Paper · Apr 20, 2021

Sharp bounds for the anisotropic -capacity of Euclidean compact sets

Ruixuan Li, Changwei Xiong

math.DGWider flowsarXiv:2104.09905
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Abstract

We prove various sharp bounds for the anisotropic -capacity () of compact sets in the Euclidean space (). For example, using the inverse anisotropic mean curvature flow (IAMCF), we get an upper bound of Szegö type (1931) for when is a smooth, star-shaped and -mean convex hypersurface in (). Moreover, for such a surface in , by introducing the anisotropic Hawking mass and studying its monotonicity property along IAMCF, we obtain an upper bound of Bray–Miao type (2008) for .

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Submitted
Apr 20, 2021
Subjects
math.DG, math.AP

Comments: 26 pages. Comments are very welcome

Mean curvature flowAnisotropic & nonlocal flowsPhysics

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    Topics Anisotropic & nonlocal flows · Same authors Ruixuan Li, Changwei Xiong · Shared terms “capacity”, “compact set”, “upper bound”

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    Topics Anisotropic & nonlocal flows · Shared terms “inverse anisotropic”, “IAMCF”

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    Topics Anisotropic & nonlocal flows · Shared terms “inverse anisotropic”, “compact set”, “star shaped”

  7. Threshold dynamics approximation schemes for anisotropic mean curvature flows with a forcing term

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    Topics Anisotropic & nonlocal flows · Shared terms “compact set”

  8. Some results on anisotropic fractional mean curvature flows

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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.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Anisotropic & nonlocal flows Wider flows When surface energy depends on direction, as in crystals, mean curvature flow becomes anisotropic, and the Wulff shape replaces the round sphere. Start with: Crystalline variational problems (Taylor, 1978) Physics Ricci flow first appeared in physics, as the leading-order renormalization-group flow of the string sigma model (Friedan, 1980). Start with: Nonlinear Models in 2+ε Dimensions (Friedan, 1980)

In the guidebook

  1. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.