Paper · Apr 16, 2020

Stationary sets and asymptotic behavior of the mean curvature flow with forcing in the plane

Nicola Fusco, Vesa Julin, Massimiliano Morini

math.APWider flowsarXiv:2004.07734v2
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Abstract

We consider the flat flow solutions of the mean curvature equation with a forcing term in the plane. We prove that for every constant forcing term the stationary sets are given by a finite union of disks with equal radii and disjoint closures. On the other hand for every bounded forcing term tangent disks are never stationary. Finally in the case of an asymptotically constant forcing term we show that the only possible long time limit sets are given by disjoint unions of disks with equal radii and possibly tangent.

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Submitted
Apr 16, 2020
Last revised
Apr 17, 2020 (v2)
Subjects
math.AP
Mean curvature flow

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  1. Stationary sets of the mean curvature flow with a forcing term

    Vesa Julin and Joonas NiinikoskiWider flows

    Topics Mean curvature flow · Same author Vesa Julin · Shared terms “stationary sets”, “forcing term”, “constant forcing”

  2. The Asymptotics of the Area-Preserving Mean Curvature and the Mullins-Sekerka Flow in Two Dimensions

    Vesa Julin, Massimiliano Morini, Marcello Ponsiglione et al.Wider flows

    Topics Mean curvature flow · Same authors Vesa Julin, Massimiliano Morini · Shared terms “finite union”, “disjoint”, “flat flow”

  3. Quantitative Alexandrov theorem and asymptotic behavior of the volume preserving mean curvature flow

    Vesa Julin and Joonas NiinikoskiWider flows

    Topics Mean curvature flow · Same author Vesa Julin · Shared terms “asymptotic behavior”, “disjoint”, “flat flow”

  4. Existence and uniqueness for anisotropic and crystalline mean curvature flows

    Antonin Chambolle, Massimiliano Morini, Matteo Novaga et al.Wider flows

    Topics Mean curvature flow · Same author Massimiliano Morini · Shared terms “forcing term”, “bounded forcing”, “flat flow”

  5. Long time behaviour of discrete volume preserving mean curvature flows

    Massimiliano Morini, Marcello Ponsiglione and Emanuele SpadaroWider flows

    Topics Mean curvature flow · Same author Massimiliano Morini · Shared terms “finite union”, “disjoint”, “long time”

  6. Consistency of the flat flow solution to the volume preserving mean curvature flow

    Vesa Julin and Joonas NiinikoskiWider flows

    Topics Mean curvature flow · Same author Vesa Julin · Shared terms “flat flow”

  7. Flat flow solution to the mean curvature flow with volume constraint

    Vesa JulinWider flows

    Topics Mean curvature flow · Same author Vesa Julin · Shared terms “flat flow”

  8. Minimizing Movements for Anisotropic and Inhomogeneous Mean Curvature Flows

    Antonin Chambolle, Daniele De Gennaro and Massimiliano MoriniWider flows

    Topics Mean curvature flow · Same author Massimiliano Morini

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)

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.