Universal inequality and upper bounds of eigenvalues for non-integer poly-Laplacian on a bounded domain

Chen, Hua; Zeng, Ao

doi:10.1007/s00526-017-1220-y

Cited by 9 publications

(13 citation statements)

References 30 publications

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“…Li-Yau [28] improved the constant C = N N +2 C N , and with that constant (1.14) is also called Berezin-Li-Yau inequality because this constant is achieved with the help of Legendre transform as in the Berezin's earlier paper [2]. The Berezin-Li-Yau inequality then is generalized in [11][12][13]25,29,31], for degenerate elliptic operators in [9,22,38] for the fractional Laplacian (−∆) s defined in (1.4) and the inequality reads…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Bounds for eigenvalues of the Dirichlet problem for the logarithmic Laplacian

Chen

Верон

2022

Advances in Calculus of Variations

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We provide bounds for the sequence of eigenvalues { λ i ⁢ ( Ω ) } i {\{\lambda_{i}(\Omega)\}_{i}} of the Dirichlet problem L Δ ⁢ u = λ ⁢ u ⁢ in ⁢ Ω , u = 0 ⁢ in ⁢ ℝ N ∖ Ω , L_{\Delta}u=\lambda u\text{ in }\Omega,\quad u=0\text{ in }\mathbb{R}^{N}% \setminus\Omega, where L Δ {L_{\Delta}} is the logarithmic Laplacian operator with Fourier transform symbol 2 ⁢ ln ⁡ | ζ | {2\ln\lvert\zeta\rvert} . The logarithmic Laplacian operator is not positively defined if the volume of the domain is large enough. In this article, we obtain the upper and lower bounds for the sum of the first k eigenvalues by extending the Li–Yau method and Kröger’s method, respectively. Moreover, we show the limit of the quotient of the sum of the first k eigenvalues by k ⁢ ln ⁡ k {k\ln k} is independent of the volume of the domain. Finally, we discuss the lower and upper bounds of the k-th principle eigenvalue, and the asymptotic behavior of the limit of eigenvalues.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Bounds for eigenvalues of the Dirichlet problem for the logarithmic Laplacian

Chen

Верон

2022

Advances in Calculus of Variations

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“…or s > 1 2 and s ∈ Q. For s > 1 and s ∈ Q, the results in [8] give the following upper bound for the ratio…”

mentioning

confidence: 91%

“…Note that (1.17) only give the upper bound of ratio λ k+1 λ1 for s = 1 2 . Additionally, we mention the recent survey of Chen and Zeng [8], which studied the universal inequalities for the non-integer poly-Laplacian (−△)…”

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confidence: 99%

Estimates the upper bounds of Dirichlet eigenvalues for fractional Laplacian

Chen¹,

Chen²

2022

DCDS

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<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset\mathbb{R}^n \; (n\geq 2) $\end{document}</tex-math></inline-formula> be a bounded domain with continuous boundary <inline-formula><tex-math id="M2">\begin{document}$ \partial\Omega $\end{document}</tex-math></inline-formula>. In this paper, we study the Dirichlet eigenvalue problem of the fractional Laplacian which is restricted to <inline-formula><tex-math id="M3">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M4">\begin{document}$ 0<s<1 $\end{document}</tex-math></inline-formula>. Denoting by <inline-formula><tex-math id="M5">\begin{document}$ \lambda_{k} $\end{document}</tex-math></inline-formula> the <inline-formula><tex-math id="M6">\begin{document}$ k^{th} $\end{document}</tex-math></inline-formula> Dirichlet eigenvalue of <inline-formula><tex-math id="M7">\begin{document}$ (-\triangle)^{s}|_{\Omega} $\end{document}</tex-math></inline-formula>, we establish the explicit upper bounds of the ratio <inline-formula><tex-math id="M8">\begin{document}$ \frac{\lambda_{k+1}}{\lambda_{1}} $\end{document}</tex-math></inline-formula>, which have polynomially growth in <inline-formula><tex-math id="M9">\begin{document}$ k $\end{document}</tex-math></inline-formula> with optimal increase orders. Furthermore, we give the explicit lower bounds for the Riesz mean function <inline-formula><tex-math id="M10">\begin{document}$ R_{\sigma}(z) = \sum_{k}(z-\lambda_{k})_{+}^{\sigma} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M11">\begin{document}$ \sigma\geq 1 $\end{document}</tex-math></inline-formula> and the trace of the Dirichlet heat kernel of fractional Laplacian.</p>

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“…Some detail information on Lévy process will be found in a textbook [36] and a good survey [1]. We also refer the readers to an excellent article [16], where some important and interesting literatures on fractional Laplacian have been given by Chen and Zeng. We assume that φ ∈…”

Section: Introductionmentioning

confidence: 99%

“…is much less studied, but it has recently received more attention. Here, we only refer the reader to papers [10,16,26,42,46,47] and the references therein. In particular, Chen and Zeng investigated the eigenvalues with higher order of fractional Laplacian.…”

Section: Introductionmentioning

confidence: 99%

Universal Bounds for Fractional Laplacian on a Bounded Open Domain in $\mathbb{R}^{n}$

Zeng¹

2021

Preprint

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Let Ω be a bounded open domain on the Euclidean space R n and Q + be the set of all positive rational numbers. In 2017, Chen and Zeng investigated the eigenvalues with higher order of the fractional Laplacian (−∆) s | Ω for s > 0 and s ∈ Q + , and they obtained a universal inequality of Yang type( Universal inequality and upper bounds of eigenvalues for non-integer poly-Laplacian on a bounded domain, Calculus of Variations and Partial Differential Equations, (2017) 56:131 ). In the spirit of Chen and Zeng's work, we study the eigenvalues of fractional Laplacian, and establish an inequality of eigenvalues with lower order under the same condition. Also, our eigenvalue inequality is universal and generalizes the eigenvalue inequality for the poly-harmonic operators given by Jost et al.(Universal bounds for eigenvalues of polyharmonic operator. Trans. Amer. Math. Soc. 363(4), 1821-1854).

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Universal inequality and upper bounds of eigenvalues for non-integer poly-Laplacian on a bounded domain

Cited by 9 publications

References 30 publications

Bounds for eigenvalues of the Dirichlet problem for the logarithmic Laplacian

Bounds for eigenvalues of the Dirichlet problem for the logarithmic Laplacian

Estimates the upper bounds of Dirichlet eigenvalues for fractional Laplacian

Universal Bounds for Fractional Laplacian on a Bounded Open Domain in $\mathbb{R}^{n}$

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