Related papers: Variance-Refined In-Diameter Lower Bound for the F…

We study the problem of minimizing the second Dirichlet eigenvalue for the Laplacian operator among sets of given perimeter. In two dimensions, we prove that the optimum exists, is convex, regular, and its boundary contains exactly two…

A stability result in terms of the perimeter is obtained for the first Dirichlet eigenvalue of the Laplacian operator. In particular, we prove that, once we fix the dimension $n\geq2$, there exists a constant $c>0$, depending only on $n$,…

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…

Let $m$ be a bounded function and $\alpha$ a nonnegative parameter. This article is concerned with the first eigenvalue $\lambda\_\alpha(m)$ of the drifted Laplacian type operator $\mathcal L\_m$ given by $\mathcal L\_m(u)=…

We present a fractional counterpart of a generalized Kohler-Jobin inequality, showing that, among all bounded, open sets $\Omega\subset \mathbb{R}^N$ with Lipschitz boundary, having the same fractional torsional rigidity, the first…

Suppose that $M$ is a compact Riemannian manifold with boundary and $u$ is an $L^2$-normalized Dirichlet eigenfunction with eigenvalue $\lambda$. Let $\psi$ be its normal derivative at the boundary. Scaling considerations lead one to expect…

Let $M$ be a closed hypersurface in a noncompact rank-1 symmetric space $(\bar{\mathbb{M}}, ds^2)$ with $-4 \leq K_{\bar{\mathbb{M}}} \leq -1,$ or in a complete, simply connected Riemannian manifold $\mathbb{M}$ such that $0 \leq…

We investigate the asymptotic behavior of the eigenvalues of the Laplacian with homogeneous Robin boundary conditions, when the (positive) Robin parameter is diverging. In this framework, since the convergence of the Robin eigenvalues to…

Let $\Sigma$ be a closed embedded minimal hypersurface in the unit sphere $\mathbb{S}^{m+1}$ and let $\Lambda=\max\limits_{\Sigma}|A|$ be the norm of its second fundamental form. In this work we prove that the first eigenvalue of the…

We prove a lower bound estimate for the first non-zero eigenvalue of the Witten-Laplacian on compact Riemannian manifolds. As an application, we derive a lower bound estimate for the diameter of compact gradient shrinking Ricci solitons.…

In this paper we study the problem of prescribing Dirichlet eigenvalues on an arbitrary compact manifold $M$ of dimension $n\geq 3$ with a non-empty smooth boundary $\partial M$. We show that for any finite increasing sequence of real…

We give a new estimate on the lower bound for the first Dirichlet eigenvalue for a compact manifold with positive Ricci curvature in terms of the in-diameter and the lower bound of the Ricci curvature. The result improves the previous…

We establish an explicit lower bound of the first eigenvalue of the Laplacian on K\"ahler manifolds based off the comparison results of Li and Wang. The lower bound will depend on the diameter, dimension, holomorphic sectional curvature and…

For an $n$-dimensional polytope $\Omega$ in $\mathbb{R}^{n}$, we study lower bounds for eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. In the asymptotic formula on the average of the first $k$ eigenvalues, Li and Yau…

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…

Let $\Sigma$ be a closed, embedded, oriented hypersurface in a closed oriented Riemannian manifold $N$. Under a lower bound on the Ricci curvature and an upper bound on the sectional curvature of $N$, we establish a lower bound for the…

For any bounded regular domain $\Omega$ of a real analytic Riemannian manifold $M$, we denote by $\lambda_{k}(\Omega)$ the $k$-th eigenvalue of the Dirichlet Laplacian of $\Omega$. In this paper, we consider $\lambda_k$ and as a functional…

Let $G$ be a compact connected Lie group and let $K$ be a closed subgroup of $G$. In this paper we study whether the functional $g\mapsto \lambda_1(G/K,g)\operatorname{diam}(G/K,g)^2$ is bounded among $G$-invariant metrics $g$ on $G/K$.…

For a bounded domain $\Omega$ with a piecewise smooth boundary in a complete Riemannian manifold $M$, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. By making use of a fact that eigenfunctions form an orthonormal…

In this paper, we give a sharp lower bound for the first (nonzero) Neumann eigenvalue of Finsler-Laplacian in Finsler manifolds in terms of diameter, dimension, weighted Ricci curvature.