scholarly journals Lower Bounds of Dirichlet Eigenvalues for General Grushin Type Bi-Subelliptic Operators

2019 ◽  
Vol 35 (1) ◽  
pp. 66-84
Author(s):  
Hua Chen
Keyword(s):  
Lower Bounds ◽  
Dirichlet Eigenvalues ◽  
Subelliptic Operators

Lower bounds for the first Dirichlet eigenvalue of the Laplacian for domains in hyperbolic space

2015 ◽  
Vol 160 (2) ◽  
pp. 191-208 ◽  
Author(s):  
SERGEI ARTAMOSHIN
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Negative Curvature ◽  
Connected Space ◽  
Constant Negative Curvature ◽  
Dirichlet Eigenvalue ◽  
Dirichlet Eigenvalues ◽  
First Dirichlet Eigenvalue ◽  
A New Technique ◽  
Simply Connected

AbstractWe consider domains in a simply connected space of constant negative curvature and develop a new technique that improves existing classical lower bound for Dirichlet eigenvalues obtained by H. P. McKean as well as the lower bounds recently obtained by A. Savo.

scholarly journals Estimates the upper bounds of Dirichlet eigenvalues for fractional Laplacian

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Hua Chen ◽  
Hong-Ge Chen
Keyword(s):  
Eigenvalue Problem ◽  
Lower Bounds ◽  
Fractional Laplacian ◽  
Upper Bounds ◽  
Dirichlet Eigenvalue ◽  
Dirichlet Eigenvalues ◽  
Dirichlet Eigenvalue Problem ◽  
Mean Function ◽  
Continuous Boundary ◽  
Dirichlet Heat Kernel

<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&lt;s&lt;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>

scholarly journals Brownian Motions and Heat Kernel Lower Bounds on Kähler and Quaternion Kähler Manifolds

2020 ◽  
Author(s):  
Fabrice Baudoin ◽  
Guang Yang
Keyword(s):  
Lower Bound ◽  
Heat Kernel ◽  
Lower Bounds ◽  
Heat Kernels ◽  
Comparison Theorems ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Brownian Motions ◽  
Dirichlet Eigenvalues ◽  
Metric Balls

Abstract We study the radial parts of the Brownian motions on Kähler and quaternion Kähler manifolds. Thanks to sharp Laplacian comparison theorems, we deduce as a consequence a sharp Cheeger–Yau-type lower bound for the heat kernels of such manifolds and also sharp Cheng’s type estimates for the Dirichlet eigenvalues of metric balls.

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