Eigenvalue estimate for the weighted p-Laplacian

Wang, Lin Feng

doi:10.1007/s10231-011-0195-0

Cited by 23 publications

(11 citation statements)

References 16 publications

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“…Many people devote to study the weighted Laplacian under conditions about the m-dimensional Bakry-Émery curvature [6][7][8]. In particular, the author proved in [9] that when the m-dimensional Bakry-Émery curvature is bounded from below by Ric m ≥ −(m − 1)K for some constant K ≥ 0, then the bottom of the L 2 μ spectrum is bounded above by…”

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confidence: 98%

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A splitting theorem for the weighted measure

Wang

2011

Ann Glob Anal Geom

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Let M be an n-dimensional complete noncompact Riemannian manifold, h be a smooth function on M and dμ = e h dV be the weighted measure. In this article, we prove that when the spectrum of the weighted Laplacian μ has a positive lower bound λ 1 (M) > 0 and the m(m > n)-dimensional Bakry-Émery curvature is bounded from below by − m−1 m−2 λ 1 (M), then M splits isometrically as R × N whenever it has two ends with infinite weighted volume, here N is an (n − 1)-dimensional compact manifold.

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confidence: 98%

“…Further results about the weighted Laplacian can be found in [6][7][8][9] and the references therein.…”

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confidence: 98%

A splitting theorem for the weighted measure

Wang

2011

Ann Glob Anal Geom

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“…When p = 2, the weighted p-Laplacian degenerates or is singular at points ∇f = 0. In this case ε-regularization technique is usually applied by replacing the linearized operator with its approximate operator, see [14,22] for examples. For ε > 0, we define an approximate operator L φ,ε :…”

Section: Regularization Procedures and Basic Lemmamentioning

confidence: 99%

Monotonicity formulas for the first eigenvalue of the weighted p-Laplacian under the Ricci-harmonic flow

2019

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Let p,φ be the weighted p-Laplacian defined on a smooth metric measure space. We study the evolution and monotonicity formulas for the first eigenvalue, λ 1 = λ(p,φ), of p,φ under the Ricci-harmonic flow. We derive some monotonic quantities involving the first eigenvalue, and as a consequence, this shows that λ 1 is monotonically nondecreasing and almost everywhere differentiable along the flow existence.

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“…For example second author in [2], has studied the evolution for the first eigenvalue of p-Laplacian along the Yamabe flow and also in [3] shown the monotonicity of eigenvalues of Witten-Laplace operator along the Ricci-Bourguignon flow. Also for more details in a case of p-Laplacian operator, Wang in [17], has shown the eigenvalue estimate for the weighted p-Laplacian and later in [18] shown the gradient estimate on the weighted p-Laplace heat equation. Beside what mentioned before A. Abolarinwa in [1], has studied the evolution and monotonicity of the first eigenvalue of p-Laplacian under the Ricci-harmonic flow and also you can find some useful results in eigenvalue monotonicity of the p-Laplace operator under the Ricci flow in [20], also Cao and Songbo Hou have worked on monotonicity of the first eigenvalue under Ricci flow and you can see their results in [8,12].…”

Section: Introductionmentioning

confidence: 99%