Paper · Mar 20, 2018

Gradient estimates and Liouville type theorems for Poisson equations

Nguyen Thac Dung, Nguyen Ngoc Khanh

math.DGarXiv:1803.07251
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Abstract

In this paper, we will address to the following parabolic equation on a smooth metric measure space with Bakry-Émery curvature bounded from below. Here is a differentiable function defined in . Our motivation is originally inspired by gradient estimates of Allen-Cahn and Fisher equations (). In this paper, we show new gradient estimates for these equations. As their applications, we obtain Liouville type theorems for positive or bounded solutions to the above equation when either (the Fisher equation) or; (the Allen-Cahn equation); or (the equation involving gradient Ricci solitons).

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Submitted
Mar 20, 2018
Subjects
math.DG

Comments: 10 pages

Ricci solitonsLimit spaces & RCDPhase fields & multiphase flows

Its neighbourhood

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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 Elliptic gradient estimates for a nonlinear heat equation and applications (Jia-Yong Wu, 2016)7Wu 2016 Convergence of the Allen-Cahn Equation to multi-phase mean-curvature flow (Tim Laux and Thilo Simon, 2016)5Laux 2016 On the Entropy of Parabolic Allen-Cahn Equation (Ao Sun, 2018)2Sun 2018 Convergence rates of the Allen-Cahn equation to mean curvature flow: A short proof based on relative entropies (Julian Fischer, Tim Laux and Theresa M. Simon, 2020)8Fischer 2020 Convergence of Allen-Cahn equations to De Giorgi's multiphase mean curvature flow (Pascal Steinke, 2023)6Steinke 2023 Gradient estimates for a nonlinear parabolic equation on smooth metric measure spaces with evolving metrics and potentials (Ali Taheri and Vahideh Vahidifar, 2023)1Taheri 2023 L^p Liouville type theorems for harmonic functions on gradient Ricci solitons (Yong Luo, 2023)4Luo 2023 Three circles theorems and Liouville type theorems (Run-Qiang Jian and Zhu-Hong Zhang, 2024)3Jian 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Gradient estimates for a nonlinear parabolic equation on smooth metric measure spaces with evolving metrics and potentials

    Ali Taheri and Vahideh Vahidifar

    Topics Limit spaces & RCD · Shared terms “gradient estimates”, “smooth metric”, “Liouville type”

  2. On the Entropy of Parabolic Allen-Cahn Equation

    Ao SunWider flows

    Topics Phase fields & multiphase flows · Shared terms “Allen Cahn”, “Cahn equation”

  3. Three circles theorems and Liouville type theorems

    Run-Qiang Jian and Zhu-Hong Zhang

    Topics Ricci solitons · Shared terms “Liouville type”, “type theorem”, “curvature bounded”

  4. Liouville type theorems for harmonic functions on gradient Ricci solitons

    Yong Luo

    Topics Ricci solitons · Shared terms “Liouville type”, “type theorem”, “gradient Ricci”

  5. Convergence of the Allen-Cahn Equation to multi-phase mean-curvature flow

    Tim Laux and Thilo SimonWider flows

    Topics Phase fields & multiphase flows · Shared terms “Allen Cahn”, “Cahn equation”

  6. Convergence of Allen-Cahn equations to De Giorgi's multiphase mean curvature flow

    Pascal SteinkeWider flows

    Topics Phase fields & multiphase flows · Shared terms “Allen Cahn”, “Cahn equation”

  7. Elliptic gradient estimates for a nonlinear heat equation and applications

    Jia-Yong Wu

    Topics Limit spaces & RCD, Ricci solitons · Shared terms “gradient estimates”, “smooth metric”, “Liouville”

  8. Convergence rates of the Allen-Cahn equation to mean curvature flow: A short proof based on relative entropies

    Julian Fischer, Tim Laux and Theresa M. SimonWider flows

    Topics Phase fields & multiphase flows · Shared terms “Allen Cahn”, “Cahn equation”

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.

Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997) Phase fields & multiphase flows Interfaces moving by curvature can be approximated by smooth phase fields: the Allen–Cahn equation's transition layers converge to mean curvature… Start with: Two-Dimensional Motion of Idealized Grain Boundaries (Mullins, 1956)

In the guidebook

  1. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.