Related papers: Inequalities for the Steklov Eigenvalues

In this note we establish an expression for the Steklov spectrum of warped products in terms of auxiliary Steklov problems for drift Laplacians with weight induced by the warping factor. As an application, we show that a compact manifold…

Let $(\Omega,g)$ be a compact, analytic Riemannian manifold with analytic boundary $\partial \Omega = M.$ We give $L^2$-lower bounds for Steklov eigenfunctions and their restrictions to interior hypersurfaces $H \subset \Omega^{\circ}$ in a…

We prove that the isoperimetric profile of a convex domain $\Omega$ with compact closure in a Riemannian manifold $(M^{n+1},g)$ satisfies a second order differential inequality which only depends on the dimension of the manifold and on a…

We consider Steklov eigenvalues of three-dimensional, nearly-spherical domains. In previous work, we have shown that the Steklov eigenvalues are analytic functions of the domain perturbation parameter. Here, we compute the first-order term…

Antonio Ros gave a lower bound for the first eigenvalue $\lambda_1$ of $\Delta$ of a $P$-manifold $(M, g)$ in terms of the lower bound on the Ricci curvature $Ric_M$ and asked what happened when this lower bound was achieved. In this paper…

Let $(M,g)$ be a Riemannian manifold with a distinguished point $O$ and assume that the geodesic distance $d$ from $O$ is an isoparametric function. Let $\Omega\subset M$ be a bounded domain, with $O \in \Omega$, and consider the problem…

In this article, we establish a geometric lower bound for the first positive eigenvalue $\lambda^{(1)}_{1}$ of the rough Laplacian acting on $1$-forms for closed $2n$-dimensional Riemannian manifolds with nonvanishing Euler characteristic.…

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…

Let $\Omega\subset\mathbb{R}^{n}$ be a smooth bounded domain and $m\in C(\overline{\Omega})$ be a sign-changing weight function. For $1<p<\infty$, consider the eigenvalue problem $$ \left\{ \begin{array} [c]{ll} -\Delta_{p}u=\lambda…

} In this article, we put forward a Neumann eigenvalue problem for the bi-harmonic operator $\Delta^2$ on a bounded smooth domain $\Om$ in the Euclidean $n$-space ${\bf R}^n$ ($n\ge2$) and then prove that the corresponding first non-zero…

In the present paper, we study the variational properties of Steklov transmission eigenvalues, which can be seen as eigenvalues of the sum of two Dirichlet-to-Neumann operators on two different sides of a given curve contained in a surface.…

We consider the lower bound of nodal sets of Steklov eigenfunctions on smooth Riemannian manifolds with boundary--the eigenfunctions of the Dirichlet-to-Neumann map. Let $N_\lambda$ be its nodal set. Assume that zero is a regular value of…

We consider the Laplacian with attractive Robin boundary conditions, \[ Q^\Omega_\alpha u=-\Delta u, \quad \dfrac{\partial u}{\partial n}=\alpha u \text{ on } \partial\Omega, \] in a class of bounded smooth domains…

Given two compact Riemannian manifolds with boundary $M_1$ and $M_2$ such that their respective boundaries $\Sigma_1$ and $\Sigma_2$ admit neighborhoods $\Omega_1$ and $\Omega_2$ which are isometric, we prove the existence of a constant…

In this paper we obtain several results concerning the optimization of higher Steklov eigenvalues both in two and higher dimensional cases. We first show that the normalized (by boundary length) $k$-th Steklov eigenvalue on the disk is not…

Let $\Omega$ be a bounded domain with $C^2$-smooth boundary in an $n$-dimensional oriented Riemannian manifold. It is well-known that for the bi-harmonic equation $\Delta^2 u=0$ in $\Omega$ with the $0$-Dirichlet boundary condition, there…

We consider the magnetic Steklov eigenvalue problem on compact Riemannian manifolds with boundary for generic magnetic potentials and establish various results concerning the spectrum. We provide equivalent characterizations of magnetic…

We prove the existence of a family of compact subdomains $\Omega$ of the flat cylinder $\mathbb{R}^N\times \mathbb{R}/2\pi\mathbb{Z}$ for which the Neumann eigenvalue problem for the Laplacian on $\Omega$ admits eigenfunctions with constant…

Let $\Omega\subset \mathbb R^2$ be a bounded planar domain, with piecewise smooth boundary $\partial \Omega$. For $\sigma>0$, we consider the Robin boundary value problem \[ -\Delta f =\lambda f, \qquad \frac{\partial f}{\partial n} +…

Let $T$ be a finite tree with leaf set $\dO$ as the boundary and let $\lambda_2$ be the first nontrivial Steklov eigenvalue. Let $D$ and $\ell$ be the maximum vertex degree and the number of leaves, respectively. Motivated by the spectral…