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 Eigenvalue bounds of mixed Steklov problems

2019 ◽  
Vol 22 (02) ◽  
pp. 1950008 ◽  
Author(s):  
Asma Hassannezhad ◽  
Ari Laptev
Keyword(s):  
Eigenvalue Problem ◽  
Fractional Laplacian ◽  
Eigenvalue Problems ◽  
Asymptotic Results ◽  
Riesz Means ◽  
Dirichlet Eigenvalue ◽  
Eigenvalue Bounds ◽  
Steklov Eigenvalue ◽  
Dirichlet Eigenvalue Problem ◽  
Neumann Eigenvalue

We study bounds on the Riesz means of the mixed Steklov–Neumann and Steklov–Dirichlet eigenvalue problem on a bounded domain [Formula: see text] in [Formula: see text]. The Steklov–Neumann eigenvalue problem is also called the sloshing problem. We obtain two-term asymptotically sharp lower bounds on the Riesz means of the sloshing problem and also provide an asymptotically sharp upper bound for the Riesz means of mixed Steklov–Dirichlet problem. The proof of our results for the sloshing problem uses the average variational principle and monotonicity of sloshing eigenvalues. In the case of Steklov–Dirichlet eigenvalue problem, the proof is based on a well-known bound on the Riesz means of the Dirichlet fractional Laplacian, and an inequality between the Dirichlet and Navier fractional Laplacian. The two-term asymptotic results for the Riesz means of mixed Steklov eigenvalue problems are discussed in the Appendix which in particular show the asymptotic sharpness of the bounds we obtain.

scholarly journals Eigenvalues for a combination between local and nonlocal p-Laplacians

2019 ◽  
Vol 22 (5) ◽  
pp. 1414-1436 ◽  
Author(s):  
Leandro M. Del Pezzo ◽  
Raúl Ferreira ◽  
Julio D. Rossi
Keyword(s):  
Eigenvalue Problem ◽  
Bounded Domain ◽  
Limit Problem ◽  
First Eigenvalue ◽  
Order Zero ◽  
Dirichlet Eigenvalue ◽  
The First Eigenvalue ◽  
Homogeneous Operator ◽  
Dirichlet Eigenvalue Problem

Abstract In this paper we study the Dirichlet eigenvalue problem $$\begin{array}{} \displaystyle -\Delta_p u-\Delta_{J,p}u =\lambda|u|^{p-2}u \text{ in } \Omega,\quad u=0 \, \text{ in } \, \Omega^c=\mathbb{R}^N\setminus\Omega. \end{array}$$ Here Ω is a bounded domain in ℝN, Δpu is the standard local p-Laplacian and ΔJ,pu is a nonlocal p-homogeneous operator of order zero. We show that the first eigenvalue (that is isolated and simple) satisfies $\begin{array}{} \displaystyle \lim_{p\to\infty} \end{array}$(λ1)1/p = Λ where Λ can be characterized in terms of the geometry of Ω. We also find that the eigenfunctions converge, u∞ = $\begin{array}{} \displaystyle \lim_{p\to\infty} \end{array}$ up, and find the limit problem that is satisfied in the limit.

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 The Ratio of Eigenvalues of the Dirichlet Eigenvalue Problem for Equations with One-Dimensionalp-Laplacian

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Gabriella Bognár ◽  
Ondřej Došlý
Keyword(s):  
Eigenvalue Problem ◽  
Dirichlet Eigenvalue ◽  
One Dimensional ◽  
Single Well ◽  
Dirichlet Eigenvalue Problem

We establish an estimate for the ratio of eigenvalues of the Dirichlet eigenvalue problem for the equation with one-dimensionalp-Laplacian involving a nonnegative unimodal (single-well) potential.

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