Paper · Dec 21, 2021

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

Sebastian Hensel, Tim Laux

math.APWider flowsarXiv:2112.11150
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Abstract

We study weak solutions to mean curvature flow satisfying Young's angle condition for general contact angles . First, we construct BV solutions using the Allen-Cahn approximation with boundary contact energy as proposed by Owen and Sternberg. Second, we prove the weak-strong uniqueness and stability for this solution concept. The main ingredient for both results is a relative energy, which can also be interpreted as a tilt excess.

Comments: 24 pages

Mean curvature flowWeak & singular flowsStability & uniquenessFree boundary & capillaryPhase fields & multiphase flows

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The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Weak-strong uniqueness for the mean curvature flow of double bubbles (Sebastian Hensel and Tim Laux, 2021)2Hensel 2021 A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness (Sebastian Hensel and Tim Laux, 2021)1Hensel 2021 Weak-strong uniqueness for volume-preserving mean curvature flow (Tim Laux, 2022)7Laux 2022 Diffuse-interface approximation and weak-strong uniqueness of anisotropic mean curvature flow (Tim Laux, Kerrek Stinson and Clemens Ullrich, 2022)3Laux 2022 Quantitative convergence of the nonlocal Allen–Cahn equation to volume-preserving mean curvature flow (Milan Kroemer and Tim Laux, 2023)5Kroemer 2023 Stability of multiphase mean curvature flow beyond circular topology changes (Julian Fischer, Sebastian Hensel, Alice Marveggio et al., 2024)4Fischer 2024 Convergence of a heterogeneous Allen-Cahn equation to weighted mean curvature flow (Likhit Ganedi, Alice Marveggio and Kerrek Stinson, 2024)8Ganedi 2024 Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow (Tim Laux and Keisuke Takasao, 2026)6Laux 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness

    Sebastian Hensel and Tim LauxWider flows PDF

    Topics Phase fields & multiphase flows, Weak & singular flows, Stability & uniqueness · Same authors Sebastian Hensel, Tim Laux · Shared terms “weak strong”, “Allen Cahn”, “strong uniqueness”

  2. Weak-strong uniqueness for the mean curvature flow of double bubbles

    Sebastian Hensel and Tim LauxWider flows PDF

    Topics Phase fields & multiphase flows, Weak & singular flows, Stability & uniqueness · Same authors Sebastian Hensel, Tim Laux · Shared terms “weak strong”, “strong uniqueness”

  3. Diffuse-interface approximation and weak-strong uniqueness of anisotropic mean curvature flow

    Tim Laux, Kerrek Stinson and Clemens UllrichWider flows PDF

    Topics Phase fields & multiphase flows, Weak & singular flows, Stability & uniqueness · Same author Tim Laux · Shared terms “weak strong”, “strong uniqueness”, “Allen Cahn”

  4. Stability of multiphase mean curvature flow beyond circular topology changes

    Julian Fischer, Sebastian Hensel, Alice Marveggio et al.Wider flows PDF

    Topics Phase fields & multiphase flows, Weak & singular flows, Stability & uniqueness · Same author Sebastian Hensel · Shared terms “weak strong”, “strong uniqueness”, “relative energy”

  5. Quantitative convergence of the nonlocal Allen–Cahn equation to volume-preserving mean curvature flow

    Milan Kroemer and Tim LauxWider flows PDF

    Topics Phase fields & multiphase flows, Weak & singular flows, Stability & uniqueness · Same author Tim Laux · Shared terms “Allen”, “Cahn”, “weak strong”

  6. Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow

    Tim Laux and Keisuke TakasaoWider flows PDF

    Topics Phase fields & multiphase flows, Weak & singular flows · Same author Tim Laux · Shared terms “Allen Cahn”

  7. Weak-strong uniqueness for volume-preserving mean curvature flow

    Tim LauxWider flows PDF

    Topics Weak & singular flows, Stability & uniqueness · Same author Tim Laux · Shared terms “weak strong”, “strong uniqueness”, “relative”

  8. Convergence of a heterogeneous Allen-Cahn equation to weighted mean curvature flow

    Likhit Ganedi, Alice Marveggio and Kerrek StinsonWider flows PDF

    Topics Phase fields & multiphase flows, Weak & singular flows, Stability & uniqueness · Shared terms “Allen Cahn”, “weak strong”, “strong uniqueness”

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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.

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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) 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.