Paper · Apr 18, 2018

Motion of level sets by inverse anisotropic mean curvature

Francesco Della Pietra, Nunzia Gavitone, Chao Xia

math.APWider flowsarXiv:1804.06639
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Abstract

In this paper we consider the weak formulation of the inverse anisotropic mean curvature flow, in the spirit of Huisken-Ilmanen. By using approximation method involving Finsler-p-Laplacian, we prove the existence and uniqueness of weak solutions.

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Submitted
Apr 18, 2018
Subjects
math.AP
Mean curvature flowAnisotropic & nonlocal flowsWeak & singular flowsStability & uniqueness

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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 2016, 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 A level set crystalline mean curvature flow of surfaces (Yoshikazu Giga and Norbert Požár, 2016)3Giga 2016 Existence and uniqueness for anisotropic and crystalline mean curvature flows (Antonin Chambolle, Massimiliano Morini, Matteo Novaga et al., 2017)5Chambolle 2017 Non-uniqueness of closed embedded non-smooth hypersurfaces with constant anisotropic mean curvature (Yoshiki Jikumaru and Miyuki Koiso, 2019)7Jikumaru 2019 Minimizing movements for forced anisotropic mean curvature flow of partitions with mobilities (Giovanni Bellettini, Antonin Chambolle and Shokhrukh Yu. Kholmatov, 2020)4Bellettini 2020 Anisotropic and crystalline mean curvature flow of mean-convex sets (Antonin Chambolle and Matteo Novaga, 2020)8Chambolle 2020 A variational approach to the volume-preserving anisotropic mean curvature flow in 2D (Andrea Kubin, Domenico Angelo La Manna and Enrico Pasqualetto, 2025)6Kubin 2025 Asymptotic behaviour of the weak inverse anisotropic mean curvature flow (Chaoqun Gao, Yong Wei and Rong Zhou, 2025)1Gao 2025 Higher regularity of the inverse anisotropic mean curvature flow (Chaoqun Gao, Yong Wei and Rong Zhou, 2026)2Gao 2026
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  1. Asymptotic behaviour of the weak inverse anisotropic mean curvature flow

    Chaoqun Gao, Yong Wei and Rong ZhouWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Shared terms “inverse anisotropic”, “Ilmanen”, “Huisken”

  2. Higher regularity of the inverse anisotropic mean curvature flow

    Chaoqun Gao, Yong Wei and Rong ZhouWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Shared terms “inverse anisotropic”, “Ilmanen”, “Huisken”

  3. A level set crystalline mean curvature flow of surfaces

    Yoshikazu Giga and Norbert PožárWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows, Stability & uniqueness · Shared terms “level set”, “formulation”, “approximation”

  4. Minimizing movements for forced anisotropic mean curvature flow of partitions with mobilities

    Giovanni Bellettini, Antonin Chambolle and Shokhrukh Yu. KholmatovWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows

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

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

    Topics Anisotropic & nonlocal flows, Weak & singular flows, Stability & uniqueness · Shared terms “level set”, “formulation”, “approximation”

  6. A variational approach to the volume-preserving anisotropic mean curvature flow in 2D

    Andrea Kubin, Domenico Angelo La Manna and Enrico PasqualettoWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows

  7. Non-uniqueness of closed embedded non-smooth hypersurfaces with constant anisotropic mean curvature

    Yoshiki Jikumaru and Miyuki KoisoWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows, Stability & uniqueness

  8. Anisotropic and crystalline mean curvature flow of mean-convex sets

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    Topics Anisotropic & nonlocal flows, Stability & uniqueness

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) 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)

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.