scholarly journals Estimates for lower bounds of eigenvalues of the poly-Laplacian and the quadratic polynomial operator of the Laplacian

2013 ◽  
Vol 143 (6) ◽  
pp. 1147-1162 ◽  
Author(s):  
Qing-Ming Cheng ◽  
He-Jun Sun ◽  
Guoxin Wei ◽  
Lingzhong Zeng
Keyword(s):  
Lower Bounds ◽  
Eigenvalue Problems ◽  
Quadratic Polynomial ◽  
Polynomial Operator ◽  
Dirichlet Eigenvalue

In this paper, we investigate the Dirichlet eigenvalue problems of the poly-Laplacian with any order and the quadratic polynomial operator of the Laplacian. We give some estimates for lower bounds of the sums of their first k eigenvalues.

scholarly journals EIGENVALUE ESTIMATES FOR HIGHER ORDER ELLIPTIC EQUATIONS

1994 ◽  
Vol 25 (3) ◽  
pp. 267-278
Author(s):  
HSU-TUNG KU ◽  
MEI-CHIN KU ◽  
XIN-MIN ZHANG
Keyword(s):  
Lower Bound ◽  
Elliptic Equations ◽  
Eigenvalue Problems ◽  
Higher Order ◽  
Bounded Domains ◽  
Dirichlet Eigenvalue ◽  
Eigenvalue Estimates

In this paper, we obtain good lower bound estimates of eigenvalues for various Dirichlet eigenvalue problems of higher order elliptic equations on bounded domains in $\mathbb{R}^n$.

scholarly journals New Refinements and Improvements of Jordan’s Inequality

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 284 ◽  
Author(s):  
Lina Zhang ◽  
Xuesi Ma
Keyword(s):  
Lower Bounds ◽  
Upper Bounds ◽  
Quadratic Polynomial ◽  
Quartic Polynomial ◽  
Polynomial Bounds ◽  
Jordan’S Inequality

The polynomial bounds of Jordan’s inequality, especially the cubic and quartic polynomial bounds, have been studied and improved in a lot of the literature; however, the linear and quadratic polynomial bounds can not be improved very much. In this paper, new refinements and improvements of Jordan’s inequality are given. We present new lower bounds and upper bounds for strengthened Jordan’s inequality using polynomials of degrees 1 and 2. Our bounds are tighter than the previous results of polynomials of degrees 1 and 2. More importantly, we give new improvements of Jordan’s inequality using polynomials of degree 5, which can achieve much tighter bounds than those previous methods.

Upper and Lower Bounds for Special Eigenvalues

1959 ◽  
Vol 26 (2) ◽  
pp. 246-250
Author(s):  
F. C. Appl ◽  
C. F. Zorowski
Keyword(s):  
Exact Solution ◽  
Power Series ◽  
Lower Bounds ◽  
Comparison Theorem ◽  
Eigenvalue Problems ◽  
Buckling Load ◽  
Upper And Lower Bounds ◽  
Elastic Buckling ◽  
Variable Section ◽  
Exact Eigenvalue

Abstract A method for finding upper and lower bounds for the fundamental eigenvalue in special eigenvalue problems is presented. The method is systematic and is shown to provide convergence from above and below to the exact eigenvalue under certain conditions. The method is based on the relatively well-known enclosure or comparison theorem of Collatz, and makes use of a power series to approximate the eigenfunction. The method is applied to two examples concerning the critical-elastic buckling load of variable-section columns with pinned ends. Results for the first example compare well with the exact solution, which is known; the second example is presented as an addition to the literature.

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 Multiple positive solutions for discrete p-Laplacian equations with potential term

2013 ◽  
Vol 7 (2) ◽  
pp. 327-342 ◽  
Author(s):  
Jong-Ho Kim ◽  
Jea-Hyun Park ◽  
June-Yub Lee
Keyword(s):  
Positive Solutions ◽  
Eigenvalue Problems ◽  
Existence Of Solutions ◽  
Growth Conditions ◽  
Multiple Positive Solutions ◽  
Source Terms ◽  
Dirichlet Eigenvalue ◽  
Nonlinear Source ◽  
Potential Term ◽  
Laplacian Equations

We study the existence of solutions to nonlinear discrete boundary value problems with the discrete p-Laplacian, potential, and nonlinear source terms. Using variational methods, we demonstrate that there exist at least two positive solutions. The existence strongly depends on the smallest positive eigenvalue of Dirichlet eigenvalue problems and the growth conditions of the source terms.

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>

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