Related papers: Sharp upper bound and a comparison theorem for the…

We obtain upper and lower bounds for Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space. A very general upper bound is proved, which depends only on the geometry of the fixed boundary and on the measure of the…

In this article, we prove an isoperimetric inequality for the harmonic mean of the first $(n-1)$ nonzero Steklov eigenvalues on bounded domains in $n$-dimensional Hyperbolic space. Our approach to prove this result also gives a similar…

In this paper, we address the problem of maximizing the Steklov eigenvalues with a diameter constraint. We provide an estimate of the Steklov eigenvalues for a convex domain in terms of its diameter and volume and we show the existence of…

In this paper, we study the upper bounds for discrete Steklov eigenvalues on trees via geometric quantities. For a finite tree, we prove sharp upper bounds for the first nonzero Steklov eigenvalue by the reciprocal of the size of the…

It was conjectured by Escobar [J. Funct. Anal. 165 (1999), 101--116] that for an $n$-dimensional ($n\geq 3$) smooth compact Riemannian manifold with boundary, which has nonnegative Ricci curvature and boundary principal curvatures bounded…

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…

Given a compact surface with boundary, we introduce a family of functionals on the space of its Riemannian metrics, defined via eigenvalues of a Steklov-type problem. We prove that each such functional is uniformly bounded from above, and…

In recent years, eigenvalue optimization problems have received a lot of attention, in particular, due to their connection with the theory of minimal surfaces. In the present paper we prove that on any orientable surface there exists a…

Let $\Omega$ be a star-shaped bounded domain in $(\mathbb{S}^{n}, ds^{2})$ with smooth boundary. In this article, we give a sharp lower bound for the first non-zero eigenvalue of the Steklov eigenvalue problem in $\Omega.$ This result is…

We study the problem of maximizing the first nontrivial Steklov eigenvalue of the Laplace-Beltrami Operator among subdomains of fixed volume of a Riemannian manifold. More precisely, we study the expansion of the corresponding profile of…

We show that metrics that maximize the k-th Steklov eigenvalue on surfaces with boundary arise from free boundary minimal surfaces in the unit ball. We prove several properties of the volumes of these minimal submanifolds. For free boundary…

We investigate the question of sharp upper bounds for the Steklov eigenvalues of a hypersurface of revolution of the Euclidean space with two boundary components isometric to two copies of $\mathbb{S}^{n-1}$. For the case of the first non…

We prove some sharp isoperimetric type inequalities for domains with smooth boundary on Riemannian manifolds. For example, using generalized convexity, we show that among all domains with a lower bound $l$ for the cut distance and Ricci…

Given a closed Riemannian manifold $M$ and $b\geq2$ closed connected submanifolds $N_j\subset M$ of codimension at least $2$, we prove that the first non-zero eigenvalue of the domain $\Omega_\varepsilon\subset M$ obtained by removing the…

We consider mixed Steklov-Dirichlet eigenvalue problem on smooth bounded domains in Riemannian manifolds. Under certain symmetry assumptions on multiconnected domains in $\mathbb{R}^{n}$ with a spherical hole, we obtain isoperimetric…

We prove a sharp upper bound and a lower bound for the first nonzero eigenvalue of the Wentzell-Laplace operator on compact manifolds with boundary and an isoperimetric inequality for the same eigenvalue in the case where the manifold is a…

In this note, an isoperimetric inequality for the harmonic mean of lower order Steklov eigenvalues is proved on bounded domains in noncompact rank-$1$ symmetric spaces. This work extends result of \cite{BR.01} and \cite{V.21} proved for…

Given a Riemmanian manifold, we provide a new method to compute a sharp upper bound for the first eigenvalue of the Laplacian for the Dirichlet problem on a geodesic ball of radius less than the injectivity radius of the manifold. This…

Let $U\subset \mathbb{R}^n$ ($n\geq 3$) be an exterior Euclidean domain with smooth boundary $\partial U$. We consider the Steklov eigenvalue problem on $U$. First we derive a sharp lower bound for the first eigenvalue in terms of the…

In this paper we find an upper bound for the first Steklov eigenvalue for a surface of revolution with boundary consisting of two spheres of different radii. Moreover, we prove that in some cases this boundary is sharp.