'Quasi-optimal error estimates for the mean curvature flow with a forcing term', Differential Integral Equations 8, 735-752.
11. G. Bellettini and M. Paolini (1996), 'Anisotropic motion by mean curvature in the context of Finsler geometry', Hokkaido Math. J. 25, 537-566.
12. G. Bellettini, M. Novaga and M. Paolini (1999), 'Facet-breaking for three dimensional crystals evolving by mean curvature', Interfaces Free Bound. 1, 39-55.
13. G. Bellettini, M. Novaga and M. Paolini (2001), 'Characterization of facet-breaking for non-smooth mean curvature flow in the convex case', Interfaces Free Bound. 3, 415-446.
14. G. Bellettini, M. Paolini and C. Verdi (1990), 'Γ-convergence of discrete approximations to interfaces with prescribed mean curvature', Rendconti Mat. Acc. Lincei 9, 317-328.
15. J. F. Blowey and C. M. Elliott (1991), 'The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy, Part I: Mathematical Analysis', European J. Appl. Math. 2, 233-280.
16. J. F. Blowey and C. M. Elliott (1993a), 'The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy, Part II: Numerical Analysis', European J. Appl. Math. 3, 147-179.
17. J. F. Blowey and C. M. Elliott (1993b), Curvature dependent phase boundary motion and parabolic obstacle problems, in Proc. IMA Workshop on Degenerate Diffusion, 1991 (W. Ni, L. Peletier and J. L. Vasquez, eds) Springer, IMA 47, pp. 19-60.
18. J. F. Blowey and C. M. Elliott (1994), A phase field model with a double obstacle potential, in Motion by Mean Curvature and Related Topics (G. Buttazzo and A. Visintin, ed.), de Gruyter, New York, pp. 1-22.
19. K. A. Brakke (1978), The Motion of a Surface by its Mean Curvature, Princeton University Press, NJ.
20. K. A. Brakke (1992), 'The surface evolver', Exp. Math. 1, 141-165.
21. R. J. Braun, S. R. Coriell, G. B. McFadden and A. A. Wheeler (1993), 'Phase field models for anisotropic interfaces', Phys. Rev. E 48, 2016-2024.
22. E. Burman and M. Picasso (2003), 'Anisotropic, adaptive finite elements for the computation of a solutal dendrite', Interfaces Free Bound. 5, 103-127.
23. G. Caginalp (1986), 'An analysis of a phase field model of a free boundary', Arch. Rat. Mech. Anal. 92, 206-245.
24. J. W. Cahn and J. E. Hilliard (1958), 'Free energy of a nonuniform system I: Interfacial free energy', J. Chem. Phys. 28, 258-267.
25. J. W. Cahn and J. E. Taylor (1994), 'Surface motion by surface diffusion', Acta Metall. Mater. 42, 1045-1063.
26. J. W. Cahn, C. M. Elliott and A. Novick-Cohen (1996), 'The Cahn-Hilliard equation with a concentration dependent mobility: motion by minus the Laplacian of the mean curvature', European J. Appl. Math. 7, 287-301.
27. V. Caselles, R. Kimmel, G. Sapiro and C. Sbert (1997), 'Minimal surfaces: a geometric three dimensional segmentation approach', Numer. Math. 77, 423-451.
28. A. Chambolle (2004), 'An algorithm for mean curvature motion', Interfaces Free Bound. 6, 195-218.
29. X. Chen (1992), 'Generation and propagation of interface in reaction diffusion equations', J. Differential Equations 96, 116-141.
30. X. Chen (1994), 'Spectrum for the Allen-Cahn, Cahn-Hilliard and phase field equations for generic interfaces', Comm. Partial Differential Equations 19, 1371-1395.
31. X. Chen and C. M. Elliott (1994), 'Asymptotics for a parabolic double obstacle problem', Proc. R. Soc. London A 444, 429-445.
32. X. Chen, C. M. Elliott, A. R. Gardiner and J. J. Zhao (1998), 'Convergence of numerical solutions to the Allen-Cahn equation', Appl. Anal. 69, 47-56.
33. Y.-G. Chen, Y. Giga and S. Goto (1991), 'Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations', J. Differential Geom. 33, 749-786.
34. Y.-G. Chen, Y. Giga, Y. T. Hitaka and M. Honma (1994), A stable difference scheme for computing motion of level surfaces by the mean curvature, in Proceedings of the Global Analysis Research Center Symposium, Seoul, Korea (D. Kim et al., eds), pp. 1-19.
35. D. L. Chopp (1993), 'Computing minimal surfaces via level set curvature flow', J. Comput. Phys. 106, 77-91.
36. D. L. Chopp and J. A. Sethian (1999), 'Motion by intrinsic Laplacian of curvature', Interfaces Free Bound. 1, 107-123.
37. U. Clarenz, U. Diewald, G. Dziuk, M. Rumpf and R. Rusu (2004), 'A finite element method for surface restoration with smooth boundary conditions', Computer Aided Geometric Design 21 (5), 427-445.
38. B. D. Coleman, R. S. Falk and M. Moakher (1995), 'Stability of cylindrical bodies in the theory of surface diffusion', Phys. D 89, 123-135.
39. M. G. Crandall and P.-L. Lions (1996), 'Convergent difference schemes for nonlinear parabolic equations and mean curvature motion', Numer. Math. 75, 17-41.
40. M. G. Crandall, H. Ishii and P.-L. Lions (1992), 'User's guide to viscosity solutions of second order partial differential equations', Bull. Amer. Math. Soc. 27, 1-67.
41. K. Deckelnick (1997), 'Weak solutions of the curve shortening flow', Calc. Var. 5, 489-510.
42. K. Deckelnick (2000), 'Error analysis for a difference scheme approximating mean curvature flow', Interfaces Free Bound. 2, 117-142.
43. K. Deckelnick and G. Dziuk (1994), On the approximation of the curve shortening flow, in Calculus of Variations, Applications and Computations: Pont-à-Mousson, 1994 (C. Bandle, J. Bemelmans, M. Chipot, J. Saint Jean Paulin and I. Shafrir, eds), Pitman Research Notes in Mathematics Series, pp. 100-108.
44. K. Deckelnick and G. Dziuk (1995), 'Convergence of a finite element method for non-parametric mean curvature flow', Numer. Math. 72, 197-222.
45. K. Deckelnick and G. Dziuk (1999), 'Discrete anisotropic curvature flow of graphs', Math. Model. Numer. Anal. 33, 1203-1222.
46. K. Deckelnick and G. Dziuk (2000), 'Error estimates for a semi implicit fully discrete finite element scheme for the mean curvature flow of graphs', Interfaces Free Bound. 2, 341-359.
47. K. Deckelnick and G. Dziuk (2001), Convergence of numerical schemes for the approximation of level set solutions to mean curvature flow, in Numerical Methods for Viscosity Solutions and Applications (M. Falcone and C. Makridakis, eds), Vol. 59 of Series Adv. Math. Appl. Sciences, pp. 77-94.
48. K. Deckelnick and G. Dziuk (2002a), 'A fully discrete numerical scheme for weighted mean curvature flow', Numer. Math. 91, 423-452.
49. K. Deckelnick and G. Dziuk (2002b), A finite element level set method for anisotropic mean curvature flow with space dependent weight, Abschlußband SFB 256, Bonn.
50. K. Deckelnick and C. M. Elliott (1998), 'Finite element error bounds for curve shrinking with prescribed normal contact to a fixed boundary', IMA J. Numer. Anal. 18, 635-654.
51. K. Deckelnick and C. M. Elliott (2004), 'Uniqueness and error analysis for Hamilton-Jacobi equations with discontinuities', Interfaces Free Bound. 6, 329-349.
52. K. Deckelnick, G. Dziuk and C. M. Elliott (2003a), 'Error analysis of a semidiscrete numerical scheme for diffusion in axially symmetric surfaces', SIAM J. Numer. Anal. 41, 2161-2179.
53. K. Deckelnick, G. Dziuk and C. M. Elliott (2003b), Fully discrete semi-implicit second order splitting for anisotropic surface diffusion of graphs, Preprint No. 33/2003, Universität Magdeburg, SIAM J. Numer. Anal., to appear.
54. K. Deckelnick, C. M. Elliott and V. M. Styles (2001), 'Numerical diffusion-induced grain boundary motion', Interfaces Free Bound. 3, 393-414.
55. P. de Mottoni and M. Schatzman (1995), 'Geometric evolution of developed interfaces', Trans. Amer. Math. Soc. 347, 1533-1589.
56. M. Droske and M. Rumpf (2004), 'A level set formulation for Willmore flow', Interfaces Free Bound. 6, 361-378.
57. G. Dziuk (1988), Finite elements for the Beltrami operator on arbitrary surfaces, in Partial Differential Equations and Calculus of Variations (S. Hildebrandt and R. Leis, eds), Vol. 1357 of Lecture Notes in Mathematics, Springer, pp. 142-155.
58. G. Dziuk (1991), 'An algorithm for evolutionary surfaces', Numer. Math. 58, 603-611.
59. G. Dziuk (1994), 'Convergence of a semi-discrete scheme for the curve shortening flow', Math. Models Methods Appl. Sci. 4, 589-606.
60. G. Dziuk (1999a), Numerical schemes for the mean curvature flow of graphs, in IUTAM Symposium on Variations of Domains and Free-Boundary Problems in Solid Mechanics (P. Argoul, M. Frémond and Q. S. Nguyen, eds), Kluwer, Dordrecht/Boston/London, pp. 63-70.
61. G. Dziuk (19996), 'Discrete anisotropic curve shortening flow', SIAM J. Numer. Anal. 36, 1808-1830.
62. n: existence and computation', SIAM J. Math. Anal. 33, 1228-1245.
63. K. Ecker (2002) Lectures on regularity for mean curvature flow, Preprint No. 23/2002, Universität Freiburg.
64. K. Ecker and G. Huisken (1989), 'Mean curvature evolution of entire graphs', Ann. of Math. 130, 453-471.
65. C. M. Elliott (1997), Approximation of curvature dependent interface motion, in The State of the Art in Numerical Analysis (I. S. Duff et al., ed.), Vol. 63 of Inst. Math. Appl. Conf. Ser., New Ser., Clarendon Press, Oxford, pp. 407-440.
66. C. M. Elliott and H. Garcke (1997), 'Existence results for diffusive surface motion laws', Adv. Math. Sci. Appl. 7, 467-490.
67. C. M. Elliott and A. R. Gardiner (1996), Double obstacle phase field computations of dendritic growth, University of Sussex, CMAIA Research Report 96-15.
68. C. M. Elliott and J. R. Ockendon (1982), Weak and Variational Methods for Moving Boundary Problems, Pitman, London.
69. C. M. Elliott and R. Schätzle (1996), 'The limit of the anisotropic double-obstacle Allen-Cahn equation', Proc. Roy. Soc. Edinburgh 126, 1217-1234.
70. C. M. Elliott and R. Schätzle (1997), The limit of the fully anisotropic double-obstacle Allen-Cahn equation in the non-smooth case, SIAM J. Math. Anal. 28, 274-303.
71. C. M. Elliott and A. M. Stuart (1993), 'The global dynamics of discrete semilinear parabolic equations', SIAM J. Numer. Anal. 30, 1622-1663.
72. C. M. Elliott and V. M. Styles (2003), 'Computations of bidirectional grain boundary dynamics in thin metallic films', J. Comput. Phys. 187, 524-543.
73. C. M. Elliott, A. R. Gardiner and T. Kuhn (1996), Generalizes double obstacle phase field approximation of the anisotropic mean curvature flow, University of Sussex, CMAIA Research Report 96-17.
74. C. M. Elliott, A. R. Gardiner and R. Schätzle (1998), 'Crystalline curvature flow of a graph in a variational setting', Adv. Math. Sci. Appl. 8, 425-460.
75. C. M. Elliott, M. Paolini and R. Schätzle (1996), 'Interface estimates for the fully anisotropic Allen-Cahn equation and anisotropic mean curvature flow', Math. Models Methods Appl. Sci. 6, 1103-1118.
76. J. Escher, U. F. Mayer and G. Simonett (1998), 'The surface diffusion flow for immersed hypersurfaces', SIAM J. Math. Anal. 29, 1419-1433.
77. L. C. Evans (1993), 'Convergence of an algorithm for mean curvature motion', Indiana Univ. Math. J. 42, 533-557.
78. L. C. Evans and J. Spruck (1991), 'Motion of level sets by mean curvature I', J. Differential Geom. 33, 636-681.
79. L. C. Evans and J. Spruck (1992a), 'Motion of level sets by mean curvature II', Trans. Amer. Math. Soc. 330, 321-332.
80. L. C. Evans and J. Spruck (1992b), 'Motion of level sets by mean curvature IIP, J. Geom. Anal. 2, 121-150.
81. L. C. Evans and J. Spruck (1995), 'Motion of level sets by mean curvature IV', J. Geom. Anal. 5, 77-114.
82. L. C. Evans, H. M. Soner and P. E. Souganidis (1992), 'Phase transition and generalised motion by mean curvature', Comm. Pure Appl. Math. 45, 1097-1123.
83. X. Feng and A. Prohl (2003), 'Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows', Numer. Math. 94, 33-65.
84. F. Fierro, R. Goglione and M. Paolini (1994), 'Finite element minimization of curvature functionals with anisotropy', Calcolo 3-4, 191-210.
85. M. Fried (1999), Niveauflächen zur Berechnung zweidimensionaler Dendrite, Dissertation Freiburg.
86. M. Fried (2004), 'A level set based finite element algorithm for the simulation of dendritic growth', Comput. Vis. Sci. 7 97-110.
87. M. Gage (1993), 'Evolving plane curves by curvature in relative geometries', Duke Math. J. 72, 441-466.
88. M. Gage and R. S. Hamilton (1996), 'The heat equation shrinking convex plane curves', J. Differential Geom. 23 69-96.
89. H. Garcke and V. M. Styles (2004), 'Bi-directional diffusion induced grain boundary motion with triple junctions', Interfaces Free Bound. 6, 271-294.
90. Y. Giga (2002), Surface evolution equations: a level set method, Hokkaido University Technical Report Series in Mathematics No. 71.
91. Y. Giga and K. Ito (1998), 'On pinching of curves moved by surface diffusion', Commun. Appl. Anal. 2(3), 393-405.
92. D. Gilbarg and N. S. Trudinger (1998), Elliptic Partial Differential Equations of Second Order, Grundlehren der mathematischen Wissenschaften, Springer.
93. P. M. Girao (1995), 'Convergence of a crystalline algorithm for the motion of a simple closed convex curve by weighted curvature', SIAM J. Numer. Anal. 32, 886-899.
94. M. A. Grayson (1987), 'The heat equation shrinks embedded plane curves to round points', J. Differential Geom. 26, 285-314.
95. M. A. Grayson (1989) 'A short note on the evolution of a surface by its mean curvature', Duke Math. J. 58 (3), 555-558.
96. M. E. Gurtin (1993), Thermomechanics of Evolving Phase Boundaries in the Plane, Oxford Mathematical Monographs, Oxford University Press.
97. C. Herring (1951), Surface diffusion as a motivation for sintering, in The Physics of Powder Metallurgy (W. E. Kingston, ed.), McGraw-Hill, New York, pp. 143-179.
98. G. Huisken (1984), 'Flow by mean curvature of convex surfaces into spheres', J. Differential Geom. 20, 237-266.
99. G. Huisken (1989), 'Non-parametric mean curvature evolution with boundary conditions', J. Differential Equations 77, 369-378.
100. T. Ilmanen (1993), 'Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature', J. Differential Geom. 38, 417-461.
101. C. Johnson and V. Thom'ee (1975), 'Error estimates for a finite element approximation of a minimal surface', Math. Comp. 29, 343-349.
102. M. A. Katsoulakis and A. T. Kho (2001), 'Stochastic curvature flows: asymptotic derivation, level set formulation and numerical experiments', Interfaces Free Bound. 3, 265-290.
103. M. Kimura (1997), 'Numerical analysis of moving boundary problems using the boundary tracking method', Japan J. Indust. Appl. Math. 14, 373-398.
104. R. Kornhuber and R. Krause (2003), On multi-grid methods for vector Allen-Cahn equations, in Domain Decomposition Methods in Science and Engineering (I. Herrera et al., eds), UNAM, Mexico City, Mexico, pp. 307-314.
105. T. Kuhn (1996), Approximation of anisotropic and advected mean curvature flows by phase field models, DPhil thesis, University of Sussex.
106. T. Kühn (1998), 'Convergence of a fully discrete approximation for advected mean curvature flows', IMA J. Numer. Anal. 18, 595-634.
107. E. Kuwert and R. Schätzle (2004a), 'The Willmore flow with small initial energy', J. Differential Geom. 57, 409-441.
108. E. Kuwert and R. Schätzle (2004b), 'Removability of point singularities of Willmore surfaces', Ann. of Math. 159, 1-43.
109. O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva (1968), Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, RI.
110. G. Lieberman (1986), 'The first initial-boundary value problem for quasilinear second order parabolic equations', Ann. Scuola Norm. Sup. Pisa 13, 347-387.
111. G. B. McFadden, A. A. Wheeler, R. J. Braun, S. R. Coriell and R. F. Sekerka (1993), 'Phase-field models for anisotropic interfaces', Phys. Rev. E 48 (3), 2016-2024.
112. U. F. Mayer and G. Simonett (2000), 'Self-intersections for the surface diffusion and the volume-preserving mean curvature flow', Differential Integral Equations 13, 1189-1199.
113. U. F. Mayer and G. Simonett (2002), 'A numerical scheme for axi-symmetric solutions of curvature driven free boundary problems, with applications to the Willmore flow', Interfaces Free Bound. 4, 89-109.
114. B. Merriman, J. K. Bence and S. Osher (1994), 'Motion of multiple junctions: A level set approach', J. Comput. Phys. 112, 343-363.
115. K. Mikula and J. Kacur (1996), 'Evolution of convex plane curves describing anisotropic motions of phase interfaces', SIAM J. Sci. Comput. 17 (6), 1302-1327.
116. K. Mikula and D. Sevcovic (2001), 'Evolution of plane curves driven by a nonlinear function of curvature and anisotropy', SIAM J. Appl. Math. 61 (5), 1473-1501.
117. W. W. Mullins (1957), 'Theory of thermal grooving', J. Appl. Phys. 28, 333-339.
118. F. A. Nichols and W. W. Mullins (1965), 'Surface-(interface-) and volume-diffusion contributions to morphological changes driven by capillarity', Trans. Metall. Soc., AIME 233, 1840-1847.
119. R. H. Nochetto and C. Verdi (1996a), 'Convergence of double obstacle problems to the geometric motions of fronts', SIAM J. Math. Anal. 26, 1514-1526.
120. R. H. Nochetto and C. Verdi (1996b), 'Combined effect of explicit time-stepping and quadrature for curvature driven flows', Numer. Math. 74, 105-136.
121. R. H. Nochetto and C. Verdi (1997), 'Convergence past singularities for a fully discrete approximation of curvature driven interfaces', SIAM J. Numer. Anal. 34, 490-512.
122. R. H. Nochetto, M. Paolini and C. Verdi (1993), 'Sharp error analysis for curvature dependent evolving fronts', Math. Models Methods Appl. Sci. 3, 771-723.
123. R. H. Nochetto, M. Paolini and C. Verdi (1994), 'Optimal interface error estimates for the mean curvature flow', Ann. Scuola Norm. Sup. Pisa Cl. Sci. 21, 193-212.
124. R. H. Nochetto, M. Paolini and C. Verdi (1996), 'A dynamic mesh algorithm for curvature dependent evolving interfaces', J. Comput. Phys. 123, 296-310.
125. V. I. Oliker and N. N. Uraltseva (1993), 'Evolution of nonparametric surfaces with speed depending on curvature II: The mean curvature case', Comm. Pure Appl. Math. 46, No 1, 97-135.
126. S. Osher and R. Fedkiw (2003), Level Set Methods and Dynamic Implicit Surfaces, Vol. 153 of Applied Mathematical Sciences, Springer.
127. S. Osher and J. A. Sethian (1988), 'Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations', J. Comput. Phys. 79, 12-49.
128. M. Paolini (1995), An efficient algorithm for computing anisotropic evolution by mean curvature, in Proceedings of the International Conference on Curvature Flows and Related Topics held in Levico, Italy, June 27-July 2nd, 1994 (A. Damlamian et al., eds), Vol. 5 of GAKUTO Int. Ser., Math. Sci. Appl., Gakkotosho, Tokyo, pp. 199-213.
129. M. Paolini (1997), 'A quasi-optimal error estimate for a discrete singularly perturbed approximation to the prescribed curvature problem', Math. Comp. 66, 45-67.
130. M. Paolini and C. Verdi (1992), 'Asymptotic and numerical analysis of the mean curvature flow with a space dependent relaxation parameter', Asymptot. Anal. 5, 553-574.
131. A. Polden (1996), Curves and surfaces of least total curvature and fourth-order flows, PhD dissertation, Universität Tübingen, Germany.
132. T. Preußer and M. Rumpf (2000), A level set method for anisotropic geometric diffusion in 3D image processing, Report SFB 256 Bonn, 37.
133. J. Rubinstein, P. Sternberg and J. B. Keller (1989), 'Fast reaction, slow diffusion and curve shortening', SIAM J. Appl. Math. 49, 116-133.
134. R. Rusu (2001), An algorithm for the elastic flow of surfaces, Preprint Math. Fak. Univ. Freiburg 01-35.
135. A. Schmidt (1996), 'Computation of three dimensional dendrites with finite elements', J. Comput. Phys. 125, 293-312.
136. J. A. Sethian (1990), 'Numerical algorithms for propagating interfaces: Hamilton-Jacobi equations and conservation laws', J. Differential Geom. 31 (1), 131-161.
137. J. A. Sethian (1999), Level Set Methods and Fast Marching Methods, Vol. 3 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press.
138. G. Simonett (2001), 'The Willmore flow near spheres', Differential Integral Equations 14, 1005-1014.
139. H. M. Soner (1993), 'Motion of a set by the curvature of its boundary', J. Differential Equations 101, 313-372.
140. J. E. Taylor (1992), 'Mean curvature and weighted mean curvature', Acta Metall. Mater. 40, 1475-1485.
141. J. E. Taylor, J. W Cahn and C. A. Handwerker (1992), 'Geometric models of crystal growth', Acta Metall. Mater. 40, 1443-1474.
142. C. Truesdell (1983), 'The influence of elasticity on analysis: The classical heritage', Bull. Amer. Math. Soc. (N.S.) 9, 293-310.
143. A. Veeser (1999), 'Error estimates for semi-discrete dendritic growth', Interfaces Free Bound. 1, 227-254.
144. A. Visintin (1996), Models of Phase Transitions, Vol. 28 of Progress in Nonlinear Differential Equations, Birkhäuser, Boston.
145. N. J. Walkington (1996), 'Algorithms for computing motion by mean curvature', SIAM J. Numer. Anal. 33, 2215-2238.
146. A. A. Wheeler and G. B. McFadden (1996), 'A ξ-vector formulation of anisotropic phase-field models: 3-D Asymptotics', European J. Appl. Math. 7, 367-381.
147. T. J. Willmore (1993), Riemannian Geometry, Clarendon, Oxford.
148. E. Yokoyama and R. F. Sekerka (1992), 'A numerical study of the combined effect of anisotropic surface tension and interface kinetics on pattern formation during the growth of two-dimensional crystals', J. Crystal Growth 125, 389-403.