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Generalized motion of noncompact hypersurfaces with velocity having arbitrary growth on the curvature tensor. (English) Zbl 0837.35066
The level set approach to the evolution of surfaces is based on characterizing the surfaces as level sets of the solution of certain fully nonlinear degenerate PDEs. The basic tool of this approach is the theory of viscosity solutions. In this paper, the authors extend an earlier work by L. C. Evans and J. Spruck [J. Differ. Geom. 33, No. 3, 635-681 (1991; Zbl 0726.53029)] as well as by Y.-G. Chen, Y. Giga and S. Goto [ibid. 33, No. 3, 749-786 (1991; Zbl 0715.35037)] to the case where the normal velocity is a general continuous function of the normal and the curvature tensor. Applications in geometry include the case of Gaussian curvature. The main difficulty is that these problems lead to PDEs with singularities of higher order. The authors overcome this difficulty by extending the class of admissible test functions in the definition of viscosity solutions. They are able to establish a comparison principle and an existence result. Their method allows them to treat the generalized evolution of noncompact hypersurfaces.

35K55 Nonlinear parabolic equations
35K65 Degenerate parabolic equations
Full Text: DOI
[1] L. ALVAREZ, F. GUICHARD, P. -L. LIONS AND J. -M. MOREL, Axioms and fundamental equations of image processing, Arch. Rat. Mech. Anal. 123 (1992), 199-257. · Zbl 0788.68153
[2] G. BARLES, H. M. SONER AND P. E. SOUGANIDIS, Front propagation and phase field theory, SIA J. Control Optim. 31 (1993), 439^69. · Zbl 0785.35049
[3] Y. -G. CHEN, Y. GIGA AND S. GOTO, Uniqueness and existence of viscosity solutions of generalize mean curvature flow equations, J. Differential Geom. 33 (1991), 749-786. · Zbl 0696.35087
[4] M. G. CRANDALL, H. ISHII AND P. -L. LIONS, User’s guide to viscosity solutions of second orde partial differential equations, Bull. Amer. Math. Soc. 27 (1992), 1-67. · Zbl 0755.35015
[5] L. C. EVANS AND J. SPRUCK, Motion of level sets by mean curvature I, J. Differential Geom. 3 (1991), 635-681. · Zbl 0726.53029
[6] L. C. EVANS, H. M. SONER AND P. E. SOUGANIDIS, Phase transitions and generalized motion b mean curvature, Comm. Pure Appl. Math. 45 (1992), 1097-1123. · Zbl 0801.35045
[7] Y. GIGA AND S. GOTO, Motion of hypersurfaces and geometric equations, J. Math. Soc. Japa 44(1992), 99-111. · Zbl 0739.53005
[8] Y. GIGA, S. GOTO, H. ISH AND M. -H. SATO, Comparison principle and convexity preservin properties for singular degenerate parabolic equations on unbounded domains, Indiana Univ. Math. J. 40 (1992), 443-470. · Zbl 0836.35009
[9] S. GOTO, Generalized motion of hypersurfaces whose growth speed depends superlinearly o curvature tensor, Differential Integral Equations 7 (1994), 323-343. · Zbl 0808.35007
[10] T. ILMANEN, Generalized flow of sets by mean curvature on a manifold, Indiana Univ. Math J. 41 (1992), 671-705. · Zbl 0759.53035
[11] P. -L. LIONS, Axiomatic derivation of image processing models, · Zbl 0939.68912
[12] M. KATSOULAKIS AND P. E. SOUGANIDIS, Interacting particle systems and generalized mea curvature evolution, Arch. Rat. Mech. Anal. 127 (1994), 133-157. · Zbl 0808.76002
[13] M. KATSOULAKIS AND P. E. SOUGANIDIS, Generalized motion by mean curvature as a microscopi limit of stochastic Ising models with long range interactions and Glauber dynamics, Comm. Math. Phys., · Zbl 0821.60095
[14] D. S. MITRINOVIC, Analytic inequalities, Springer-Verlag, Berlin, 1970 · Zbl 0199.38101
[15] H. M. SONER, Motion of a set by their mean curvature of its boundary, J. Differential Equation 101 (1993), 313-372. · Zbl 0769.35070
[16] P. E. SOUGANIDIS, Interface dynamics in phase transitions, Proceedings ICM 94, · Zbl 0845.35045
[17] N. S. TRUDINGER, The Dirichlet problem for the prescribed curvature equations, Arch. Rat Mech. Anal. III (1990), 153-170. · Zbl 0721.35018
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