Upper bounds for the Steklov eigenvalues on trees

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 of the total volume of $M$ and the volume of its boundary. We provide examples illustrating the necessity of these geometric quantities in the lower bound. Our result can be seen as a counterpart of the lower bound for the first nonzero Laplace eigenvalue on closed pinched negatively curved manifolds of dimension at least three proved by Schoen in 1982. The proof is composed of certain key elements. We provide a uniform lower bound for the first eigenvalue of the Steklov–Dirichlet problem on a neighborhood of the boundary of $M$ and show that it provides an obstruction to having a small first nonzero Steklov eigenvalue. As another key element of the proof, we give a tubular neighborhood theorem for totally geodesic hypersurfaces in a pinched negatively curved manifold. We give an explicit dependence for the width function in terms of the volume of the boundary and the pinching constant.

Suppose M is homeomorphic to 2-dimensional sphere and let \(f_i\), \(1\le i\le 3\), be the first eigenfunctions of the Laplacian on M. Cheng proved (Proc Am Math Soc 55(2):379–381, 1976) that if \(\sum _{i=1}^3 f_i^2\) is a constant, then M is isometric to a sphere of constant curvature. In view of the similarities between eigenvalues of the Laplacian and Steklov eigenvalue, we study eigenfuction of the first nonzero Steklov eigenvalue of a 2-dimensional compact manifold with boundary \(\partial M\). Suppose M is a domain equipped with the flat metric g, and let f be an eigenfunction of the first nonzero Steklov’s eigenvalue. We prove that if \(\nabla _g f\) is parallel along \(\partial M\), the Lie bracket of the vector field orthogonal to \(\nabla _g f\) and the tangent to \(\partial M\) is zero, and \(|\nabla _g f|^2\) is constant in M, then (M, g) is isometric to the disk equipped with the flat metric. We also prove that if the eigenfunctions \(f_1, f_2\) corresponding to the first nonzero Steklov eigenvalue satisfy that \(f_1^2+f_2^2\) is constant on \(\partial M\) and \(\varphi _*(g)\) is a Riemannian metric conformal to the flat metric of \(D^2\) where \(\varphi =(f_1,f_2)\), then (M, g) is isometric to the disk equipped with the flat metric. In another direction, we prove that a simply connected domain in the hyperbolic space \(\mathbb {H}^3\) such that its Steklov eigenvalues are the same as a geodesic ball must be isometric to the geodesic ball.

We consider a properly embedded minimal hypersurfacewith free boundary in a compact n-dimensional Riemannian manifold M be with nonnegative Ricci curvature and strictly convex boundary. Here, we obtain a new estimate from below for the first nonzero Steklov eigenvalue.

Shape optimization for the Steklov problem in higher dimensions

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 analysis. In particular, we present modernized proofs of the classical inequalities due to Szegö and Weinstock for the first nonzero Neumann and Steklov eigenvalues. We also extend the inequality for the second nonzero Neumann eigenvalue, obtained recently by Nadirashvili and the authors, to nonhomogeneous membranes with log‐subharmonic densities. In the homogeneous case, we show that this inequality is strict, which implies that the maximum of the second nonzero Neumann eigenvalue is not attained in the class of simply connected membranes of a given mass. The same is true for the second nonzero Steklov eigenvalue, as follows from our results on the Hersch–Payne–Schiffer inequalities. Copyright © 2009 John Wiley & Sons, Ltd.

We construct surfaces with arbitrarily large multiplicity for their first nonzero Steklov eigenvalue. The proof is based on a technique by Burger and Colbois originally used to prove a similar result for the Laplacian spectrum. We start by constructing surfaces $S_p$ with a specific subgroup of isometry $G_p:= \mathbb {Z}_p \rtimes \mathbb {Z}_p^*$ for each prime p . We do so by gluing surfaces with boundary following the structure of the Cayley graph of $G_p$ . We then exploit the properties of $G_p$ and $S_p$ in order to show that an irreducible representation of high degree (depending on p ) acts on the eigenspace of functions associated with $\sigma _1(S_p)$ , leading to the desired result.

R\\'esum\\'eIn this article, we give computable lower bounds for the first non-zero Steklov eigenvalue σ1 of a compact connected 2-dimensional Riemannian manifold M with several cylindrical boundary components. These estimates show how the geometry of M away from the boundary affects this eigenvalue. They involve geometric quantities specific to manifolds with boundary such as the extrinsic diameter of the boundary. In a second part, we give lower and upper estimates for the low Steklov eigenvalues of a hyperbolic surface with a geodesic boundary in terms of the length of some families of geodesics. This result is similar to a well known result of Schoen, Wolpert and Yau for Laplace eigenvalues on a closed hyperbolic surface.

We prove a lower bound for the k-th Steklov eigenvalues in terms of an isoperimetric constant called the k-th Cheeger-Steklov constant in three different situations: finite spaces, measurable spaces, and Riemannian manifolds. These lower bounds can be considered as higher order Cheeger type inequalities for the Steklov eigenvalues. In particular it extends the Cheeger type inequality for the first nonzero Steklov eigenvalue previously studied by Escobar in 1997 and by Jammes in 2015 to higher order Steklov eigenvalues. The technique we develop to get this lower bound is based on considering a family of accelerated Markov operators in the finite and measurable situations and of mass concentration deformations of the Laplace-Beltrami operator in the manifold setting which converges uniformly to the Steklov operator. As an intermediary step in the proof of the higher order Cheeger type inequality, we define the Dirichlet-Steklov connectivity spectrum and show that the Dirichlet connectivity spectra of this family of operators converges to (or is bounded by) the Dirichlet-Steklov spectrum uniformly. Moreover, we obtain bounds for the Steklov eigenvalues in terms of its Dirichlet-Steklov connectivity spectrum which is interesting in its own right and is more robust than the higher order Cheeger type inequalities. The Dirichlet-Steklov spectrum is closely related to the Cheeger-Steklov constants. Resume Pour tout k P N, une borne inferieure pour la k-ieme valeur propre de Steklov en termes d'une constante isoperimetrique, appelee la k-ieme constante de Cheeger-Steklov, est obtenue dans trois situations differentes : espaces finis, espaces mesurables et varietes riemanniennes. Ces bornes inferieures peuvent etre considerees comme des inegalites de type Cheeger d'ordre superieur pour les valeurs propres de Steklov. En particulier, elles etendent l'inegalite de type Cheeger pour la premi ere valeur propre non nulle de Steklov etudiee par Escobar en 1997 et par Jammes en 2015. La technique developpee pour obtenir ces bornes inferieure utilise une famille d'operateurs de Markov acceleres dans les situations finies et mesurables et une famille d'operateurs de Laplace-Beltrami deformes et concentres pres de lafronti ere. Lors d'une etape intermediaire de la preuve de l'inegalite de type Cheeger d'ordre superieur, nous definissons le spectre de connectivite de Dirichlet-Steklov et nous montrons que les spectres de connectivite de Dirichlet de cette famille d'operateurs convergent uniformement vers (ou sont bornes par) le spectre de Dirichlet-Steklov. De plus, nous obtenons des bornes pour les valeurs propres de Steklov en termes du spectre de connectivite de Dirichlet-Steklov, ce dernier etant interessant en lui-meme. Il est aussi plus robuste que les inegalites de type Cheeger d'ordre superieur. Le spectre de Dirichlet-Steklov et les constantes de Cheeger-Steklov sont etroitement lies.

On a two-dimensional compact Riemannian manifold with boundary, we prove that the first nonzero Steklov eigenvalue is nondecreasing along the unnormalized geodesic curvature flow if the initial metric has positive geodesic curvature and vanishing Gaussian curvature. Using the normalized geodesic curvature flow, we also obtain some estimate for the first nonzero Steklov eigenvalue. On the other hand, we prove that the compact soliton of the geodesic curvature flow must be the trivial one.

Using methods in the spirit of deterministic homogenisation theory we obtain convergence of the Steklov eigenvalues of a sequence of domains in a Riemannian manifold to weighted Laplace eigenvalues of that manifold. The domains are obtained by removing small geodesic balls that are asymptotically densely uniformly distributed as their radius tends to zero. We use this relationship to construct manifolds that have large Steklov eigenvalues. In dimension two, and with constant weight equal to 1, we prove that Kokarev’s upper bound of 8pi for the first nonzero normalised Steklov eigenvalue on orientable surfaces of genus 0 is saturated. For other topological types and eigenvalue indices, we also obtain lower bounds on the best upper bound for the eigenvalue in terms of Laplace maximisers. For the first two eigenvalues, these lower bounds become equalities. A surprising consequence is the existence of free boundary minimal surfaces immersed in the unit ball by first Steklov eigenfunctions and with area strictly larger than 2pi . This was previously thought to be impossible. We provide numerical evidence that some of the already known examples of free boundary minimal surfaces have these properties and also exhibit simulations of new free boundary minimal surfaces of genus zero in the unit ball with even larger area. We prove that the first nonzero Steklov eigenvalue of all these examples is equal to 1, as a consequence of their symmetries and topology, so that they are consistent with a general conjecture by Fraser and Li. In dimension three and larger, we prove that the isoperimetric inequality of Colbois–El Soufi–Girouard is sharp and implies an upper bound for weighted Laplace eigenvalues. We also show that in any manifold with a fixed metric, one can construct by varying the weight a domain with connected boundary whose first nonzero normalised Steklov eigenvalue is arbitrarily large.

We study upper bounds for the first non-zero eigenvalue of the Steklov problem defined on finite graphs with boundary. For finite graphs with boundary included in a Cayley graph associated to a group of polynomial growth, we give an upper bound for the first non-zero Steklov eigenvalue depending on the number of vertices of the graph and of its boundary. As a corollary, if the graph with boundary also satisfies a discrete isoperimetric inequality, we show that the first non-zero Steklov eigenvalue tends to zero as the number of vertices of the graph tends to infinity. This extends recent results of Han and Hua, who obtained a similar result in the case of mathbb {Z}^n. We obtain the result using metric properties of Cayley graphs associated to groups of polynomial growth.

We prove that among all doubly connected domains of ℝnof the formB1\B̅2, whereB1andB2are open balls of fixed radii such thatB̅2⊂B1, the first nonzero Steklov eigenvalue achieves its maximal value uniquely when the balls are concentric. Furthermore, we show that the ideas of our proof also apply to a mixed boundary conditions eigenvalue problem found in literature.

In this paper, two interesting eigenvalue comparison theorems for the first non-zero Steklov eigenvalue of the Laplacian have been established for manifolds with radial sectional curvature bounded from above. Besides, sharper bounds for the first non-zero eigenvalue of the Wentzell eigenvalue problem of the weighted Laplacian, which can be seen as a natural generalization of the classical Steklov eigenvalue problem, have been obtained.

We give some sharp lower bounds of the first eigenvalue for the Hodge Laplacian acting on differential forms on the boundary of a Riemannian manifold. We also give some sharp estimates for the first nonzero Steklov eigenvalue for differential forms.

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 of the boundary is 2, and asymptotically sharp as the diameter of the boundary tends to infinity in the other cases. We also investigate the case of weighted graphs and compare our result to the Cheeger inequality.