Paper · Oct 16, 2024

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

Kiichi Tashiro

math.DGWider flowsarXiv:2410.12283v4
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Abstract

In the present paper, we study the existence of Brakke-type weak mean curvature flow satisfying a prescribed contact angle condition for a general angle via Ilmanen's regularization. The main ingredients of the result are the extension of Ilmanen's regularization to the capillarity and the derivation of the first variation estimates for the interior and wetted boundary varifolds separately.

Comments: The paper consists of 28 pages and includes 2 figures. I have corrected minor mistakes from the third version. I have also corrected the admissible class used in the minimization problem of Ilmanen's functional (this does not affect the rest part of this paper). To appear in Ann. Sc. Norm. Super. Pisa, Cl. Sci

Mean curvature flowWeak & singular flowsFree boundary & capillary

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    Kiichi TashiroWider flows PDF

    Topics Weak & singular flows · Same author Kiichi Tashiro · Shared terms “elliptic regularization”, “Brakke”

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    Shokhrukh KholmatovWider flows PDF

    Topics Free boundary & capillary, Weak & singular flows · Shared terms “prescribed contact”, “contact angle”

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    Tokuhiro Eto and Yoshikazu GigaWider flows PDF

    Topics Free boundary & capillary, Weak & singular flows · Shared terms “prescribed contact”, “contact angle”, “angle condition”

  4. Convergence of a minimizing movement scheme for contact-angle mean curvature flow in a smooth bounded domain

    Tokuhiro Eto and Jiwoong JangWider flows PDF

    Topics Free boundary & capillary, Weak & singular flows · Shared terms “contact angle”, “prescribed contact”, “ingredient”

  5. A Mean Curvature Flow with Prescribed Contact Angles in a High Dimensional Cylinder

    Zhenghuan Gao, Bendong Lou and Jinju XuWider flows PDF

    Topics Free boundary & capillary · Shared terms “prescribed contact”, “contact angle”

  6. BV solutions for mean curvature flow with constant contact angle: Allen-Cahn approximation and weak-strong uniqueness

    Sebastian Hensel and Tim LauxWider flows PDF

    Topics Free boundary & capillary, Weak & singular flows · Shared terms “contact angle”, “flow satisfying”, “angle condition”

  7. Existence of flat flows for volume-preserving mean curvature flow with contact angle

    Jiwoong JangWider flows PDF

    Topics Free boundary & capillary, Weak & singular flows · Shared terms “contact angle”, “angle condition”

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

    Can Cui and Nung Kwan YipWider flows PDF

    Topics Free boundary & capillary · Shared terms “contact angle”, “prescribed contact”

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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002) 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. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  2. 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.