Related papers: Metric upper bounds for Steklov and Laplace eigenv…

We prove inequalities for Laplace eigenvalues on Riemannian manifolds generalising to higher eigenvalues two classical inequalities for the first Laplace eigenvalue - the inequality in terms of the $L^2$-norm of mean curvature, due to…

We prove several results about the multiplicity of the first Steklov eigenvalues on compact surfaces with boundary. We improve some bounds on the multiplicity, especially for the first eigenvalue, and we prove they are sharp on some…

On a closed weighted Riemannian manifold with nonnegative Bakry-\'{E}mery Ricci curvature, it is shown that the ratio of the $k$-th to first eigenvalues of the weighted Laplacian is dominated by $641k^2$, using an argument via the Cheeger…

In this paper, we give a sharp lower bound for the first eigenvalue of the basic Laplacian acting on basic $1$-forms defined on a compact manifold whose boundary is endowed with a Riemannian flow. The limiting case gives rise to a…

In the present paper we develop an approach to obtain sharp spectral asymptotics for Steklov type problems on planar domains with corners. Our main focus is on the two-dimensional sloshing problem, which is a mixed Steklov-Neumann boundary…

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…

In this paper, we obtain optimal upper bounds for all the Neumann eigenvalues in two situations (that are closely related). First we consider a one-dimensional Sturm-Liouville eigenvalue problem where the density is a function $h(x)$ whose…

We give an overview of results on shape optimization for low eigenvalues of the Laplacian on bounded planar domains with Neumann and Steklov boundary conditions. These results share a common feature: they are proved using methods of complex…

Let $\Delta$ and $B$ be the maximum vertex degree and a subset of vertices in a graph $G$ respectively. In this paper, we study the first (non-trivial) Steklov eigenvalue $\sigma_2$ of $G$ with boundary $B$. Using metrical deformation via…

We study solutions of uniformly elliptic PDE with Lipschitz leading coefficients and bounded lower order coefficients. We extend previous results of A. Logunov concerning nodal sets of harmonic functions and, in particular, prove polynomial…

We study Riemannian manifolds with boundary under a lower $N$-weighted Ricci curvature bound for $N$ at most $1$, and under a lower weighted mean curvature bound for the boundary. We examine rigidity phenomena in such manifolds with…

In this paper, we study the Steklov eigenvalue of a Riemannian manifold (M, g) with smooth boundary. For compact M , we establish a Cheeger-type inequality for the first Steklov eigenvalue by the isocapacitary constant. For non-compact M ,…

In this paper, we obtain geometric upper bounds for the first eigenvalue $\lambda_1(J)$ of the Jacobi operator for both closed and compact with boundary hypersurfaces having constant mean curvature (CMC). As an application, we derive new…

Under various elliptic boundary conditions, we obtain lower eigenvalue estimates for Dirac operators by using Hormander's weighted $L^2$-technique. Lower bounds in terms of the volume of the underlying manifolds are also deduced from the…

We obtain inequalities for all Laplace eigenvalues of Riemannian manifolds with an upper sectional curvature bound, whose rudiment version for the first Laplace eigenvalue was discovered by Berger in 1979. We show that our inequalities…

We prove sharp $L^p$ estimates for the Steklov eigenfunctions on compact manifolds with boundary in terms of their $L^2$ norms on the boundary. We prove it by establishing $L^p$ bounds for the harmonic extension operators as well as the…

We give lower bounds for the first non-zero Steklov eigenvalue on connected graphs. These bounds depend on the extrinsic diameter of the boundary and not on the diameter of the graph. We obtain a lower bound which is sharp when the cardinal…

In this work, we study compact Riemannian manifolds with boundary satisfying V-static-type equations. By combining a generalized Reilly formula with Steklov-type boundary value problems, we derive integral inequalities for geometric…

In this short note, we show the rigidity of a trace estimate for Steklov eigenvalues with respect to functions in our previous work (Trace and inverse trace of Steklov eigenvalues. J. Differential Equations 261 (2016), no. 3, 2026--2040.).…

The aim of this paper is give a simple proof of some results in \cite{Jun Ling-2006-IJM} and \cite{JunLing-2007-AGAG}, which are very deep studies in the sharp lower bound of the first eigenvalue in the Laplacian operator on compact…