×
Ilmanen, Tom; Sternberg, Peter; Ziemer, William P.

Equilibrium solutions to generalized motion by mean curvature. (English) Zbl 0941.35028

J. Geom. Anal. 8, No. 5, 845-858 (1998).
We consider viscosity equilibria to the mean curvature level set flow with a Dirichlet condition. The main result shows that almost every level set of an equilibrium solution is analytic off of a singular set of Hausdorff dimension at most \(n-8\) and that these level sets are stationary and stable with respect to the area functional. A key tool developed is a maximum principle for solutions to obstacle problems where the obstacle consists of (viscosity) minimal surfaces. Convergence to equilibrium as \(t\to\infty\) is also established for the associated initial-boundary value problem.
Reviewer: Peter Sternberg (Bloomington)

Cited in 4 Documents

MSC:

35J60 Nonlinear elliptic equations
49Q05 Minimal surfaces and optimization
35K55 Nonlinear parabolic equations

Keywords:

level set flow; viscosity solutions; stable varifolds; minimal surfaces; obstacle problems
PDF BibTeX XML
Full Text: DOI

References:

[1] Bombieri, E.; De Giorgi, E.; Giusti, E., Minimal cones and the Bernstein problem, Invent. Math., 7, 243-268 (1969) · Zbl 0183.25901
[2] Chen, Y.-G.; Giga, Y.; Goto, S., Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Diff. Geom., 33, 749-786 (1991) · Zbl 0696.35087
[3] Evans, L. C.; Spruck, J., Motion of level sets by mean curvature I, J. Diff. Geom., 33, 635-681 (1991) · Zbl 0726.53029
[4] Evans, L. C.; Spruck, J., Motion of level sets by mean curvature II, Trans. A.M.S., 330, 321-332 (1992) · Zbl 0776.53005
[5] Evans, L. C.; Spruck, J., Motion of level sets by mean curvatureIII, J. Geom. Anal., 2, 121-150 (1992) · Zbl 0768.53003
[6] Evans, L. C.; Spruck, J., Motion of level sets by mean curvature IV, J. Geom. Anal., 5, 77-114 (1995)
[7] Federer, H., Geometric Measure Theory (1969), New York: Springer-Verlag, New York · Zbl 0176.00801
[8] Gilbarg, D.; Trudinger, N., Elliptic Partial Differential Equations of Second Order (1983), New York: Springer-Verlag, New York · Zbl 0562.35001
[9] Hardt, R.; Zhou, X., An evolution problem for linear growth functionals, Comm. P.D.E., 19, 11-12, 1879-1907 (1994) · Zbl 0811.35061
[10] Ilmanen, T. Elliptic regularization and partial regularity for motion by mean curvature,Mem. A.M.S., 520, (1994). · Zbl 0798.35066
[11] Ilmanen, T., The level-set flow on a manifold, Proc. Symp. Pure Math., 54, 193-204 (1993) · Zbl 0827.53014
[12] Ilmanen, T., A strong maximum principle for singular minimal hypersurfaces, Calc. Var., 4, 443-467 (1996) · Zbl 0863.49030
[13] Simon, L. Lectures on geometric measure theory,Proc. Centre Math. Anal., Austr. Nat. Univ.,3, (1983). · Zbl 0546.49019
[14] Soner, M., Motion of a set by the curvature of its boundary, J. Diff. Eqs., 101, 313-372 (1993) · Zbl 0769.35070
[15] Schoen, R.; Simon, L., Regularity of stable minimal hypersurfaces, CPAM, 34, 741-797 (1981) · Zbl 0497.49034
[16] Sternberg, P.; Ziemer, W. P., Generalized motion by curvature with a Dirichlet condition, J. Diff. Eqs., 114, 580-600 (1994) · Zbl 0810.35048
[17] Sternberg, P.; Ziemer, W. P.; Ni, W.-M., The Dirichlet problem for functions of least gradient, Degenerate Diffusions (1993), New York: Springer-Verlag, New York · Zbl 0818.49024
[18] Sternberg, P.; Williams, G.; Ziemer, W. P., The constrained least gradient problem inR^n, Trans. A.M.S., 339, 403-432 (1993) · Zbl 0788.49034
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.