Curvature Driven Interface Evolution

  • Harald GarckeEmail author
Survey Article


Curvature driven surface evolution plays an important role in geometry, applied mathematics and in the natural sciences. In this paper geometric evolution equations such as mean curvature flow and its fourth order analogue motion by surface diffusion are studied as examples of gradient flows of the area functional. Also in many free boundary problems the motion of an interface is given by an evolution law involving curvature quantities. We will introduce the Mullins-Sekerka flow and the Stefan problem with its anisotropic variants and discuss their properties.

In phase field models the area functional is replaced by a Ginzburg-Landau functional leading to a diffuse interface model. We derive the Allen-Cahn equation, the Cahn-Hilliard equation and the phase field system as gradient flows and relate them to sharp interface evolution laws.


Mean curvature flow Gradient flow Surface diffusion Mullins-Sekerka problem Stefan problem Crystal growth Phase field equation Allen-Cahn equation Cahn-Hilliard equation 

Mathematics Subject Classification (2000)

53C44 53K93 35K91 35R35 35K55 49Q20 53A10 80A22 82B24 



Figures 13, 5, 812 are numerical computations by Robert Nürnberg (Imperial College, London) and they were performed in the context of the work [8, 9, 10, 11, 12, 13]. Figure 6 has been provided by Ulrich Weikard. Helmut Abels, Klaus Deckelnick, Daniel Depner, Hans-Christoph Grunau, Claudia Hecht, Barbara Niethammer and Matthias Röger made helpful suggestions which improved the presentation. I would like to express my gratitude to all the above mentioned colleagues for their contributions and to Eva Rütz for typing my often very rough notes.


  1. 1.
    Abels, H., Wilke, M.: Convergence to equilibrium for the Cahn-Hilliard equation with a logarithmic free energy. Nonlinear Anal. 67(11), 3176–3193 (2007) MathSciNetzbMATHGoogle Scholar
  2. 2.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. Clarendon Press, Oxford (2000) zbMATHGoogle Scholar
  3. 3.
    Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel (2005), viii+333 pp. zbMATHGoogle Scholar
  4. 4.
    Alikakos, N.D., Bates, P.W., Chen, X.: Convergence of the Cahn-Hilliard equation to the Hele-Shaw model. Arch. Ration. Mech. Anal. 128(2), 165–205 (1994) MathSciNetzbMATHGoogle Scholar
  5. 5.
    Almgren, F., Taylor, J.E., Wang, L.: Curvature-driven flows: a variational approach. SIAM J. Control Optim. 31(2), 387–438 (1993) MathSciNetzbMATHGoogle Scholar
  6. 6.
    Angenent, S.B., Gurtin, M.E.: Multiphase thermomechanics with interfacial structure. II. Evolution of an isothermal interface. Arch. Ration. Mech. Anal. 108, 323–391 (1989) MathSciNetzbMATHGoogle Scholar
  7. 7.
    Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing. Partial Differential Equations and the Calculus of Variations with a Foreword by Olivier Faugeras. Applied Mathematical Sciences, vol. 147. Springer, New York (2002), xxvi+286 pp. Google Scholar
  8. 8.
    Barrett, J.W., Garcke, H., Nürnberg, R.: A parametric finite element method for fourth order geometric evolution equations. J. Comput. Phys. 222(1), 441–462 (2007) MathSciNetzbMATHGoogle Scholar
  9. 9.
    Barrett, J.W., Garcke, H., Nürnberg, R.: On the parametric finite element approximation of evolving hypersurfaces in \(\mathbb{R}^{3}\). J. Comput. Phys. 227(9), 4281–4307 (2008) MathSciNetzbMATHGoogle Scholar
  10. 10.
    Barrett, J.W., Garcke, H., Nürnberg, R.: Parametric approximation of surface clusters driven by isotropic and anisotropic surface energies. Interfaces Free Bound. 12(2), 187–234 (2010) MathSciNetzbMATHGoogle Scholar
  11. 11.
    Barrett, J.W., Garcke, H., Nürnberg, R.: On stable parametric finite element methods for the Stefan problem and the Mullins-Sekerka problem with applications to dendritic growth. J. Comput. Phys. 229(18), 6270–6299 (2010) MathSciNetzbMATHGoogle Scholar
  12. 12.
    Barrett, J.W., Garcke, H., Nürnberg, R.: Numerical computations of facetted pattern formation in snow crystal growth. Phys. Rev. E 86(1), 011604 (2012) Google Scholar
  13. 13.
    Barrett, J.W., Garcke, H., Nürnberg, R.: A variational formulation of anisotropic geometric evolution equations in higher dimensions. Numer. Math. 109(1), 1–44 (2008) MathSciNetzbMATHGoogle Scholar
  14. 14.
    Bellettini, G.: An introduction to anisotropic and crystalline mean curvature flow. In: Proceedings of Minisemester on Evolution of Interfaces. Sapporo 210. Hokkaido University Technical Report Series in Math., vol. 145, pp. 102–159 (2010) Google Scholar
  15. 15.
    Bellettini, G., Novaga, M., Paolini, M.: Facet-breaking for three-dimensional crystals evolving by mean curvature. Interfaces Free Bound. 1, 39–55 (1999) MathSciNetzbMATHGoogle Scholar
  16. 16.
    Bellettini, G., Novaga, M., Paolini, M.: On a crystalline variational problem, part I: First variation and global L regularity. Arch. Ration. Mech. Anal. 157, 165–191 (2001) MathSciNetzbMATHGoogle Scholar
  17. 17.
    Bellettini, G., Paolini, M.: Anisotropic motion by mean curvature in the context of Finsler geometry. Hokkaido Math. J. 25(3), 537–566 (1996) MathSciNetzbMATHGoogle Scholar
  18. 18.
    Bernoff, A.J., Bertozzi, A.L., Witelski, T.P.: Axisymmetric surface diffusion: dynamics and stability of self-similar pinchoff. J. Stat. Phys. 93(3–4), 725–776 (1998) MathSciNetzbMATHGoogle Scholar
  19. 19.
    Blatt, S.: Loss of convexity and embeddedness for geometric evolution equations of higher order. J. Evol. Equ. 10(1), 21–27 (2010) MathSciNetzbMATHGoogle Scholar
  20. 20.
    Blowey, J.F., Elliott, C.M.: The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy. I. Mathematical analysis. Eur. J. Appl. Math. 2(3), 233–280 (1991) MathSciNetzbMATHGoogle Scholar
  21. 21.
    Braides, A.: Γ-Convergence for Beginners. Oxford Lecture Series in Mathematics and Its Applications, vol. 22. Oxford University Press, London (2005), xii+217 pp. Google Scholar
  22. 22.
    Brakke, K.A.: The Motion of a Surface by Its Mean Curvature. Mathematical Notes, vol. 20. Princeton University Press, Princeton (1978), i+252 pp. zbMATHGoogle Scholar
  23. 23.
    Brochet, D., Chen, X., Hilhorst, D.: Finite dimensional exponential attractor for the phase field model. J. Anal. Appl. 49, 197–212 (1993) MathSciNetzbMATHGoogle Scholar
  24. 24.
    Bronsard, L., Garcke, H., Stoth, B.: A multi-phase Mullins-Sekerka system: matched asymptotic expansions and an implicit time discretisation for the geometric evolution problem. Proc. R. Soc. Edinb. A 128, 481–506 (1998) MathSciNetzbMATHGoogle Scholar
  25. 25.
    Bronsard, L., Kohn, R.V.: Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics. J. Differ. Equ. 90(2), 211–237 (1991) MathSciNetzbMATHGoogle Scholar
  26. 26.
    Bronsard, L., Stoth, B.: Volume-preserving mean curvature flow as a limit of a nonlocal Ginzburg-Landau equation. SIAM J. Math. Anal. 28(4), 769–807 (1997) MathSciNetzbMATHGoogle Scholar
  27. 27.
    Brokate, M., Sprekels, J.: Hysteresis and Phase Transitions. Applied Mathematical Sciences, vol. 121. Springer, New York (1996), x+357 pp. zbMATHGoogle Scholar
  28. 28.
    Caginalp, G.: An analysis of a phase field model of a free boundary. Arch. Ration. Mech. Anal. 92, 205–245 (1986) MathSciNetzbMATHGoogle Scholar
  29. 29.
    Caginalp, G.: Stefan and Hele-Shaw type models as asymptotic limits of the phase-field equations. Phys. Rev. A 39(3), 5887–5896 (1989) MathSciNetzbMATHGoogle Scholar
  30. 30.
    Caginalp, G., Chen, X.: Convergence of the phase field model to its sharp interface limits. Eur. J. Appl. Math. 9, 417–445 (1998) MathSciNetzbMATHGoogle Scholar
  31. 31.
    Caginalp, G., Fife, P.C.: Dynamics of layered interfaces arising from phase boundaries. SIAM J. Appl. Math. 48(3), 506–518 (1988) MathSciNetGoogle Scholar
  32. 32.
    Cahn, J.W., Elliott, C.M., Novick-Cohen, A.: The Cahn-Hilliard equation with a concentration dependent mobility: motion by minus the Laplacian of the mean curvature. Eur. J. Appl. Math. 7(3), 287–301 (1996) MathSciNetzbMATHGoogle Scholar
  33. 33.
    Cao, F.: Geometric Curve Evolution and Image Processing. Lecture Notes in Mathematics, vol. 1805. Springer, Berlin (2003) zbMATHGoogle Scholar
  34. 34.
    Chan, T.F., Shen, J.: Image Processing and Analysis. SIAM, Philadelphia (2005), xxi+184 pp. zbMATHGoogle Scholar
  35. 35.
    Chen, L.-Q.: Phase-field models for microstructure evolution. Annu. Rev. Mater. Res. 32, 113–140 (2002) Google Scholar
  36. 36.
    Chen, X.: Generation and propagation of interfaces for reaction-diffusion equations. J. Differ. Equ. 96(1), 116–141 (1992) zbMATHGoogle Scholar
  37. 37.
    Chen, X.: The Hele-Shaw problem and area-preserving curve shortening motion. Arch. Ration. Mech. Anal. 123, 117–151 (1993) zbMATHGoogle Scholar
  38. 38.
    Chen, X., Hilhorst, D., Logak, E.: Mass conserving Allen-Cahn equation and volume preserving mean curvature flow. Interfaces Free Bound. 12(4), 527–549 (2010) MathSciNetzbMATHGoogle Scholar
  39. 39.
    Chen, X., Hong, J., Yi, F.: Existence, uniqueness and regularity of classical solutions of the Mullins-Sekerka problem. Commun. Partial Differ. Equ. 21, 1705–1727 (1996) MathSciNetzbMATHGoogle Scholar
  40. 40.
    Chen, X., Reitich, F.: Local existence and uniqueness of the classical Stefan problem with surface tension and dynamical undercooling. J. Math. Anal. Appl. 162, 350–362 (1992) MathSciNetGoogle Scholar
  41. 41.
    Chen, Y.G., Giga, Y., Goto, S.L.: Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Differ. Geom. 33, 749–786 (1991) MathSciNetzbMATHGoogle Scholar
  42. 42.
    Chou, K.-S.: A blow-up criterion for the curve shortening flow by surface diffusion. Hokkaido Math. J. 32(1), 1–19 (2003) MathSciNetzbMATHGoogle Scholar
  43. 43.
    Constantin, P., Pugh, M.: Global solutions for small data to the Hele-Shaw problem. Nonlinearity 6, 393–415 (1993) MathSciNetzbMATHGoogle Scholar
  44. 44.
    Cummings, L.J., Richardson, G., Ben Amar, M.: Models of void electromigration. Eur. J. Appl. Math. 12(2), 97–134 (2001) MathSciNetzbMATHGoogle Scholar
  45. 45.
    Dai, S., Niethammer, B., Pego, R.L.: Crossover in coarsening rates for the monopole approximation of the Mullins-Sekerka model with kinetic drag. Proc. R. Soc. Edinb., Sect. A, Math. 140(03), 553–571 (2010) MathSciNetzbMATHGoogle Scholar
  46. 46.
    Dal Maso, G.: An Introduction to Γ-Convergence. Progress in Nonlinear Differential Equations and Their Applications, vol. 8. Birkhäuser, Boston (1993), xiv+339 pp. Google Scholar
  47. 47.
    Davi, F., Gurtin, M.E.: On the motion of a phase interface by surface diffusion. Z. Angew. Math. Phys. 41(6), 782–811 (1990) MathSciNetzbMATHGoogle Scholar
  48. 48.
    Deckelnick, K., Elliott, C.M.: Local and global existence results for anisotropic Hele-Shaw flows. Proc. R. Soc. Edinb. A 129, 265–294 (1999) MathSciNetzbMATHGoogle Scholar
  49. 49.
    Deckelnick, K., Dziuk, G., Elliott, C.M.: Computation of geometric partial differential equations and mean curvature flow. Acta Numer. 14, 139–232 (2005) MathSciNetzbMATHGoogle Scholar
  50. 50.
    De Mottoni, P., Schatzman, M.: Geometrical evolution of developed interfaces. Trans. Am. Math. Soc. 347(5), 1533–1589 (1995) zbMATHGoogle Scholar
  51. 51.
    Dierkes, U., Hildebrandt, S., Sauvigny, F.: Minimal Surfaces. Revised and Enlarged, 2nd edn. Grundlehren der Mathematischen Wissenschaften, vol. 339. Springer, Heidelberg (2010), xvi+688 pp. Google Scholar
  52. 52.
    Dinghas, A.: Über einen geometrischen Satz von Wulff für die Gleichgewichtsform von Kristallen. Z. Kristallogr. 105, 304–314 (1944) MathSciNetzbMATHGoogle Scholar
  53. 53.
    Duchon, J., Robert, R.: Evolution d’une interface par capillarité et diffusion de volume I. Existence locale en temps. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 361–378 (1984) MathSciNetzbMATHGoogle Scholar
  54. 54.
    Dupaix, C., Hilhorst, D., Kostin, I.N.: The viscous Cahn-Hilliard equation as a limit of the phase field model: lower semicontinuity of the attractor. J. Dyn. Differ. Equ. 11(2), 333–353 (1999). (English summary) MathSciNetzbMATHGoogle Scholar
  55. 55.
    Dziuk, G.: An algorithm for evolutionary surfaces. Numer. Math. 58, 603–611 (1991) MathSciNetzbMATHGoogle Scholar
  56. 56.
    Eck, C., Garcke, H., Knabner, P.: Mathematische Modellierung. Springer, Berlin (2011). xiv+513 pp., Revised second edn. zbMATHGoogle Scholar
  57. 57.
    Ecker, K.: Regularity Theory for Mean Curvature Flow. Progress in Nonlinear Differential Equations and Their Applications, vol. 57. Birkhäuser, Boston (2004), xiv+165 pp. zbMATHGoogle Scholar
  58. 58.
    Ecker, K.: Heat equations in geometry and topology. Jahresber. Dtsch. Math.-Ver. 110(3), 117–141 (2008) MathSciNetzbMATHGoogle Scholar
  59. 59.
    Efendiev, M.A., Gajewski, H., Zelik, S.: The finite dimensional attractor for a 4th order system of Cahn-Hilliard type with a supercritical nonlinearity. Adv. Differ. Equ. 7(9), 1073–1100 (2002) MathSciNetzbMATHGoogle Scholar
  60. 60.
    Eilks, C., Elliott, C.M.: Numerical simulation of dealloying by surface dissolution via the evolving surface finite element method. J. Comput. Phys. 227(23), 9727–9741 (2008) MathSciNetzbMATHGoogle Scholar
  61. 61.
    Elliott, C.M.: The Cahn-Hilliard model for the kinetics of phase separation. In: Mathematical Models for Phase Change Problems, Óbidos, 1988. Internat. Ser. Numer. Math., vol. 88, pp. 35–73. Birkhäuser, Basel (1989) Google Scholar
  62. 62.
    Elliott, C.M., Garcke, H.: On the Cahn-Hilliard equation with degenerate mobility. SIAM J. Math. Anal. 27(2), 404–423 (1996) MathSciNetzbMATHGoogle Scholar
  63. 63.
    Elliott, C.M., Garcke, H.: Existence results for diffusive surface motion laws. Adv. Math. Sci. Appl. 7(1), 467–490 (1997) MathSciNetzbMATHGoogle Scholar
  64. 64.
    Elliott, C.M., Maier-Paape, S.: Losing a graph with surface diffusion. Hokkaido Math. J. 30, 297–305 (2001) MathSciNetzbMATHGoogle Scholar
  65. 65.
    Elliott, C.M., Ockendon, J.R.: Weak and Variational Methods for Moving Boundary Problems. Pitman, Boston (1982) zbMATHGoogle Scholar
  66. 66.
    Elliott, C.M., Zheng, S.: On the Cahn-Hilliard equation. Arch. Ration. Mech. Anal. 96(4), 339–357 (1986) MathSciNetzbMATHGoogle Scholar
  67. 67.
    Elliott, C.M., Zheng, S.: Global existence and stability of solutions to the phase field equations. In: Free Boundary Value Problems, Oberwolfach, 1989. Internat. Ser. Numer. Math., vol. 95, pp. 46–58. Birkhäuser, Basel (1990) Google Scholar
  68. 68.
    Escher, J.: The Dirichlet-Neumann operator on continuous functions. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 4(21), 235–266 (1994) MathSciNetGoogle Scholar
  69. 69.
    Escher, J.: Funktionalanalytische Methoden bei freien Randwertaufgaben. Jahresber. Dtsch. Math.-Ver. 109(4), 195–219 (2007) MathSciNetzbMATHGoogle Scholar
  70. 70.
    Escher, J., Simonett, G.: Classical solutions for Hele-Shaw models with surface tension. Adv. Differ. Equ. 2(4), 619–642 (1997) MathSciNetzbMATHGoogle Scholar
  71. 71.
    Escher, J., Simonett, G.: A center manifold analysis for the Mullins-Sekerka model. J. Differ. Equ. 143(2), 267–292 (1998) MathSciNetzbMATHGoogle Scholar
  72. 72.
    Escher, J., Mayer, U., Simonett, G.: The surface diffusion flow for immersed hypersurfaces. SIAM J. Math. Anal. 29(6), 1419–1433 (1998) MathSciNetzbMATHGoogle Scholar
  73. 73.
    Escher, J., Prüss, J., Simonett, G.: Analytic solutions for a Stefan problem with Gibbs-Thomson correction. J. Reine Angew. Math. 563, 1–52 (2003) MathSciNetzbMATHGoogle Scholar
  74. 74.
    Evans, L.C., Soner, H.M., Souganidis, P.E.: Phase transitions and generalized motion by mean curvature. Commun. Pure Appl. Math. 45(9), 1097–1123 (1992) MathSciNetzbMATHGoogle Scholar
  75. 75.
    Evans, C., Spruck, J.: Motion by mean curvature I. J. Differ. Geom. 33, 635–681 (1991) MathSciNetzbMATHGoogle Scholar
  76. 76.
    Fife, P.C.: Models for phase separation and their mathematics. In: Mimura, M., Nishida, T. (eds.) Nonlinear Partial Differential Equations and Applications. KTK, Tokyo (1993) Google Scholar
  77. 77.
    Fife, P.C.: Barrett Lecture Notes (1991). University of Tennessee Google Scholar
  78. 78.
    Fonseca, I.: The Wulff theorem revisited. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 432(1884), 125–145 (1991) MathSciNetzbMATHGoogle Scholar
  79. 79.
    Fonseca, I., Müller, S.: A uniqueness proof for the Wulff theorem. Proc. R. Soc. Edinb. A 119(1–2), 125–136 (1991) zbMATHGoogle Scholar
  80. 80.
    Friedman, A.: Variational Principles and Free Boundary Problems. Wiley/Interscience, New York (1982) zbMATHGoogle Scholar
  81. 81.
    Gage, M., Hamilton, R.S.: The heat equation shrinking convex plane curves. J. Differ. Geom. 23, 69–95 (1986) MathSciNetzbMATHGoogle Scholar
  82. 82.
    Garcke, H.: Mechanical effects in the Cahn-Hilliard model: a review on mathematical results. In: Miranville, A. (ed.) Mathematical Methods and Models in Phase Transitions, pp. 43–77. Nova Science, New York (2005) Google Scholar
  83. 83.
    Garcke, H.: Kepler, Kristalle und Computer. Mathematik und numerische Simulationen helfen Kristallwachstum zu verstehen. MDMV 20, 219–228 (2012) MathSciNetzbMATHGoogle Scholar
  84. 84.
    Garcke, H., Nestler, B., Stinner, B.: A diffuse interface model for alloys with multiple components and phases. SIAM J. Appl. Math. 64(3), 775–799 (2004) MathSciNetzbMATHGoogle Scholar
  85. 85.
    Garcke, H., Schaubeck, S.: Existence of weak solutions for the Stefan problem with anisotropic Gibbs-Thomson law. Adv. Math. Sci. Appl. 21(1), 255–283 (2011) MathSciNetzbMATHGoogle Scholar
  86. 86.
    Garcke, H., Sturzenhecker, T.: The degenerate multi-phase Stefan problem with Gibbs-Thomson law. Adv. Math. Sci. Appl. 8(2), 929–941 (1998) MathSciNetzbMATHGoogle Scholar
  87. 87.
    Garcke, H., Wieland, S.: Surfactant spreading on thin viscous films: nonnegative solutions of a coupled degenerate system. SIAM J. Math. Anal. 37(6), 2025–2048 (2006) MathSciNetzbMATHGoogle Scholar
  88. 88.
    Giga, Y.: Anisotropic curvature effects in interface dynamics. Sūgaku Expo. 52, 113–117 (2000). English translation, Sugaku Expositions 16, 135–152 (2003) MathSciNetzbMATHGoogle Scholar
  89. 89.
    Giga, Y.: Singular diffusivity—facets, shocks and more. In: Hill, J.M., Moore, R. (eds.) Applied Math. Entering the 21st Century, pp. 121–138. ICIAM, Sydney (2003). SIAM, Philadelphia 2004 Google Scholar
  90. 90.
    Giga, Y.: Surface Evolution Equations. A Level Set Approach. Monographs in Mathematics, vol. 99. Birkhäuser, Basel (2006), xii+264 pp. zbMATHGoogle Scholar
  91. 91.
    Giga, M.-H., Giga, Y.: On the role of kinetic and interfacial anisotropy in the crystal growth theory. Preprint (2013) Google Scholar
  92. 92.
    Giga, M.-H., Giga, Y.: Very singular diffusion equations: second and fourth order problems. Jpn. J. Ind. Appl. Math. 27, 323–345 (2010) MathSciNetzbMATHGoogle Scholar
  93. 93.
    Giga, Y., Ito, K.: Loss of convexity of simple closed curves moved by surface diffusion. In: Topics in Nonlinear Analysis. Progr. Nonlinear Differential Equations Appl., vol. 35, pp. 305–320. Birkhäuser, Basel (1999) Google Scholar
  94. 94.
    Giga, Y., Ito, K.: On pinching of curves moved by surface diffusion. Commun. Appl. Anal. 2(3), 393–406 (1998) MathSciNetzbMATHGoogle Scholar
  95. 95.
    Giga, Y., Rybka, P.: Quasi-static evolution of 3-D crystals grown from supersaturated vapor. J. Differ. Equ. 15, 1–15 (2003) MathSciNetGoogle Scholar
  96. 96.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Grundlehren der mathematischen Wissenschaften, vol. 224. Springer, Berlin (1998), xiii+517 pp. Google Scholar
  97. 97.
    Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics, vol. 80. Birkhäuser, Basel (1984), xii+240 pp. zbMATHGoogle Scholar
  98. 98.
    Grayson, M.A.: The heat equation shrinks embedded plane curves to round points. J. Differ. Geom. 26, 285–314 (1987) MathSciNetzbMATHGoogle Scholar
  99. 99.
    Gurtin, M.E.: Configurational Forces as Basic Concepts of Continuum Physics. Applied Mathematical Sciences, vol. 137. Springer, New York (2000), vii+249 pp. Google Scholar
  100. 100.
    Gurtin, M.G.: Thermodynamics of Evolving Phase Boundaries in the Plane. Clarendon, Oxford (1993) Google Scholar
  101. 101.
    Hadžić, M., Guo, Y.: Stability in the Stefan problem with surface tension (I). Commun. Partial Differ. Equ. 35(2), 201–244 (2010) zbMATHGoogle Scholar
  102. 102.
    Hanzawa, E.: Classical solutions of the Stefan problem. Tohoku Math. J. 33(3), 297–335 (1981) MathSciNetzbMATHGoogle Scholar
  103. 103.
    Hildebrandt, S., Tromba, A.: The Parsimonious Universe. Shape and Form in the Natural World. Copernicus, New York (1996), xiv+330 Google Scholar
  104. 104.
    Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20(1), 237–266 (1984) MathSciNetzbMATHGoogle Scholar
  105. 105.
    Huisken, G.: The volume preserving mean curvature flow. J. Reine Angew. Math. 382, 35–48 (1987) MathSciNetzbMATHGoogle Scholar
  106. 106.
    Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Differ. Geom. 31, 285–299 (1990) MathSciNetzbMATHGoogle Scholar
  107. 107.
    Huisken, G.: Local and global behaviour of hypersurfaces moving by mean curvature. Proc. Symp. Pure Math. 54, 175–191 (1993) MathSciNetGoogle Scholar
  108. 108.
    Ilmanen, T.: Convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature. J. Differ. Geom. 38(2), 417–461 (1993) MathSciNetzbMATHGoogle Scholar
  109. 109.
    Ilmanen, T.: Elliptic regularization and partial regularity for motion by mean curvature. Mem. Amer. Math. Soc. 108(520) (1994), x+90 pp. Google Scholar
  110. 110.
    Kohn, R.V., Otto, F.: Upper bounds on coarsening rates. Commun. Math. Phys. 229(3), 375–395 (2002) MathSciNetzbMATHGoogle Scholar
  111. 111.
    Kraus, C.: The degenerate and non-degenerate Stefan problem with inhomogeneous and anisotropic Gibbs-Thomson law. Eur. J. Appl. Math. 22(5), 393–422 (2011) MathSciNetzbMATHGoogle Scholar
  112. 112.
    Libbrecht, K.G.: The Snowflake. Winter’s Secret Beauty (2003). Voyageur Press Google Scholar
  113. 113.
    Libbrecht, K.G.: Morphogenesis on Ice: the physics of snow crystals. Engineering & Science 1 (2001) Google Scholar
  114. 114.
    Lifshitz, I.M., Slyozov, V.V.: The kinetics of precipitation from supersaturated solid solutions. J. Phys. Chem. Solids 19, 35–50 (1961) Google Scholar
  115. 115.
    Luckhaus, S.: Solutions for the two-phase Stefan problem with the Gibbs–Thomson law for the melting temperature. Eur. J. Appl. Math. 1(2), 101–111 (1990) MathSciNetzbMATHGoogle Scholar
  116. 116.
    Luckhaus, S., Sturzenhecker, T.: Implicit time discretization for the mean curvature flow equation. Calc. Var. Partial Differ. Equ. 3(2), 253–271 (1995) MathSciNetzbMATHGoogle Scholar
  117. 117.
    Mantegazza, C.: Lecture Notes on Mean Curvature Flow. Progress in Mathematics, vol. 290. Springer, Basel (2011), xii+166 pp. zbMATHGoogle Scholar
  118. 118.
    Mayer, U.F.: Two-sided Mullins-Sekerka flow does not preserve convexity. Electr. J. Differ. Equ. 1, 171–179 (1998) Google Scholar
  119. 119.
    Mayer, U.F.: A numerical scheme for moving boundary problems that are gradient flows for the area functional. Eur. J. Appl. Math. 11, 61–80 (2000) zbMATHGoogle Scholar
  120. 120.
    Mayer, U.F., Simonett, G.: Self-intersections for the surface diffusion and the volume-preserving mean curvature flow. Differ. Integral Equ. 13(7–9), 1189–1199 (2000) MathSciNetzbMATHGoogle Scholar
  121. 121.
    Meirmanov, A.M.: The Stefan Problem. De Gruyter, Berlin (1992) zbMATHGoogle Scholar
  122. 122.
    Mielke, A., Theil, F., Levitas, V.I.: A variational formulation of rate-independent phase transformations using an extremum principle. Arch. Ration. Mech. Anal. 162(2), 137–177 (2002) MathSciNetzbMATHGoogle Scholar
  123. 123.
    Miranville, A., Zelik, S.: Robust exponential attractors for Cahn-Hilliard type equations with singular potentials. Math. Methods Appl. Sci. 27(5), 545–582 (2004) MathSciNetzbMATHGoogle Scholar
  124. 124.
    Modica, L.: The gradient theory of phase transitions and the minimal interface criterion. Arch. Ration. Mech. Anal. 98, 123–142 (1987) MathSciNetzbMATHGoogle Scholar
  125. 125.
    Modica, L., Mortola, S.: Un esempio di Γ-convergenza. Boll. Unione Mat. Ital, B 14(5), 285–299 (1977) MathSciNetzbMATHGoogle Scholar
  126. 126.
    Morgan, F.: Geometric Measure Theory. A Beginner’s Guide, 4th edn. Elsevier, Amsterdam (2009) zbMATHGoogle Scholar
  127. 127.
    Mucha, P.: Regular solutions to a monodimensional model with discontinuous elliptic operator. Interfaces Free Bound. 14, 145–152 (2012) MathSciNetzbMATHGoogle Scholar
  128. 128.
    Mucha, P.: On weak solutions to the Stefan problem with Gibbs-Thomson correction. Differ. Integral Equ. 20(7), 769–792 (2007) MathSciNetzbMATHGoogle Scholar
  129. 129.
    Mucha, P., Rybka, P.: A note on a model system with sudden directional diffusion. J. Stat. Phys. 146, 975–988 (2012) MathSciNetzbMATHGoogle Scholar
  130. 130.
    Mullins, W.W.: Two-dimensional motion of idealized grain boundaries. J. Appl. Phys. 27, 900–904 (1956) MathSciNetGoogle Scholar
  131. 131.
    Mullins, W.W.: Theory of thermal grooving. J. Appl. Phys. 28, 333–339 (1957) Google Scholar
  132. 132.
    Niethammer, B.: Derivation of the LSW-theory for Ostwald ripening by homogenization methods. Arch. Ration. Mech. Anal. 147(2), 119–178 (1999) MathSciNetzbMATHGoogle Scholar
  133. 133.
    Niethammer, B., Otto, F.: Ostwald ripening: the screening length revisited. Calc. Var. PDE 13(1), 33–68 (2001) MathSciNetzbMATHGoogle Scholar
  134. 134.
    Niethammer, B., Pego, R.L.: Non-self-similar behavior in the LSW theory of Ostwald ripening. J. Stat. Phys. 95(5–6), 867–902 (1999) MathSciNetzbMATHGoogle Scholar
  135. 135.
    Novick-Cohen, A.: On the Viscous Cahn-Hilliard Equation, Edinburgh, 1985–1986. Material Instabilities in Continuum Mechanics, pp. 329–342. Oxford Sci. Publ., New York (1988) Google Scholar
  136. 136.
    Novick-Cohen, A.: The Cahn-Hilliard equation: mathematical and modeling perspectives. Adv. Math. Sci. Appl. 8(2), 965–985 (1998) MathSciNetzbMATHGoogle Scholar
  137. 137.
    Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Applied Mathematical Sciences, vol. 153. Springer, New York (2003), xiv+273 pp. zbMATHGoogle Scholar
  138. 138.
    Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26(1–2), 101–174 (2001) MathSciNetzbMATHGoogle Scholar
  139. 139.
    Pego, R.L.: Front migration in the nonlinear Cahn-Hilliard equation. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 422(1863), 261–278 (1989) MathSciNetzbMATHGoogle Scholar
  140. 140.
    Penrose, O., Fife, P.C.: Thermodynamically consistent models of phase-field type for the kinetics of phase transitions. Physica D 43(1), 44–62 (1990) MathSciNetzbMATHGoogle Scholar
  141. 141.
    Plotnikov, P.I., Starovoitov, V.N.: Stefan problem with surface tension as a limit of the phase field model. Differ. Equ. 29(3), 395–404 (1993) MathSciNetGoogle Scholar
  142. 142.
    Prüss, J., Simonett, G.: Stability of equilibria for the Stefan problem with surface tension. SIAM J. Math. Anal. 40(2), 675–698 (2008) MathSciNetzbMATHGoogle Scholar
  143. 143.
    Prüss, J., Simonett, G., Zacher, R.: On normal stability for nonlinear parabolic equations. Discrete Contin. Dyn. Syst. 2009(suppl.), 612–621 (2009). 7th AIMS Conference on Dynamical Systems, Differential Equations and Applications zbMATHGoogle Scholar
  144. 144.
    Prüss, J., Simonett, G.: On the two-phase Navier-Stokes equations with surface tension. Interfaces Free Bound. 12(3), 311–345 (2010) MathSciNetzbMATHGoogle Scholar
  145. 145.
    Radkevich, E.V.: The Gibbs-Thompson correction and conditions for the existence of a classical solution of the modified Stefan problem. Dokl. Akad. Nauk SSSR 316(6), 1311–1315 (1991). Translation in Soviet Math. Dokl. 43(1), 274–278 (1991) Google Scholar
  146. 146.
    Ritoré, M., Sinestrari, C.: Mean Curvature Flow and Isoperimetric Inequalities. Birkhäuser, Basel (2010) zbMATHGoogle Scholar
  147. 147.
    Röger, M.: Existence of weak solutions for the Mullins-Sekerka flow. SIAM J. Math. Anal. 37(1), 291–301 (2005) MathSciNetzbMATHGoogle Scholar
  148. 148.
    Rossi, R., Savaré, G.: Gradient flows of non convex functionals in Hilbert spaces and applications. ESAIM Control Optim. Calc. Var. 12, 564–614 (2006) MathSciNetzbMATHGoogle Scholar
  149. 149.
    Rubinstein, L.I.: The Stefan Problem. AMS Translation, vol. 27. Am. Math. Soc., Providence (1971) Google Scholar
  150. 150.
    Rybka, P., Hoffmann, K.-H.: Convergence of solutions to Cahn-Hilliard equation. Commun. Partial Differ. Equ. 24(5–6), 1055–1077 (1999) MathSciNetzbMATHGoogle Scholar
  151. 151.
    Sapiro, G.: Geometric Partial Differential Equations and Image Analysis. Cambridge University Press, Cambridge (2001) zbMATHGoogle Scholar
  152. 152.
    Schätzle, R.: The quasistationary phase field equations with Neumann boundary conditions. J. Differ. Equ. 162(2), 473–503 (2000) zbMATHGoogle Scholar
  153. 153.
    Schätzle, R.: Hypersurfaces with mean curvature given by an ambient Sobolev function. J. Differ. Geom. 58(3), 371–420 (2001) zbMATHGoogle Scholar
  154. 154.
    Schmidt, A.: Computation of three dimensional dendrites with finite elements. J. Comput. Phys. 195, 293–312 (1996) Google Scholar
  155. 155.
    Sethian, J.A.: Level Set Methods and Fast Marching Methods. Evolving Interfaces in Computational Geometry Fluid Mechanics, Computer Vision, and Materials Science, 2nd edn. Cambridge Monographs on Applied and Computational Mathematics, vol. 3. Cambridge University Press, Cambridge (1999), xx+378 zbMATHGoogle Scholar
  156. 156.
    Sethian, J.A.: Curvature and the evolution of fronts. Commun. Math. Phys. 101(4), 487–499 (1985) MathSciNetzbMATHGoogle Scholar
  157. 157.
    Soner, H.M.: Motion of a set by the curvature of its boundary. J. Differ. Equ. 101, 313–372 (1993) MathSciNetzbMATHGoogle Scholar
  158. 158.
    Soner, H.M.: Convergence of the phase field equations to the Mullins-Sekerka problem with kinetic undercooling. Arch. Ration. Mech. Anal. 131, 139–197 (1995) MathSciNetzbMATHGoogle Scholar
  159. 159.
    Solonnikov, V.A.: Unsteady flow of a finite mass of a fluid bounded by a free surface. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 152 (1986), Kraev. Zadachi Mat. Fiz. i Smezhnye Vopr. Teor. Funktsii 18, 137–157, 183–184; translation in J. Soviet Math. 40(5), 672–686 (1988) Google Scholar
  160. 160.
    Spencer, B.J., Voorhees, P.W., Davis, S.H.: Morphological instability in epitaxially strained dislocation-free solid films. Phys. Rev. Lett. 67, 3696–3699 (1991) Google Scholar
  161. 161.
    Steinbach, I.: Phase-field models in materials science. Model. Simul. Mater. Sci. Eng. 17, 073001 (2009) MathSciNetGoogle Scholar
  162. 162.
    Stoth, B.: Convergence of the Cahn-Hilliard equation to the Mullins-Sekerka problem in spherical symmetry. J. Differ. Equ. 125(1), 154–183 (1996) MathSciNetzbMATHGoogle Scholar
  163. 163.
    Stoth, B.: A sharp interface limit of the phase field equations: one-dimensional and axisymmetric. Eur. J. Appl. Math. 7(6), 603–633 (1996) MathSciNetzbMATHGoogle Scholar
  164. 164.
    Taylor, J.E.: Crystalline variational problems. Bull. Am. Math. Soc. 84(4), 568–588 (1978) zbMATHGoogle Scholar
  165. 165.
    Taylor, J.E.: Constructions and conjectures in crystalline nondifferential geometry. In: Lawson, B., Tanenblat, K. (eds.) Differential Geometry. Proceedings of the Conference on Differential Geometry, Rio de Janeiro, 1991. Pitman Monographs Surveys Pure Appl. Math., vol. 52, pp. 321–336 (1991) Google Scholar
  166. 166.
    Taylor, J.E.: Mean curvature and weighted mean curvature. Acta Metall. Mater. 40(7), 1475–1485 (1992) Google Scholar
  167. 167.
    Taylor, J.E., Cahn, J.W.: Linking anisotropic sharp and diffuse surface motion laws via gradient flows. J. Stat. Phys. 77(1–2), 183–197 (1994) MathSciNetzbMATHGoogle Scholar
  168. 168.
    Taylor, J.E., Cahn, J.W., Handwerker, C.A.: Geometric models of crystal growth. Acta Metall. Mater. 40(7), 1443–1474 (1992) Google Scholar
  169. 169.
    Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences, vol. 68. Springer, New York (1988), xxi+648 pp. zbMATHGoogle Scholar
  170. 170.
    Villani, C.: Optimal Transport. Old and New. Grundlehren der Mathematischen Wissenschaften, vol. 338. Springer, Berlin (2009), xxii+973 pp. zbMATHGoogle Scholar
  171. 171.
    Visintin, A.: Models of Phase Transitions. Progress in Nonlinear Differential Equations and Their Applications, vol. 28. Birkhäuser, Boston (1996) zbMATHGoogle Scholar
  172. 172.
    Voorhees, P.W.: The theory of Ostwald ripening. J. Stat. Phys. 38, 231–252 (1985) Google Scholar
  173. 173.
    Voorhees, P.W.: Ostwald ripening of two-phase mixtures. Annu. Rev. Mater. Sci. 22, 197–215 (1992) Google Scholar
  174. 174.
    Wagner, C.: Theorie der Alterung von Niederschlägen durch Umlösen. Z. Elektrochem. 65, 581–594 (1961) Google Scholar
  175. 175.
    Wang, S.-L., Sekerka, R.F., Wheeler, A.A., Murray, B.T., Coriell, S.R., Braun, R.J., McFadden, G.B.: Thermodynamically-consistent phase-field models for solidification. Physica D: Nonlinear Phenomena 69(1–2), 189–200 (1993) MathSciNetzbMATHGoogle Scholar
  176. 176.
    Wheeler, G.: Surface diffusion flow near spheres. Calc. Var. Partial Differ. Equ. 44(1–2), 131–151 (2012) zbMATHGoogle Scholar
  177. 177.
    Wheeler, G.: On the Curve Diffusion Flow of Closed Plane Curves. Annali di Matematica Pura ed Applicata (2012) Google Scholar
  178. 178.
    White, B.: Evolution of curves and surfaces by mean curvature. In: Proceedings of the International Congress of Mathematicians, vol. 1, Beijing, 2002, pp. 525–538 (2002) Google Scholar
  179. 179.
    Wulff, G.: Zur Frage der Geschwindigkeit des Wachstums und der Auflösung der Kristallflächen. Z. Kristallogr. 34, 449–530 (1901) Google Scholar
  180. 180.
    Zheng, S.: Asymptotic behavior of solutions to the Cahn-Hilliard equation. Appl. Anal. 23, 165–184 (1986) MathSciNetzbMATHGoogle Scholar

Copyright information

© Deutsche Mathematiker-Vereinigung and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität RegensburgRegensburgGermany

Personalised recommendations