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A Class of Parabolic Equations Driven by the Mean Curvature Flow

Part of: Classical differential geometry Parabolic equations and systems Partial differential equations on manifolds; differential operators Global differential geometry

Published online by Cambridge University Press:  30 August 2018

Anderson L. A. de Araujo  and
Marcelo Montenegro
Anderson L. A. de Araujo*
Affiliation:
Departamento de Matemática Universidade Federal de Viçosa, CCE, Avenida PH Rolfs, s/n CEP 36570-900 Viçosa, MG, Brazil (anderson.araujo@ufv.br)
Marcelo Montenegro
Affiliation:
Departamento de Matemática, Rua Sérgio Buarque de Holanda, Universidade Estadual de Campinas, IMECC, 651 CEP 13083-859 Campinas, SP, Brazil (msm@ime.unicamp.br)
*
*Corresponding author.
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Abstract

We study a class of parabolic equations which can be viewed as a generalized mean curvature flow acting on cylindrically symmetric surfaces with a Dirichlet condition on the boundary. We prove the existence of a unique solution by means of an approximation scheme. We also develop the theory of asymptotic stability for solutions of general parabolic problems.

Keywords

mean curvature flowasymptotic behaviourstability

MSC classification

Primary: 35K55: Nonlinear parabolic equations
Secondary: 35K57: Reaction-diffusion equations 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.) 53A04: Curves in Euclidean space 53A05: Surfaces in Euclidean space 58J35: Heat and other parabolic equation methods
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 1 , February 2019 , pp. 135 - 163
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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References

1.Altschuler, S., Angenent, S. and Giga, Y., Mean curvature flow through singularities for surfaces of rotation, J. Geom. Anal. 5 (1995), 293–358.Google Scholar
2.Angenent, S., Parabolic equations for curves on surfaces Part I. Curves with p-integrable curvature, Ann. Math. 132 (1990), 451–483.Google Scholar
3.Angenent, S., Parabolic equations for curves on surfaces Part II. Intersections, blow-up and generalized solutions, Ann. Math. 133 (1991), 171–215.Google Scholar
4.Angenent, S., On the formation of singularities in the curve shortening flow, J. Diff. Geom. 33 (1991), 601–633.Google Scholar
5.Dziuk, G. and Kawohl, B., On rotationally symmetric mean curvature flow, J. Diff. Equ. 93 (1991), 142–149.Google Scholar
6.Escher, J. and Matioc, B. V., Neck pinching for periodic mean curvature flows, Analysis 30 (2010), 253–260.Google Scholar
7.Evans, L. C. and Spruck, J., Motion of level sets by mean curvature II, Trans. Amer. Math. Soc. 330 (1992), 321–332.Google Scholar
8.Fila, M., Kawohl, B. and Levine, H., Quenching for quasilinear equations, Commun. Partial Diff. Equ. 17 (1992), 593–614.Google Scholar
9.Friedman, A., Partial differential equations of parabolic types (Prentice-Hall, 1964).Google Scholar
10.Gage, M. and Hamilton, R. S., The heat equation shrinking convex plane curves, J. Diff. Geom. 23 (1986), 69–96.Google Scholar
11.Giga, Y., Surface evolution equations: a level set approach, Monographs in Mathematics (Birkhäuser, 2006).Google Scholar
12.Grayson, M. A., The heat equation shrinks embedded plane curves to round points, J. Diff. Geom. 26 (1987), 285–314.Google Scholar
13.Grayson, M. A., A short note on the evolution of a surface by its mean curvature, Duke Math. J. 58 (1989), 555–558.Google Scholar
14.Hamilton, R. S., The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. 7 (1982), 65–222.Google Scholar
15.Huisken, G., Flow by mean curvature of convex surfaces into spheres, J. Diff. Geom. 20 (1984), 237–266.Google Scholar
16.Huisken, G., Non-parametric mean curvature evolution with boundary conditions, J. Diff. Equ. 77 (1989), 369–378.Google Scholar
17.Huisken, G., Asymptotic behaviour for singularities of the mean curvature flow, J. Diff. Geom. 31 (1990), 285–299.Google Scholar
18.Huisken, G. and Sinestrari, C., Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta. Math. 183 (1999), 45–70.Google Scholar
19.Huisken, G. and Sinestrari, C., Mean curvature flow singularities for mean convex surfaces, Calc. Var. 8 (1999), 1–14.Google Scholar
20.Lieberman, G. M., Second order parabolic differential equations (World Scientific, 2005).Google Scholar
21.Matioc, B. V., Boundary value problems for rotationally symmetric mean curvature flows, Arch. Math. 89 (2007), 365–372.Google Scholar
22.Pao, C. V., Nonlinear parabolic and elliptic equations (Plenum, 1992).Google Scholar
23.Reynolds, A., Asymptotic behavior of solutions of nonlinear parabolic equations, J. Diff. Equ. 12 (1972), 256–261.Google Scholar
24.Soner, H. M. and Souganidis, P. E., Singularities and uniqueness of cylindrically symmetric surfaces moving by mean curvature, Commun. Partial Diff. Equ. 18 (1993), 859–894.Google Scholar