As on arXiv
Abstract
We study the obstacle problem associated to mean curvature flow. We add to the geometric vanishing-viscosity approximation of Evans and Spruck a singular perturbation that penalizes the violation of the constraint, and pass to the limit. The resulting level set formulation has unique solutions - up to fattening. Extending the work of Evans and Spruck and a work by Ullrich and one of the authors, we show that generic level sets of this flow are distributional solutions of the obstacle problem.
Comments: 25 pages, 1 figure
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Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
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Generic Mean Curvature Flow With Rough Weights
Topics Weak & singular flows · Same authors Tim Laux, Keisuke Takasao · Shared terms “generic mean”, “distributional solutions”
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Generic level sets in mean curvature flow are BV solutions
Topics Weak & singular flows · Same author Tim Laux · Shared terms “generic level”, “level set”, “Evans”
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Equivalence of weak solution concepts for mean curvature flow
Topics Weak & singular flows · Same author Tim Laux · Shared terms “Evans”, “Spruck”, “viscosity”
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Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow
Topics Weak & singular flows · Same authors Tim Laux, Keisuke Takasao
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Distributional solutions to mean curvature flow
Topics Weak & singular flows · Same author Tim Laux · Shared terms “distributional solutions”
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Weak-strong uniqueness for volume-preserving mean curvature flow
Topics Weak & singular flows · Same author Tim Laux · Shared terms “distributional solutions”
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De Giorgi varifold solutions to Mean Curvature Flow: a minimizing movements approach
Topics Weak & singular flows · Same author Tim Laux
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A level-set method for a mean curvature flow with a prescribed boundary
Topics Weak & singular flows · Shared terms “obstacle”, “level set”
Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 12C.9After Perelman
Singular flows, the Smale conjecture, and Bamler’s theory.
- 6A.10Curve Shortening and the First Geometric Flows
Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.