The gradient estimate of a Neumann eigenfunction on a compact manifold with boundary

Hu, Jingchen; Shi, Yiqian; Xu, Bin

doi:10.1007/s11401-015-0924-6

Cited by 5 publications

(4 citation statements)

References 11 publications

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“…and has been studied by Xu [34,35,36] and, subsequently, by Arnaudon, Thalmaier & Wang [1], Cheng, Thalmaier & Thompson [6], Hu, Shi & Xu [13] and Shi & Xu [21]. On compact manifolds without boundary, there are results of Xu [33] for all derivatives and by Wang & Zhou [32] for linear combinations of eigenfunctions.…”

Section: Introduction and Resultsmentioning

confidence: 99%

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A Pointwise Inequality for Derivatives of Solutions of the Heat Equation in Bounded Domains

Steinerberger

2021

Preprint

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Let u(t, x) be a solution of the heat equation in R n . Then, each k−th derivative also solves the heat equation and satisfies a maximum principle, the largest k−th derivative of u(t, x) cannot be larger than the largest k−th derivative of u(0, x). We prove an analogous statement for the solution of the heat equation on bounded domains Ω ⊂ R n with Dirichlet boundary conditions. As an application, we give a new and fairly elementary proof of the sharp growth of the second derivatives of Laplacian eigenfunction −∆φ k = λ k φ k with Dirichlet conditions on smooth domains Ω ⊂ R n .

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Section: Introduction and Resultsmentioning

confidence: 99%

“…However, the mean curvature H is bounded depending only on ∂Ω because the domain is smooth. Therefore, using the gradient estimate of Hu, Shi & Xu [13] on compact Riemannian manifolds with boundary, we get…”

Section: Proof Of Theoremmentioning

confidence: 99%

A Pointwise Inequality for Derivatives of Solutions of the Heat Equation in Bounded Domains

Steinerberger

2021

Preprint

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“…Proof. The Neumann eigenfunctions on compact manifolds satisfy the inequality ∇φ ∞ ≤ c 1 λ φ ∞ , see [18]. Applying the bound used in the proof of convergence, φ ∞ ≤ c 2 λ (n−1)/4 , we get that φ is Lipschitz with respect to the geodesic distance on X with the Lipschitz constant bounded by cλ λ…”

Section: Properties Of Isdmentioning

confidence: 98%

Intrinsic Sliced Wasserstein Distances for Comparing Collections of Probability Distributions on Manifolds and Graphs

Rustamov¹,

Majumdar²

2020

Preprint

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Collections of probability distributions arise in a variety of statistical applications ranging from user activity pattern analysis to brain connectomics. In practice these distributions are represented by histograms over diverse domain types including finite intervals, circles, cylinders, spheres, other manifolds, and graphs. This paper introduces an approach for detecting differences between two collections of histograms over such general domains. To this end, we introduce the intrinsic slicing construction that yields a novel class of Wasserstein distances on manifolds and graphs. These distances are Hilbert embeddable, which allows us to reduce the histogram collection comparison problem to the comparison of means in a high-dimensional Euclidean space. We develop a hypothesis testing procedure based on conducting t-tests on each dimension of this embedding, then combining the resulting p-values using recently proposed p-value combination techniques. Our numerical experiments in a variety of data settings show that the resulting tests are powerful and the p-values are well-calibrated. Example applications to user activity patterns, spatial data, and brain connectomics are provided.

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“…An analogous statement for Neumann eigenfunctions has been derived in [5]. Concerning Dirichlet eigenfunctions, an explicit upper constant c 2 (D) can be derived from the uniform gradient estimate of the Dirichlet semigroup in an earlier paper [10] of the third named author.…”

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confidence: 90%

Gradient Estimates on Dirichlet and Neumann Eigenfunctions

Arnaudon

Thalmaier

Wang

2018

International Mathematics Research Notices

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The gradient estimate of a Neumann eigenfunction on a compact manifold with boundary

Cited by 5 publications

References 11 publications

A Pointwise Inequality for Derivatives of Solutions of the Heat Equation in Bounded Domains

A Pointwise Inequality for Derivatives of Solutions of the Heat Equation in Bounded Domains

Intrinsic Sliced Wasserstein Distances for Comparing Collections of Probability Distributions on Manifolds and Graphs

Gradient Estimates on Dirichlet and Neumann Eigenfunctions

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