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Generalized motion by mean curvature with Neumann conditions and the Allen-Cahn model for phase transitions

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Abstract

We study asharpinterface model for phase transitions which incorporates the interaction of the phase boundaries with the walls of a container Ω. In this model, the interfaces move by their mean curvature and are normal to δΩ. We first establish local-in-time existence and uniqueness of smooth solutions for the mean curvature equation with a normal contact angle condition. We then discuss global solutions by interpreting the equation and the boundary condition in a weak (viscosity) sense. Finally, we investigate the relation of the aforementioned model with atransitionlayer model. We prove that if Ω isconvex, the transition-layer solutions converge to the sharp-interface solutions as the thickness of the layer tends to zero. We conclude with a discussion of the difficulties that arise in establishing this result in nonconvex domains.

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References

  1. Allen, S. M., and Cahn, J. W. A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening.Acta Metall. 27, 1085–1095 (1979).

    Article  Google Scholar 

  2. Anderson, R. F., and Orey, S. Small random perturbation of dynamical systems with reflecting boundary.Nagoya Math. J. 60 (1976), 189–216.

    MathSciNet  MATH  Google Scholar 

  3. Angenent, S. Parabolic equations for curves on surfaces. I: Curves with p-integrable curvature.Ann. Math. 132 (1990), 451–483; II: Intersections, blow up and generalized solutions.Ann. Math. 133 (1991), 171–215.

    Article  MathSciNet  Google Scholar 

  4. Angenent, S., and Gurtin, M. E. Multiphase thermomechanics with interfacial structure 2. Evolution of an isothermal interface.Arch. Rat. Mech. Anal. 108, 323–391 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  5. Aronson, D. G., and Weinberger, H. Multidimensional nonlinear diffusion arising in population genetics.Adv. Math. 30, 33–76(1978).

    Article  MathSciNet  MATH  Google Scholar 

  6. Barles, G. Fully nonlinear Neumann type boundary conditions for second-order elliptic and parabolic equations.J. Diff. Equations, to appear.

  7. Barles, G., Soner, H. M., and Souganidis, P. E.Front Propagation and Phase Field Theory. Center for Nonlinear Analysis, Research Report No. 92-NA-020, Department of Mathematics, Carnegie Mellon University, June 1992.

  8. Brakke, K. A.The Motion of a Surface by Its Mean Curvature. Princeton University Press, Princeton, NJ, 1978.

    MATH  Google Scholar 

  9. Bronsard, L., and Kohn, R. V. Motion by mean curvature as the singular limit of Ginzburg-Landau model.J. Diff. Equations 90, 211–237 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  10. Bronsard, L., and Reitich, F. On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation.Arch. Rat. Mech. Anal., to appear.

  11. Cahn, J. W. Critical point wetting.J. Chem. Phys. 66, 3667–3672 (1977).

    Article  Google Scholar 

  12. Chen, X. Generation and propagation of interfaces in reaction diffusion systems.J. Diff. Equations 96, 116–141 (1992).

    Article  MATH  Google Scholar 

  13. Chen, X., and Reitich, F. Local existence and uniqueness of solutions of the Stefan problem with surface tension and kinetic undercooling.J. Math. Anal. Applic. 164, 350–362 (1992).

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  15. Crandall, M. G., Ishii, H., and Lions, P.-L. User’s guide to viscosity solutions of second order partial differential equations.Bull. Amer. Math. Soc. 27, 1–67 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  16. Crandall, M. G., and Lions, P.-L. Viscosity solutions of Hamilton-Jacobi equations.Trans. Amer. Math. Soc. 277, 1–42 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  17. De Mottoni, P., and Schatzman, M. Evolution géometrique d’interfaces.C.R.A.S. 309, 453–458 (1989).

    MATH  Google Scholar 

  18. Ecker, K., and Huisken, G. Mean curvature evolution of entire graphs.Ann. Math. 130, 453–471 (1989).

    Article  MathSciNet  Google Scholar 

  19. Evans, L. C., Soner, H. M., and Souganidis, P. E. Phase transitions and generalized motion by mean curvature.Comm. Pure and Appl. Math., to appear.

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

    MathSciNet  MATH  Google Scholar 

  21. Evans, L. C., and Spruck, J. Motion of level sets by mean curvature II.Trans. Amer. Math. Soc., to appear.

  22. Freidlin, M.Functional Integration and Partial Differential Equations. Princeton University Press, Princeton, NJ, 1985.

    MATH  Google Scholar 

  23. Freidlin, M. Limit theorems for large deviations and reaction-diffusion equations.Ann. Prob. 13, 639–675 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  24. Gage, M., and Hamilton, R. The shrinking of convex plane curves by the heat equation.J. Diff. Geom. 23, 69–96 (1986).

    MathSciNet  MATH  Google Scholar 

  25. Giga, Y., and Sato, M.-H.Neumann problem for singular degenerate parabolic equations. Hokkaido University Preprint Series in Mathematics (1992).

  26. Grayson, M. The heat equation shrinks embedded plane curves to points.J. Diff. Geom. 26, 285–314 (1987).

    MathSciNet  MATH  Google Scholar 

  27. Gurtin, M. E. Multiphase thermomechanics with interfacial structure 1. Heat conduction and the capillary balance law.Arch. Rat. Mech. Anal. 104, 195–221 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  28. Huisken, G. Non parametric mean curvature evolution with boundary conditions.J. Diff. Equations 77, 369–378 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  29. Ishii, H. Perron’s method for Hamilton-Jacobi Equations.Duke Math. J. 55, 369–384 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  30. Katsoulakis, M., and Kossioris, G. In preparation.

  31. Lions, P.-L. Optimal control of diffusion processes. Part 2: Viscosity solutions and uniqueness.Comm. P.D.E. 8, 1229–1276(1983).

    Article  MATH  Google Scholar 

  32. Lions, P.-L.Quelques remarques sur les problèmes elliptiques quasilinéaires du second ordre.Jour. d’Anal. Math. 45, 234–253 (1985).

    Article  MATH  Google Scholar 

  33. Mullins, W. Two dimensional motion of idealized grain boundaries.J. Appl. Phys. 27, 900–904 (1956).

    Article  MathSciNet  Google Scholar 

  34. Osher, S., and Sethian, J. Fronts moving with curvature-dependent speed: algorithms based on Hamilton-Jacobi equations.J. Comp. Phys. 79, 12–49 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  35. Rubinstein, J., Sternberg, P., and Keller, J. B. Fast reaction, slow diffusion and curve shortening.SIAM J. Appl. Math. 49, 116–133(1989).

    Article  MathSciNet  MATH  Google Scholar 

  36. Sato, M.-H.Interface evolution with Neumann boundary condition. Hokkaido University Preprint Series in Mathematics #147(1992).

  37. Sollonikov, V. A.Boundary Value Problems in Physics. III. American Mathematical Society, Providence, RI, 1967.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Michigan State University, 48824-1027, East Lansing, Michigan

    Markos Katsoulakis, Georgios T. Kossioris & Fernando Reitich

  2. Department of Mathematics, University of Crete, P.O. Box 470, 71409, Hraklion, Crete, Greece

    Markos Katsoulakis, Georgios T. Kossioris & Fernando Reitich

  3. Department of Mathematics, North Carolina State University, Box 8205, 27695-8205, Raleigh, NC

    Markos Katsoulakis, Georgios T. Kossioris & Fernando Reitich

Authors
  1. Markos Katsoulakis
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  2. Georgios T. Kossioris
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  3. Fernando Reitich
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Communicated by David Kinderlehrer

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Katsoulakis, M., Kossioris, G.T. & Reitich, F. Generalized motion by mean curvature with Neumann conditions and the Allen-Cahn model for phase transitions. J Geom Anal 5, 255–279 (1995). https://doi.org/10.1007/BF02921677

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