Related papers: Lower bounds for Steklov eigenfunctions

We give explicit isoperimetric upper bounds for all Steklov eigenvalues of a compact orientable surface with boundary, in terms of the genus, the length of the boundary, and the number of boundary components. Our estimates generalize a…

In this paper, we study the higher Steklov eigenvalues of graphs on surfaces. We obtain the upper bound of higher Steklov eigenvalues of a finite graph $G$ with boundary $B$ and genus $g$ by using metrical deformation via probability flows.…

We study the counting function of Steklov eigenvalues on compact manifolds with boundary and obtain its upper bound involving the leading term of Weyl's law. Our estimate can be viewed as a weakened version of P\'{o}lya's Conjecture in the…

The Steklov eigenvalue problem, first introduced over 125 years ago, has seen a surge of interest in the past few decades. This article is a tour of some of the recent developments linking the Steklov eigenvalues and eigenfunctions of…

We study the problem of estimating the $L^2$ norm of Laplace eigenfunctions on a compact Riemannian manifold $M$ when restricted to a hypersurface $H$. We prove mass estimates for the restrictions of eigenfunctions $\phi_h$, $(h^2 \Delta -…

We consider the critical points of Steklov eigenfunctions on a compact, smooth $n$-dimensional Riemannian manifold $M$ with boundary $\partial M$. For generic metrics on $M$ we establish an identity which relates the sum of the indexes of a…

Let $M$ be an $m$-dimensional compact Riemannian manifold with boundary. We obtain the upper bound of the harmonic mean of the first $m$ nonzero Neumann eigenvalues and Steklov eigenvalues involving the conformal volume and relative…

We use layer potential to establish that the boundary biharmonic Steklov operators are elliptic pseudo-differential operators. Thus we are able to establish lower bounds on both the measure of boundary nodal sets and interior nodal sets for…

Let M be a compact Riemannian manifold with boundary. Let b>0 be the number of connected components of its boundary. For manifolds of dimension at least 3, we prove that it is possible to obtain an arbitrarily large (b+1)-th Steklov…

We prove a couple of new endpoint geodesic restriction estimates for eigenfunctions. In the case of general 3-dimensional compact manifolds, after a $TT^*$ argument, simply by using the $L^2$-boundedness of the Hilbert transform on $\R$, we…

Let $(\Omega, g)$ be a real analytic Riemannian manifold with real analytic boundary $\partial \Omega$. Let $\psi_{\lambda}$ be an eigenfunction of the Dirichlet-to-Neumann operator $\Lambda$ of $(\Omega, g, \partial \Omega)$ of eigenvalue…

Let $(M,g)$ be a smooth, compact Riemannian manifold and $\{\phi_h\}$ an $L^2$-normalized sequence of Laplace eigenfunctions, $-h^2\Delta_g\phi_h=\phi_h$. Given a smooth submanifold $H \subset M$ of codimension $k\geq 1$, we find conditions…

Let $(\Omega^{n+1}, g)$ be an $(n+1)$-dimensional smooth compact connected Riemannian manifold with smooth boundary $\Sigma$, satisfying that ${\text{Ric}_{\Omega}}\ge 0$ and $\Sigma$ is strictly convex, more precisely, its second…

The problem of obtaining the lower bounds on the restriction of Laplacian eigenfunctions to hypersurfaces inside a compact Riemannian manifold $(M,g)$ is challenging and has been attempted by many authors \cite{BR, GRS, Jun, ET}. This paper…

In this note, we investigate the Steklov spectrum of the warped product $[0,L]\times_h \Sigma$ equipped with the metric $dt^2+h(t)^2g_\Sigma$, where $\Sigma$ is a compact surface. We find sharp upper bounds for the Steklov eigenvalues in…

We study nodal sets of Steklov eigenfunctions in a bounded domain with $\mathcal{C}^2$ boundary. Our first result is a lower bound for the Hausdorff measure of the nodal set: we show that for $u_{\lambda}$ a Steklov eigenfunction, with…

We establish a new lower bound for the first non-zero Steklov eigenvalue of a compact Riemannian manifold with non-negative Ricci curvature and (strictly) convex boundary. Related results are also obtained under weaker geometric hypotheses.

This paper studies eigenvalues of some Steklov problems. Among other things, we show the following sharp estimtes. Let $\Omega$ be a bounded smooth domain in an $n(\geq 2)$-dimensional Hadamard manifold an let $0=\lambda_0 < \lambda_1\leq…

Let $M^n=[0,R)\times \mathbb{S}^{n-1}$ be an $n$-dimensional ($n\geq 2$) smooth Riemannian manifold equipped with the warped product metric $g=dr^2+h^2(r)g_{\mathbb{S}^{n-1}}$ and diffeomorphic to a Euclidean ball. Assume that $M$ has…

We consider the Steklov eigenvalue problem on a compact pinched negatively curved manifold $M$ of dimension at least three with totally geodesic boundaries. We obtain a geometric lower bound for the first nonzero Steklov eigenvalue in terms…