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

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.

MSC:

35J60 Nonlinear elliptic equations
49Q05 Minimal surfaces and optimization
35K55 Nonlinear parabolic equations
Full Text: DOI

References:

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