Related papers: Upper Bound Estimates for Eigenvalues of the Poly-…

We study the Dirichlet eigenvalues of the Laplacian on a convex domain in $\mathbb{R}^n$, with $n\geq 2$. In particular, we generalize and improve upper bounds for the Riesz means of order $\sigma\geq 3/2$ established in an article by…

In this paper, we use a weighted isoperimetric inequality to give a lower bound on the first Dirichlet eigenvalue of the Laplacian on a bounded domain inside a Euclidean cone. Our bound is sharp, in that only sectors realize it. This result…

A celebrated inequality by Payne relates the first eigenvalue of the Dirichlet Laplacian to the first eigenvalue of the buckling problem. Motivated by the goal of establishing a quantitative version of this inequality, we show that Payne's…

We prove a general upper bound on the $k$-th adjacency eigenvalue of a graph. For $k\ge 2$, we show that \[ \lambda_k(G)\le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1 \] for every graph $G$ on $n$ vertices. We build on a recent approach that…

Improving and extending recent results of the author, we conditionally estimate exponential sums with Dirichlet coefficients of L-functions, both over all integers and over all primes in an interval. In particular, we establish new…

Given a Riemmanian manifold, we provide a new method to compute a sharp upper bound for the first eigenvalue of the Laplacian for the Dirichlet problem on a geodesic ball of radius less than the injectivity radius of the manifold. This…

In this paper, motivated by study on universal inequalities for eigenvalues of the Dirichlet Laplacian, we prove some new inequalities for eigenvalues of the Dirichlet Laplacian on the hyperbolic space. In particular, we verify Cheng's…

In this paper, we obtain the bounds of the extreme eigenvalues of a normalized and signless Laplacian matrices using by their traces. In addition, we determine the bounds for k-th eigenvalues of normalized and signless Laplacian matrices.

We obtain upper bounds for the first Dirichlet eigenvalue of a tube around a complex submanifold $P$ of $CP^n$ which depends only on the radius of the tube, the degrees of the polynomials defining $P$ and the first eigenvalue of some model…

We study the eigenvalues of the Dirichlet Laplace operator on an arbitrary bounded, open set in $\R^d$, $d \geq 2$. In particular, we derive upper bounds on Riesz means of order $\sigma \geq 3/2$, that improve the sharp Berezin inequality…

The eigenvalue problem for the Laplacian on bounded, planar, convex domains with mixed boundary conditions is considered, where a Dirichlet boundary condition is imposed on a part of the boundary and a Neumann boundary condition on its…

We study eigenvalues of general scalar Dirichlet polyharmonic problems in domains in $\mathbb R^{d}$. We first prove a number of inequalities satisfied by the eigenvalues on general domains, depending on the relations between the orders of…

We generalize to the case of the $p-$Laplacian an old result by Hersch and Protter. Namely, we show that it is possible to estimate from below the first eigenvalue of the Dirichlet $p-$Laplacian of a convex set in terms of its inradius. We…

The purpose of the paper is to present quantitative estimates for the principal eigenvalue of discrete p-Laplacian on the set of rooted trees. Alternatively, it is studying the optimal constant of a class of weighted Hardy inequality. Three…

We give an improvement of sharp Berezin type bounds on the Riesz means $\sum_k(\Lambda-\lambda_k)_+^\sigma$ of the eigenvalues $\lambda_k$ of the Dirichlet Laplacian in a domain if $\sigma\geq 3/2$. It contains a correction term of the…

We consider the problem of minimising the $k$-th eigenvalue of the Laplacian with some prescribed boundary condition over collections of convex domains of prescribed perimeter or diameter. It is known that these minimisation problems are…

In this paper, we consider lower order eigenvalues of Laplacian operator with any order in Euclidean domains. By choosing special rectangular coordinates, we obtain two estimates for lower order eigenvalues.

In this note we present upper bounds for the variational eigenvalues of the Steklov $p$-Laplacian on domains of $\mathbb R^n$, $n\geq 2$. We show that for $1<p\leq n$ the variational eigenvalues $\sigma_{p,k}$ are bounded above in terms of…

An explicit Dirichlet series is obtained, which represents an analytic function of $s$ in the half-plane $\Re s>1/2$ except for having simple poles at points $s_j$ that correspond to exceptional eigenvalues $\lambda_j$ of the non-Euclidean…

In this paper, we prove the Generalized P\'{o}lya conjecture for the Dirichlet eigenvalues. In other words, we show that $\lambda_k(\alpha) \ge \frac{(2\pi)^{\alpha} k^{\alpha/n}}{\big(\omega_n \cdot {vol}(\Omega)\big)^{\alpha/n}}, \quad\,…