Related papers: Lower bound estimates for eigenvalues of the Lapla…

This paper is concerned with the homogenization of the Dirichlet eigenvalue problem, posed in a bounded domain $\Omega\subset\mathbb R^2$, for a vectorial elliptic operator $-\nabla\cdot A^\epsilon(\cdot)\nabla$ with $\epsilon$-periodic…

In this paper, we considered the spectrum of the Dirichlet Laplacian $\Delta_\epsilon$ on $\Omega_\epsilon=\{(x,y): -l_1<x<l_2, 0<y<\epsilon h(x)]\}$ where $ l_1,l_2>0$ and $h(x)$ is a positive analytic function having $0$ the only point…

We prove a two-term Weyl-type asymptotic formula for sums of eigenvalues of the Dirichlet Laplacian in a bounded open set with Lipschitz boundary. Moreover, in the case of a convex domain we obtain a universal bound which correctly…

By the calculation of the gap of the consecutive eigenvalues of $\Bbb S^n$ with standard metric, using the Weyl's asymptotic formula, we know the order of the upper bound of this gap is $k^{\frac{1}{n}}.$ We conjecture that this order is…

We compute estimates for eigenvalues of a class of linear second-order elliptic differential operators in divergence form (with Dirichlet boundary condition) on a bounded domain in a complete Riemannian manifold. Our estimates are based…

We prove various estimates for the first eigenvalue of the magnetic Dirichlet Laplacian on a bounded domain in two dimensions. When the magnetic field is constant, we give lower and upper bounds in terms of geometric quantities of the…

We consider eigenvalues of the Dirichlet-to-Neumann operator for Laplacian in the domain (or manifold) with edges and establish the asymptotics of the eigenvalue counting function \begin{equation*} \mathsf{N}(\lambda)= \kappa_0\lambda^d…

We establish lower bound for the first nonzero eigenvalue of the Laplacian on a closed K\"ahler manifold in terms of dimension, diameter, and lower bounds of holomorphic sectional curvature and orthogonal Ricci curvature. On compact…

We prove lower bound for the first closed or Neumann nonzero eigenvalue of the Laplacian on a compact quaternion-K\"ahler manifold in terms of dimension, diameter, and scalar curvature lower bound. It is derived as large time implication of…

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…

We consider the Laplacian in a domain squeezed between two parallel curves in the plane, subject to Dirichlet boundary conditions on one of the curves and Neumann boundary conditions on the other. We derive two-term asymptotics for…

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…

We investigate the lower bound for higher eigenvalues $\lambda_i$ of the poly-Laplace operator on a bounded domain and improve the famous Li-Yau inequality and its related results. Firstly, we consider the low dimensional cases for the…

Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold with nonempty boundary and $n\geq 2$. Assume that ${\mathrm{Ric}(M)\ge (n-1)K}$ for some ${K>0}$ and that $\partial M$ has nonnegative mean curvature with respect to the outward…

We consider the Laplacian in a domain squeezed between two parallel hypersurfaces in Euclidean spaces of any dimension, subject to Dirichlet boundary conditions on one of the hypersurfaces and Neumann boundary conditions on the other. We…

We consider the problem of maximizing the first eigenvalue of the $p$-laplacian (possibly with non-constant coefficients) over a fixed domain $\Omega$, with Dirichlet conditions along $\partial\Omega$ and along a supplementary set $\Sigma$,…

In this paper, we investigate the Dirichlet problem of Laplacian on complete Riemannian manifolds. By constructing new trial functions, we obtain a sharp upper bound of the gap of the consecutive eigenvalues in the sense of the order, which…

The present paper pioneers the study of the Dirichlet problem with $L^q$ boundary data for second order operators with complex coefficients in domains with lower dimensional boundaries, e.g., in $\Omega := \mathbb R^n \setminus \mathbb R^d$…

In this paper we study the Dirichlet eigenvalue problem $$ -\Delta_p u-\Delta_{J,p}u =\lambda|u|^{p-2}u \quad \text{ in } \Omega,\quad u=0 \quad\text{ in } \Omega^c=\mathbb{R}^N\setminus\Omega. $$ Here $\Delta_p u$ is the standard local…

Let -\Delta denote the Dirichlet Laplace operator on a bounded open set in \mathbb{R}^d. We study the sum of the negative eigenvalues of the operator -h^2 \Delta - 1 in the semiclassical limit h \to 0+. We give a new proof that yields not…