Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics.

*(English)*Zbl 0735.35072The author studies the motion of the interface between two media (with normal velocity equal to the sum of the principal curvatures) by approximation with a nonlinear parabolic problem. To justify the approximation a compactness theorem is given and exact results on the limit problem are given in the radial case with Dirichlet boundary conditions.

Reviewer: M.Biroli (Monza)

##### MSC:

35K55 | Nonlinear parabolic equations |

35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |

##### Keywords:

mean curvature; Ginzburg-Landau dynamics; motion of the interface; compactness; Dirichlet boundary conditions
PDF
BibTeX
XML
Cite

\textit{L. Bronsard} and \textit{R. V. Kohn}, J. Differ. Equations 90, No. 2, 211--237 (1991; Zbl 0735.35072)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Allen, S; Cahn, J, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta metall., 27, 1084-1095, (1979) |

[2] | {\scS. Angenant}, On the formation of singularities in the curve shortening flow, J. Differential Geom., to appear. |

[3] | {\scS. Angenant}, Parabolic equations for curves on surfaces II: Intersections, blowup, and generalized solutions, Annals of Math., to appear. |

[4] | Baldo, S, Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids, Ann. inst. H. Poincaré anal. non linéaire, 7, 37-65, (1990) |

[5] | Brakke, K, The motion of a surface by its Mean curvature, (1978), Princeton Univ. Press Princeton, NJ · Zbl 0386.53047 |

[6] | Bronsard, L, Reaction diffusion equations and motion by Mean curvature, () |

[7] | Bronsard, L; Kohn, R, On the slowness of phase boundary motion in one space dimension, Comm. pure appl. math., 43, 983-997, (1990) · Zbl 0761.35044 |

[8] | Caginalp, G; Fife, P, Dynamics of layered interfaces arising from phase boundaries, SIAM J. appl. math., 48, 506-518, (1988) |

[9] | Caginalp, G, Conserved phase field system; implications for kinetic undercooling, Phys. rev. B, 38, 789-791, (1988) |

[10] | Carr, J; Pego, R, Very slow phase separation in one dimension, (), 216-226 · Zbl 0991.35515 |

[11] | {\scJ. Carr and R. Pego}, Metastable patterns in solutions of ut = ε2uxx − f(u)″, Comm. Pure Appl. Math., to appear. · Zbl 0685.35054 |

[12] | {\scY.-G. Chen, Y. Giga, and S. Goto}, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., to appear. · Zbl 0696.35087 |

[13] | {\scY. Chen}, Weak solutions to the evolution problem for harmonic maps into spheres, preprint. |

[14] | Chen, Y; Struwe, M, Existence and partial regularity results for the heat flow for harmonic maps, Math. Z., 201, 83-103, (1989) · Zbl 0652.58024 |

[15] | DeGiorgi, E, New problems in γ-convergence and G-convergence, (), 183-194, Rome |

[16] | DeMottoni, P; Schatzman, M, Évolution géometrique d’interfaces, C.R. acad. sci. Paris Sér. I math., 309, 453-458, (1989) · Zbl 0698.35078 |

[17] | {\scL. Evans and J. Spruck}, Motion of level sets by mean curvature I, II, J. Differential Geom., to appear. · Zbl 0776.53005 |

[18] | Fonseca, I; Tartar, L, The gradient theory of phase transitions for systems with two potential wells, (), 89-102 · Zbl 0676.49005 |

[19] | Freidlin, M, Functional integration and partial differential equations, (1985), Princeton Univ. Press Princeton, NJ · Zbl 0568.60057 |

[20] | Fusco, G, A geometric approach to the dynamics of ut = ε2uxx + f(u) for small ε, (), 53-73 |

[21] | Fusco, G; Hale, J, Slow motion manifold, dormant instability and singular perturbation, J. dynamics differential equations, 1, 75-94, (1989) · Zbl 0684.34055 |

[22] | Gage, M; Hamilton, R, The shrinking of convex curves by the heat equation, J. differential geom., 23, 69-96, (1986) · Zbl 0621.53001 |

[23] | Gärtner, J, Bistable reaction-diffusion equations and excitable media, Math. nachr., 112, 125-152, (1983) · Zbl 0548.35069 |

[24] | Giusti, E, Minimal surfaces and functions of bounded variation, (1984), Birkhäuser Basel · Zbl 0545.49018 |

[25] | Grayson, M, The heat equation shrinks plane curves to points, J. differential geom., 26, 285-314, (1987) · Zbl 0667.53001 |

[26] | Gunton, J; Miguel, M.San; Sahni, P, The dynamics of first-order phase transitions, (), 267-466 |

[27] | Hamilton, R, Three manifolds with positive Ricci curvature, J. differential geom., 17, 255-306, (1982) · Zbl 0504.53034 |

[28] | Huisken, G, Flow by Mean curvature of convex surfaces into spheres, J. differential geom., 20, 237-266, (1984) · Zbl 0556.53001 |

[29] | Kohn, R; Sternberg, P, Local minimizers and singular perturbation, (), 69-84 · Zbl 0676.49011 |

[30] | Luckhaus, S; Modica, L, The Gibbs-Thompson relation within the gradient theory of phase transitions, Arch. rat. mech. anal., 107, 71-84, (1989) · Zbl 0681.49012 |

[31] | Matano, H, Convergence of solutions of one-dimensional semilinear parabolic equations, J. math. Kyoto univ., 18, 221-227, (1978) · Zbl 0387.35008 |

[32] | Modica, L, The gradient theory of phase transitions and the minimal interface criterion, Arch. rat. mech. anal., 98, 123-142, (1987) · Zbl 0616.76004 |

[33] | Modica, L; Mortola, S, Il limite nella γ-convergenza di una famiglia di funzionali ellitichi, Boll. un. mat. ital. A, 14, 526-529, (1977), (3) · Zbl 0364.49006 |

[34] | Osher, S; Sethian, J, Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, J. comput. phys., 79, 12-49, (1988) · Zbl 0659.65132 |

[35] | Pego, R, Front migration in the nonlinear Cahn-Hilliard equation, (), 261-278 · Zbl 0701.35159 |

[36] | Rubinstein, J; Sternberg, P; Keller, J, Fast reaction, slow diffusion, and curve shortening, SIAM J. appl. math., 49, 116-133, (1989) · Zbl 0701.35012 |

[37] | Rubinstein, J; Sternberg, P; Keller, J, Reaction-diffusion processes and evolution to harmonic maps, SIAM J. appl. math., 49, 1722-1733, (1989) · Zbl 0702.35128 |

[38] | Sternberg, P, The effect of a singular perturbation on nonconvex variational problems, Arch. rat. mech. anal., 101, 209-260, (1988) · Zbl 0647.49021 |

[39] | {\scP. Sternberg}, Vector-valued local minimizers of nonconvex variational problems, Rocky Mt. Math. J., to appear. · Zbl 0737.49009 |

[40] | Sethian, J, A review of recent numerical algorithms for hypersurfaces moving with curvature-dependent speed, J. differential geom., 31, 131-161, (1989) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.