A Payne-Weinberger eigenvalue estimate for wedge domains on spheres

量子点狄拉克算子的Payne-Weinberger不等式

A Payne-Weinberger inequality for quantum dot Dirac operators

针对有界单连通平面区域上的量子点狄拉克算子，通过利用其与$\boldsymbol{\bar\ni}$-Robin拉普拉斯算子的联系，建立了仅依赖区域等周亏格的第一正特征值的精确上界不等式。

An optimal anisotropic Poincaré inequality for convex domains

In this paper, we prove a sharp lower bound of the first (nonzero) eigenvalue of Finsler-Laplacian with the Neumann boundary condition. Equivalently, we prove an optimal anisotropic Poincaré inequality for convex domains, which generalizes the result of Payne-Weinberger. A lower bound of the first (nonzero) eigenvalue of Finsler-Laplacian with the Dirichlet boundary condition is also proved.

Best constants in Poincaré inequalities for convex domains

We prove a Payne-Weinberger type inequality for the $p$-Laplacian Neumann eigenvalues ($p\ge 2$). The inequality provides the sharp upper bound on convex domains, in terms of the diameter alone, of the best constants in Poincaré inequality. The key point is the implementation of a refinement of the classical Pólya-Szegö inequality for the symmetric decreasing rearrangement which yields an optimal weighted Wirtinger inequality.

On Ashbaugh-Benguria's Conjecture about Lower Order Dirichlet Eigenvalues of the Laplacian

In this paper, we prove an isoperimetric inequality for lower order eigenvalues of the Dirichlet Laplacian on bounded domains of a Euclidean space which strengthens the well-known Ashbaugh-Beguria inequality conjectured by Payne-Pólya-Weinberger on the ratio of the first two Dirichlet eigenvalues and makes an important step toward the proof of a conjecture by Ashbaugh-Benguria.

On Conformal Spectral Gap Estimates of the Dirichlet-Laplacian

We study spectral stability estimates of the Dirichlet eigenvalues of the Laplacian in non-convex domains $Ω\subset\mathbb R^2$. With the help of these estimates we obtain asymptotically sharp inequalities of ratios of eigenvalues in the frameworks of the Payne-Pólya-Weinberger inequalities. These estimates are equivalent to spectral gap estimates of the Dirichlet eigenvalues of the Laplacian in non-convex domains in terms of conformal (hyperbolic) geometry.

Eigenvalues of Euclidean wedge domains in higher dimensions

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 generalizes a lower bound of Payne and Weinberger in two dimensions.