Guaranteed lower bounds on eigenvalues of elliptic operators with a hybrid high-order method

This paper introduces a novel hybrid high-order (HHO) method to approximate the eigenvalues of a symmetric compact differential operator. The HHO method combines two gradient reconstruction operators by means of a parameter $$0<\alpha <~1$$ 0 < α < 1 and introduces a novel cell-based stabilization operator weighted by a parameter $$0<\beta <\infty $$ 0 < β < ∞ . Sufficient conditions on the parameters $$\alpha $$ α and $$\beta $$ β are identified leading to a guaranteed lower bound property for the discrete eigenvalues. Moreover optimal convergence rates are established. Numerical studies for the Dirichlet eigenvalue problem of the Laplacian provide evidence for the superiority of the new lower eigenvalue bounds compared to previously available bounds.

Estimates the upper bounds of Dirichlet eigenvalues for fractional Laplacian

<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>

Eigenvalues for a combination between local and nonlocal p-Laplacians

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.

Eigenvalue bounds of mixed Steklov problems

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.

Existence and Uniqueness of Positive Solitons for a Second-Order Difference Equation

A discrete logistic steady-state equation with both positive and negative birth rate of population will be considered. By using sub- and upper-solution method, the existence of bounded positive solutions and the existence and uniqueness of positive solitons will be established. To this end, the Dirichlet eigenvalue problem with positive and negative coefficients is considered, and a general sub- and upper-solution theorem is also obtained.

ESTIMATES FOR EIGENVALUES OF $\mathfrak L$ OPERATOR ON SELF-SHRINKERS

In this paper, we study eigenvalues of the closed eigenvalue problem of the differential operator [Formula: see text], which is introduced by Colding and Minicozzi in [Generic mean curvature flow I; generic singularities, Ann. Math.175 (2012) 755–833], on an n-dimensional compact self-shrinker in R n+p. Estimates for eigenvalues of the differential operator [Formula: see text] are obtained. Our estimates for eigenvalues of the differential operator [Formula: see text] are sharp. Furthermore, we also study the Dirichlet eigenvalue problem of the differential operator [Formula: see text] on a bounded domain with a piecewise smooth boundary in an n-dimensional complete self-shrinker in R n+p. For Euclidean space R n, the differential operator [Formula: see text] becomes the Ornstein–Uhlenbeck operator in stochastic analysis. Hence, we also give estimates for eigenvalues of the Ornstein–Uhlenbeck operator.