Paper · Dec 13, 2024

Smoothness up to the free boundary for the -Laplacian evolution equation and the -Gauss curvature flow

Albert Chau, Ben Weinkove

math.APWider flowsarXiv:2412.10144v2
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Abstract

The -Laplacian evolution equation and the -Gauss curvature flow with a flat side are degenerate parabolic equations with evolving free boundaries. We give proofs of smooth short-time existence, up to the free boundaries, using a result of the authors on linear degenerate equations on a fixed domain.

Comments: 15 pages. v2 conditions for Theorem 1.2 corrected

Other curvature flowsGauss curvature flowFree boundary & capillaryHarmonic maps & DeTurck trick

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    Genggeng Huang, Xu-Jia Wang and Yang ZhouWider flows PDF

    Topics Gauss curvature flow, Other curvature flows · Shared terms “flat side”

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    Topics Gauss curvature flow, Other curvature flows · Shared terms “flat side”

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    Topics Gauss curvature flow, Free boundary & capillary, Other curvature flows

  6. Ancient Gauss Curvature Flows of Bounded Width

    Beomjun Choi, Kyeongsu Choi and Dongjun NohWider flows PDF

    Topics Gauss curvature flow, Other curvature flows · Shared terms “flat side”

  7. Anisotropic Parabolic Obstacle Problems and the Stefan Problem: Regularity of the Evolving Free Boundary

    Ezequiel Barbosa, Rosivaldo Gonçalves and Luan de FigueiredoWider flows PDF

    Topics Free boundary & capillary · Shared terms “evolving free”, “degenerate”

  8. A Gauss curvature flow related to the Orlicz-Aleksandrov problem

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Gauss curvature flow Wider flows Convex hypersurfaces shrink with normal speed a power K^α of their Gauss curvature. Firey proposed it as a model of stones worn by waves; for α>… Start with: Shapes of worn stones (Firey, 1974) Free boundary & capillary Flows whose boundary is free to slide on a support surface, meeting it at a right angle (free boundary) or at a fixed angle (capillary). Start with: Non-parametric mean curvature evolution with boundary conditions (Huisken, 1989) Harmonic maps & DeTurck trick Ricci flow is only weakly parabolic, because it is invariant under diffeomorphisms. The DeTurck trick adds a Lie-derivative term that makes it… Start with: Harmonic Mappings of Riemannian Manifolds (Eells et al., 1964)