Abstract
This paper continues our study [CGG], [GG] of a motion of phase-boundaries whose speed locally depends on the normal vector field and curvature tensors.
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References
S. Angenent, Shrinking doughnuts, Proc. of the conference on elliptic and parabolic equations held at Gregymog, Wales, August 1989.
S. Angenent and M. Gurtin, Multiphase thermomechanics with interfacial structure. 2. Evolution of an isothermal interface, Arch. Rational Mech. Anal. 108 (1989), pp. 323–391.
K.A. Brakke, The motion of a surface by its mean curvature, Princeton University Press, 1978.
L. Bronsard and R. Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics,preprint.
G. Caginalp, The role of microscopic anisotropy in the macroscopic behavior of a phase boundary, Annals of Physics, 172 (1986), pp. 136–155.
X. Chen Generation and propagation of the interface for reaction-diffusion equation,preprint.
X.-Y. Chen, Dynamics of interfaces in reaction diffusion systems, Hiroshima Math. J. 21, to appear (1991).
Y.-G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geometry 33, to appear (1991); (Announcement: Proc. Japan Acad. Ser. A, 65 (1989), pp. 207–210 ).
L.C. Evans, H.M. Soner and P.E. Songanidis, The Allen-Cahn equation and generalized motion by mean curvature,manuscript.
L.C. Evans and J. Spruck, Motion of level sets by mean curvature I,J. Differential Geometry, to appear.
L.C. Evans and J. Spruck, Motion of level sets by mean curvature II,J. Differential Geometry, to appear..
L.C. Evans and J. Spruck, Motion of level sets by mean curvature III, preprint 1990.
Y. Giga and S. Goto, Motion of hypersurfaces and geometric equations,J. Math. Soc. Japan, to appear.
Y. Giga, S. Goto, H. Ishii and M.-H. Sato, Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains,Indiana Univ. Math. J., to appear.
Y. Ggia, S. Goto and H. Ishii, Global existence of weak solutions for interface equations coupled with diffusion equations, preprint 1990.
S. Goto, A level surface approach to interface dynamics, in preparation.
M. Grayson, A short note on the evolution of a surface by its mean curvature, Duke Math. J. 58 (1989), pp. 555–558.
M. Gurtin, Towards a nonequilibrium thermodynamics of two-phase materials, Arch. Rational Mech. Anal. 100 (1988), pp. 275–312.
M. Gurtin, Multiphase thermomechanics with interfacial structure. 1, Heat conduction and the capillary balance law, Arch. Rational Mech. Anal. 104 (1988), pp. 195–221.
R.S. Hamilton, Three manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), pp. 255–306.
G. Huisken, Asymptotic behaviour for singularities of the mean curvature flow, J. Differential Geometry 31 (1990), pp. 285–299.
T. Ilmanen, Generalized flow of sets by mean curvature on a manifold, preliminary report
B. Kawohl, Remarks on quenching, blow up and dead core, Proc. of the conference on elliptic and parabolic equations held at Gregymog, Wales, August 1989.
O.A. Ladyzhenskaya, V. Solonnikov and N. Ural’cera, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, vol. 23, AMS, 1968.
P. Demottoni and M. Schatzman, Geometrical evolution of developed interfaces, preprint; (Announcement: Evolution géometric d’interfaces, C.R. Acad. Sci. Paris, 309 (1989), pp. 453–458 ).
S. Osher and J.A. Sethian, Front propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys. 79 (1988), pp. 12–49.
J.A. Sethian, Curvature and evolution of fronts, Comm. Math. Physics 101 (1985), pp. 487–499.
H.M. Soner, Motion of a set by the curvature of its boundary, preprint 1990.
J. Taylor, Motion of curves by crystalline curvature,preliminary report.
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© 1992 Springer-Verlag New York, Inc.
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Giga, Y., Goto, S. (1992). Geometric Evolution of Phase-Boundaries. In: Gurtin, M.E., McFadden, G.B. (eds) On the Evolution of Phase Boundaries. The IMA Volumes in Mathematics and its Applications, vol 43. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9211-8_3
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DOI: https://doi.org/10.1007/978-1-4613-9211-8_3
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