Related papers: On the Yang-Yau inequality for the first Laplace e…

The Berezin--Li--Yau and the Kr\"oger inequalities show that Riesz means of order $\geq 1$ of the eigenvalues of the Laplacian on a domain $\Omega$ of finite measure are bounded in terms of their semiclassical limit expressions. We show…

Let $\Omega \subset \mathbb{R}^d$ be a bounded domain and let $\lambda_1, \lambda_2, \dots$ denote the sequence of eigenvalues of the Laplacian subject to Dirichlet boundary conditions. We consider inequalities for $\lambda_n$ that are…

Sharp upper bounds for the first eigenvalue of the Laplacian on a surface of a fixed area are known only in genera zero and one. We investigate the genus two case and conjecture that the first eigenvalue is maximized on a singular surface…

We prove explicit and sharp eigenvalue estimates for Neumann $p$-Laplace eigenvalues in domains that admit a representation in Fermi coordinates. More precisely, if $\gamma$ denotes a non-closed curve in $\mathbb{R}^2$ symmetric with…

This is a review of old and new results and methods related to the Yau conjecture on the zero set of Laplace eigenfunctions. The review accompanies two lectures given at the conference CDM 2018. We discuss the works of Donnelly and…

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 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…

This paper gives a survey of recent progress in isoparametric functions and isoparametric hypersurfaces, mainly in two directions. (1) Isoparametric functions on Riemannian manifolds, including exotic spheres. The existences and…

Let $G=(V,E)$ be a locally finite graph, $\Omega\subset V$ be a bounded domain, $\Delta$ be the usual graph Laplacian, and $\lambda_1(\Omega)$ be the first eigenvalue of $-\Delta$ with respect to Dirichlet boundary condition. Using the…

We give bounds on the first non-zero eigenvalue of the scalar Laplacian for both the Page and the Chen-LeBrun-Weber Einstein metrics. One notable feature is that these bounds are obtained without explicit knowledge of the metrics or…

We study the eigenvalues of the Dirichlet Laplace operator on an arbitrary bounded, open set in $\R^d$, $d \geq 2$. In particular, we derive upper bounds on Riesz means of order $\sigma \geq 3/2$, that improve the sharp Berezin inequality…

We establish lower bounds for the sum of the reciprocals of eigenvalues of the Laplacian. For bounded domains, our result extends the upper bound provided by Bucur and Henrot on the second Neumann eigenvalue and is related to a result by…

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…

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…

Motivated by the problem of the small-scale sign distribution of Laplace eigenfunctions, we introduce a strong notion of sign-balance for (eigen)functions, and prove that random eigenfunctions are sign-balanced above a precisely determined…

The first nontrivial eigenvalue of the Laplacian can be considered as a functional on the space of all Riemannian metrics of unit volume on a fixed surface. In this paper we prove that for the surface of genus 2 the supremum of this…

An integral inequality is derived for compact submanifolds (with or without boundary) in the unit sphere. This result leads to a characterization of spheres.

This is a survey on the recent progress in several applications of isoparametric theory, including an affirmative answer to Yau's conjecture on the first eigenvalue of Laplacian in the isoparametric case, a negative answer to Yau's 76th…

It is proved that for class $A_\gamma=\{q\in L_1[0,1]: q\geq 0, \int_0^1 q^\gamma\,dx=1\}$, where $\gamma\in (0,1)$, there exists a potential $q_*\in A_\gamma$ such that minimal eigenvalue $\lambda_1(q_*)$ of boundary problem $$…

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…