Paper · Oct 25, 2025

Anisotropic mean curvature flow with contact angle and Neumann boundary conditions in arbitrary dimensions

Can Cui, Nung Kwan Yip

math.APWider flowsarXiv:2510.22146
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Abstract

Over a bounded strictly convex domain in with smooth boundary, we establish a priori gradient estimate for an anisotropic mean curvature flow with prescribed contact angle and Neumann boundary conditions. The estimates require careful analysis of the degeneracy property of the anisotropic mean curvature operator. As a result, for both problems, we can infer that the solutions converge to one that is translation invariant in time.

Comments: 33 pages, 2 figures

Mean curvature flowAnisotropic & nonlocal flowsBoundaries & non-compactFree boundary & capillary

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    Can Cui and Nung Kwan YipWider flows PDF

    Topics Anisotropic & nonlocal flows, Free boundary & capillary, Boundaries & non-compact · Same authors Can Cui, Nung Kwan Yip · Shared terms “contact angle”, “boundary condition”, “translation invariant”

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    Topics Free boundary & capillary, Boundaries & non-compact · Shared terms “contact angle”, “prescribed contact”, “boundary condition”

  5. Existence of weak mean curvature flow with prescribed contact angle via elliptic regularization

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  8. Existence of the classical solution to the fractional mean curvature flow with capillary-type boundary conditions

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    Topics Anisotropic & nonlocal flows, Free boundary & capillary, Boundaries & non-compact · Shared terms “boundary condition”, “contact angle”

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. 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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989) 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)

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.