On the second Robin eigenvalue of the Laplacian

Eigenvalue bounds of the Robin Laplacian with magnetic field

On a compact Riemannian manifold $M$ with boundary, we give an estimate for the eigenvalues $(λ\_k(τ,α))\_k$ of the magnetic Laplacian with the Robin boundary conditions. Here, $τ$ is a positive number that defines the Robin condition and $α$ is a real differential 1-form on $M$ that represents the magnetic field. We express these estimates in terms of the mean curvature of the boundary, the parameter $τ$ and a lower bound of the Ricci curvature of $M$ (see Theorem \ref{estimate1} and Corollary \ref{corestimate}). The main technique is to use the Bochner formula established in \cite{ELMP} for …

Mean curvature bounds and eigenvalues of Robin Laplacians

We consider the Laplacian with attractive Robin boundary conditions, \[ Q^Ω_αu=-Δu, \quad \dfrac{\partial u}{\partial n}=αu \text{ on } \partialΩ, \] in a class of bounded smooth domains $Ω\in\mathbb{R}^ν$; here $n$ is the outward unit normal and $α>0$ is a constant. We show that for each $j\in\mathbb{N}$ and $α\to+\infty$, the $j$th eigenvalue $E_j(Q^Ω_α)$ has the asymptotics \[ E_j(Q^Ω_α)=-α^2 -(ν-1)H_\mathrm{max}(Ω)\,α+{\mathcal O}(α^{2/3}), \] where $H_\mathrm{max}(Ω)$ is the maximum mean curvature at $\partial Ω$. The discussion of the reverse Faber-Krahn inequality gives rise to a new ge…

具有诺伊曼或罗宾边界条件的各向异性N-拉普拉斯方程弱解的刚性

Rigidity of weak solutions for anisotropic N-Laplacian equation with Neumann or Robin boundary condition

研究\(\mathbb{R}^N\)中光滑有界凸域上各向异性\(N -\)拉普拉斯方程弱解刚性，建立关键积分不等式，在较弱假设下证明诺伊曼边界问题弱解为常数，扩展到罗宾边值问题，结果在有界和无界域均成立，填补文献空白并扩展刚性理论。

A negative mass theorem for the 2-Torus

For a closed surface M with metric g, the Robin mass m(p) at the point p is the value of the Green function G(p,q) at p=q after the logarithmic singularity has been removed. The Laplacian-mass is the average value of the Robin mass, minus the value of the Robin mass for the round sphere of the same area. The Laplacian-mass is a spectral invariant which is a natural analog of the ADM mass for asymptotically flat manifolds. We show that if M is a torus, then the minimum value of the Laplacian-mass on the conformal class of g is negative. It is attained by a (smooth) metric for which one gets a s…

Comparison of Steklov eigenvalues and Laplacian eigenvalues on graphs

In this paper, we obtain a comparison of Steklov eigenvalues and Laplacian eigenvalues on graphs and discuss its rigidity. As applications of the comparison of eigenvalues, we obtain Lichnerowicz-type estimates and some combinatorial estimates for Steklov eigenvalues on graphs.

黎曼曲面上旋量丛的伪拉普拉斯算子行列式与$\zeta^{(\rm reg)}(1)$

Determinants of pseudo-laplacians and $ζ^{({\rm reg})}(1)$ for spinor bundles over Riemann surfaces

本文研究紧黎曼曲面上埃尔米特线丛中Dolbeault拉普拉斯算子的自伴扩张，引入Robin质量概念并推导其显式表达式，以此描述标量拉普拉斯正则化$\zeta(1)$的Ricci流演化，还给出Morpurgo零亏格曲面相关结论的新证法。

Schrödinger算子在Riemann面上的第二Robin本征值界

Second Robin eigenvalue bounds for Schrödinger operators on Riemannian surfaces

本文研究了Schrödinger算子在Riemann面上的第二Robin本征值界，并推导了相关拓扑限制和本征值上界。

Bounds for eigenvalue ratios of the Laplacian

For a bounded domain $Ω$ with a piecewise smooth boundary in an $n$-dimensional Euclidean space $\mathbf{R}^{n}$, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. First we give a general inequality for eigenvalues of the Laplacian. As an application, we study lower order eigenvalues of the Laplacian and derive the ratios of lower order eigenvalues of the Laplacian.

Estimates for lower bounds of eigenvalues of the poly-Laplacian and quadratic polynomial operator of the Laplacian

In this paper, we investigate the Dirchlet eigenvalue problems of poly-Laplacian with any order and quadratic polynomial operator of the Laplacian. We give some estimates for lower bounds of the sums of their first $k$ eigenvalues which improve the previous results.

Inequalities for eigenvalues of the weighted Hodge Laplacian

In this paper, we obtain "universal" inequalities for eigenvalues of the weighted Hodge Laplacian on a compact self-shrinker of Euclidean space. These inequalities generalize the Yang-type and Levitin-Parnovski inequalities for eigenvalues of the Laplacian and Laplacian. From the recursion formula of Cheng and Yang \cite{ChengYang07}, the Yang-type inequality for eigenvalues of the weighted Hodge Laplacian are optimal in the sense of the order of eigenvalues.