Related papers: Sharp upper bound for the first eigenvalue

Ten sharp lower estimates of the first non-trivial eigenvalue of Laplacian on compact Riemannian manifolds are reviewed and compared. An improved variational formula, a general common estimate, and a new sharp one are added. The best lower…

We consider an optimization problem for the first Dirichlet eigenvalue of the $p$-Laplacian on a hypersurface in $\mathbb{R}^{2n}$, with $n \ge 2$. If $p \ge 2n-1$, then among hypersurfaces in $\mathbb{R}^{2n}$ which are $O(n) \times…

In this paper, we study the first eigenvalue of the Laplacian on doubly connected domains when Robin and Dirichlet conditions are imposed on the outer and the inner part of the boundary, respectively. We provide that the spherical shell…

In this paper, we investigate the first eigenvalues of two closed eigenvalue problems of the bi-Beltrami-Laplacian on minimal embedded isoparametric hypersurface in the unit sphere $\mathbb{S}^{n+1}(1)$. Although many mathematicians want to…

In this article, we establish a geometric lower bound for the first positive eigenvalue $\lambda^{(1)}_{1}$ of the rough Laplacian acting on $1$-forms for closed $2n$-dimensional Riemannian manifolds with nonvanishing Euler characteristic.…

In this note we give a new upper bound for the Laplacian eigenvalues of an unweighted graph. Let $G$ be a simple graph on $n$ vertices. Let $d_{m}(G)$ and $\lambda_{m+1}(G)$ be the $m$-th smallest degree of $G$ and the $m+1$-th smallest…

In the present paper we study some kinds of the problems for the bi-drifting Laplacian operator and get some sharp lower bounds for the first eigenvalue for these eigenvalue problems on compact manifolds with boundary (also called a smooth…

We study eigenvalue problems for intrinsic sub-Laplacians on regular sub-Riemannian manifolds. We prove upper bounds for sub-Laplacian eigenvalues $\lambda_k$ of conformal sub-Riemannian metrics that are asymptotically sharp as $k\to…

We derive a sharp upper bound for the first eigenvalue $\lambda_{1,p}$ of the $p$-Laplacian on asymptotically hyperbolic manifolds for $1<p<\infty$. We then prove that a particular class of conformally compact submanifolds within…

Let (M,g) be a compact Riemannian manifold of dimension >2. We show that there is a metric h conformal to g and of volume 1 such that the first positive eigenvalue the conformal Laplacian with repect to h is arbitrarily large. A similar…

We prove that the first eigenvalue of the fractional Dirichlet-Laplacian of order $s$ on a simply connected set of the plane can be bounded from below in terms of its inradius only. This is valid for $1/2<s<1$ and we show that this…

Building on seminal work of Nadirashvili and previous work of the authors, we prove the existence of metrics maximizing the area-normalized first eigenvalue of the Laplacian on every closed nonorientable surface, and give a simple new proof…

In this article, we prove that for an embedded minimal hypersurface $\Sigma^{m}$ in $S^{m+1}$, the first eigenvalue $\lambda_1$ of the Laplacian operator on $\Sigma$ satisfies: $$\lambda_1> \frac{m}{2}+G(m, |A|_{\max}, |A|_{\min} ) ,$$…

We study the eigenvalues of the magnetic Schroedinger operator associated with a magnetic potential A and a scalar potential q, on a compact Riemannian manifold M, with Neumann boundary conditions if the boundary is not empty. We obtain…

We prove an upper bound for the volume-normalized second nonzero eigenvalue of the Laplace operator on closed Riemannian manifold, in terms of the conformal volume. This bound provides effective upper bound for a large class of manifolds,…

In this note we present upper bounds for the variational eigenvalues of the $p$-Laplacian on smooth domains of complete $n$-dimensional Riemannian manifolds and Neumann boundary conditions, and on compact (boundaryless) Riemannian…

In this article, we prove an eigenvalue pinching theorem for the first eigenvalue of the Laplacian on compact hypersurfaces in a sphere. Let $(M^n,g)$ be a closed, connected and oriented Riemannian manifold isometrically immersed by $\phi$…

Let $(M^n,g)$ be a complete simply connected $n$-dimensional Riemannian manifold with curvature bounds $\operatorname{Sect}_g\leq \kappa$ for $\kappa\leq 0$ and $\operatorname{Ric}_g\geq(n-1)Kg$ for $K\leq 0$. We prove that for any bounded…

In this paper, sharp bounds for the first nonzero eigenvalues of different type have been obtained. Moreover, when those bounds are achieved, related rigidities can be characterized. More precisely, first, by applying the Bishop-type volume…

In this paper we prove that given a volume, among all domains with smooth boundary in rank-1 symmetric spaces of noncompact type, geodesic balls maximizes the first nonzero Steklov eigenvalue. We also prove a comparison result for the first…