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Asma Hassannezhad

Publications and source records attributed to Asma Hassannezhad.

18 recordsLinked to original sources
arXiv2026

Lower bounds for the low Steklov eigenvalues

For a compact, connected, orientable Riemannian manifold with $b$ boundary components, we obtain geometric lower bounds for the low Steklov eigenvalues, namely $σ_k$, $1\le k\le b-1$. Our results complement earlier results, which apply only to $σ_k$ with $k\ge b$ and depend on the geometry near the boundary, by showing how the interior geometry influences the low eigenvalues. Our result also yields lower bounds for the low Steklov eigenvalues in the setting of pinched negatively curved manifolds, thus recovering similar results in that context through an alternative proof. The proof of the main result is based on the trace inequality relating the Steklov eigenvalue to the Neumann eigenvalues of the connected subdomains of the manifold containing a boundary collar. The geometric coefficient appearing in this inequality is given by an explicit formula in terms of a quantity that can be interpreted as the electrical resistance of the boundary collar.

math.DG↗
arXiv2025

Nodal counts for the Robin problem on Lipschitz domains

We consider the Courant-sharp eigenvalues of the Robin Laplacian for bounded, connected, open sets in $\mathbb{R}^n$, $n \geq 2$, with Lipschitz boundary. We prove Pleijel's theorem which implies that there are only finitely many Courant-sharp eigenvalues in this setting as well as an improved version of Pleijel's theorem, extending previously known results that required more regularity of the boundary. In addition, we obtain an upper bound for the number of Courant-sharp Robin eigenvalues of a bounded, connected, convex, open set in $\mathbb{R}^n$ with $C^2$ boundary that is explicit in terms of the geometric quantities of the set and the norm sup of the negative part of the Robin parameter.

math.SP↗
arXiv2025

Geometric bounds for low Steklov eigenvalues of finite volume hyperbolic surfaces

We obtain geometric lower bounds for the low Steklov eigenvalues of finite-volume hyperbolic surfaces with geodesic boundary. The bounds we obtain depend on the length of a shortest multi-geodesic disconnecting the surfaces into connected components each containing a boundary component and the rate of dependency on it is sharp. Our result also identifies situations when the bound is independent of the length of this multi-geodesic. The bounds also hold when the Gaussian curvature is bounded between two negative constants and can be viewed as a counterpart of the well-known Schoen-Wolpert-Yau inequality for Laplace eigenvalues. The proof is based on analysing the behaviour of the {corresponding Steklov} eigenfunction on an adapted version of thick-thin decomposition for hyperbolic surfaces with geodesic boundary. Our results extend and improve the previously known result in the compact case obtained by a different method.

math.DG↗
arXiv2024

Tubes and Steklov eigenvalues in negatively curved manifolds

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.

math.DG↗
arXiv2023

On Pleijel's nodal domain theorem for the Robin problem

We prove an improved Pleijel nodal domain theorem for the Robin eigenvalue problem. In particular we remove the restriction, imposed in previous work, that the Robin parameter be non-negative. We also improve the upper bound in the statement of the Pleijel theorem. In the particular example of a Euclidean ball, we calculate the explicit value of the Pleijel constant for a generic constant Robin parameter and we show that it is equal to the Pleijel constant for the Dirichlet Laplacian on a Euclidean ball.

math.AP↗
arXiv2023

Applications of possibly hidden symmetry to Steklov and mixed Steklov problems on surfaces

We consider three different questions related to the Steklov and mixed Steklov problems on surfaces. These questions are connected by the techniques that we use to study them, which exploit symmetry in various ways even though the surfaces we study do not necessarily have inherent symmetry. In the spirit of the celebrated Hersch-Payne-Schiffer and Weinstock inequalities for Steklov eigenvalues, we obtain a sharp isoperimetric inequality for the mixed Steklov eigenvalues considering the interplay between the eigenvalues of the mixed Steklov-Neumann and Steklov-Dirichlet eigenvalues. In 1980, Bandle showed that the unit disk maximizes the $k$th nonzero normalized Steklov eigenvalue on simply connected domains with rotational symmetry of order $p$ when $k\le p-1$. We discuss whether the disk remains the maximizer in the class of simply connected rotationally symmetric domains when $k\geq p$. In particular, we show that for $k$ large enough, the upper bound converges to the Hersch-Payne-Schiffer upper bound. We give full asymptotics for mixed Steklov problems on arbitrary surfaces, assuming some conditions at the meeting points of the Steklov boundary with the Dirichlet or Neumann boundary.

math.SP↗
arXiv2022

Nodal count for Dirichlet-to-Neumann operators with potential

We consider Dirichlet-to-Neumann operators associated to $Δ+q$ on a Lipschitz domain in a smooth manifold, where $q$ is an $L^{\infty}$ potential. We prove a Courant-type bound for the nodal count of the extensions $u_k$ of the $k$th Dirichlet-to-Neumann eigenfunctions $ϕ_k$ to the interior satisfying $(Δ+q)u_k=0$. The classical Courant nodal domain theorem is known to hold for Steklov eigenfunctions, which are the harmonic extension of the Dirichlet-to-Neumann eigenfunctions associated to $Δ$. Our result extends it to a larger family of Dirichlet-to-Neumann operators. Our proof makes use of the duality between the Steklov and Robin problems.

math.AP↗
arXiv2021

Escobar constants of planar domains

We initiate the study of the higher-order Escobar constants $I_k(M)$, $k\geq 3$, on bounded planar domains $M$. The Escobar constants $I_k$ of the unit disk and a family of polygons are provided.

math.DG↗
arXiv2019

The Steklov and Laplacian spectra of Riemannian manifolds with boundary

Given two compact Riemannian manifolds with boundary $M_1$ and $M_2$ such that their respective boundaries $Σ_1$ and $Σ_2$ admit neighborhoods $Ω_1$ and $Ω_2$ which are isometric, we prove the existence of a constant $C$, which depends only on the geometry of $Ω_1\congΩ_2$, such that $|σ_k(M_1)-σ_k(M_2)|\leq C$ for each $k\in\mathbb{N}$. This follows from a quantitative relationship between the Steklov eigenvalues $σ_k$ of a compact Riemannian manifold $M$ and the eigenvalues $λ_k$ of the Laplacian on its boundary. Our main result states that the difference $|σ_k-\sqrt{λ_k}|$ is bounded above by a constant which depends on the geometry of $M$ only in a neighborhood of its boundary. The proofs are based on a Pohozaev identity and on comparison geometry for principal curvatures of parallel hypersurfaces. In several situations, the constant $C$ is given explicitly in terms of bounds on the geometry of $Ω_1\congΩ_2$.

math.SP↗
arXiv2018

Eigenvalue bounds of mixed Steklov problems

We study bounds on the Riesz means of the mixed Steklov-Neumann and Steklov-Dirichlet eigenvalue problem on a bounded domain $Ω$ in $\mathbb{R}^n$. The Steklov-Neumann eigenvalue problem is also called the sloshing problem. We obtain two-term asymptotically sharp lower bounds on the Riesz means of the sloshing problem and also provide an asymptotically sharp upper bound for the Riesz means of mixed Steklov-Dirichlet problem. The proof of our results for the sloshing problem uses the average variational principle and monotonicity of sloshing eigenvalues. In the case of Steklov-Dirichlet eigenvalue problem, the proof is based on a well-known bound on the Riesz means of the Dirichlet fractional Laplacian and an inequality between the Dirichlet and Navier fractional Laplacian. The two-term asymptotic results for the Riesz means of mixed Steklov eigenvalue problems are discussed in the appendix which in particular show the asymptotic sharpness of the bounds we obtain.

math.SP↗
arXiv2017

Higher order Cheeger inequalities for Steklov eigenvalues

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 mesurable 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 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.

math.SP↗
arXiv2017

A note on Kuttler-Sigillito's inequalities

We provide several inequalities between eigenvalues of some classical eigenvalue problems on domains with $C^2$ boundary in complete Riemannian manifolds. A key tool in the proof is the generalized Rellich identity on a Riemannian manifold. Our results in particular extend some inequalities due to Kutller and Sigillito from subsets of $\mathbb{R}^2$ to the manifold setting.

math.SP↗
arXiv2017

Spectral geometry of the Steklov problem on orbifolds

We consider how the geometry and topology of a compact $n$-dimensional Riemannian orbifold with boundary relates to its Steklov spectrum. In two dimensions, motivated by work of A. Girouard, L. Parnovski, I. Polterovich and D. Sher in the manifold setting, we compute the precise asymptotics of the Steklov spectrum in terms of only boundary data. As a consequence, we prove that the Steklov spectrum detects the presence and number of orbifold singularities on the boundary of an orbisurface and it detects the number each of smooth and singular boundary components. Moreover, we find that the Steklov spectrum also determines the lengths of the boundary components modulo an equivalence relation, and we show by examples that this result is the best possible. We construct various examples of Steklov isospectral Riemannian orbifolds which demonstrate that these two-dimensional results do not extend to higher dimensions. In addition, we give two-dimensional examples which show that the Steklov spectrum does \emph{not} detect the presence of interior singularities nor does it determine the orbifold Euler characteristic. In fact, a flat disk is Steklov isospectral to a cone. In another direction, we obtain upper bounds on the Steklov eigenvalues of a Riemannian orbifold in terms of the isoperimetric ratio and a conformal invariant. We generalize results of B. Colbois, A. El Soufi and A. Girouard, and the fourth author to the orbifold setting; in the process, we gain a sharpness result on these bounds that was not evident in the manifold setting. In dimension two, our eigenvalue bounds are solely in terms of the orbifold Euler characteristic and the number each of smooth and singular boundary components.

math.SP↗
arXiv2016

Zero and negative eigenvalues of the conformal Laplacian

We show that zero is not an eigenvalue of the conformal Laplacian for generic Riemannian metrics. We also discuss non-compactness for sequences of metrics with growing number of negative eigenvalues of the conformal Laplacian.

math.DG↗
arXiv2015

Sub-Laplacian eigenvalue bounds on sub-Riemannian manifolds

We study eigenvalue problems for intrinsic sub-Laplacians on regular sub-Riemannian manifolds. We prove upper bounds for sub-Laplacian eigenvalues $λ_k$ of conformal sub-Riemannian metrics that are asymptotically sharp as $k\to +\infty$. For Sasakian manifolds with a lower Ricci curvature bound, and more generally, for contact metric manifolds conformal to such Sasakian manifolds, we obtain eigenvalue inequalities that can be viewed as versions of the classical results by Korevaar and Buser in Riemannian geometry.

math.DG↗
arXiv2013

Eigenvalues of the Laplacian and extrinsic geometry

We extend the results given by Colbois, Dryden and El Soufi on the relationships between the eigenvalues of the Laplacian and an extrinsic invariant called intersection index, in two directions. First, we replace this intersection index by invariants of the same nature which are stable under small perturbations. Second, we consider complex submanifolds of the complex projective space $\mathbb{C} P^N$ instead of submanifolds of $\mathbb{R}^N$ and we obtain an eigenvalue upper bound depending only on the dimension of the submanifold which is sharp for the first non-zero eigenvalue.

math.SP↗
arXiv2013

Eigenvalues of perturbed Laplace operators on compact manifolds

We obtain upper bounds for the eigenvalues of the Schrödinger operator $L=Δ_g+q$ depending on integral quantities of the potential $q$ and a conformal invariant called the min-conformal volume. Moreover, when the Schrödinger operator $L$ is positive, integral quantities of $q$ which appear in upper bounds, can be replaced by the mean value of the potential $q$. The upper bounds we obtain are compatible with the asymptotic behavior of the eigenvalues. We also obtain upper bounds for the eigenvalues of the weighted Laplacian or the Bakry-Emery Laplacian $Δ_ϕ=Δ_g+\nabla_gϕ\cdot\nabla_g$ using two approaches: First, we use the fact that $Δ_ϕ$ is unitarily equivalent to a Schrödinger operator and we get an upper bound in terms of the $L^2$-norm of $\nabla_gϕ$ and the min-conformal volume. Second, we use its variational characterization and we obtain upper bounds in terms of the $L^\infty$-norm of $\nabla_gϕ$ and a new conformal invariant. The second approach leads to a Buser type upper bound and also gives upper bounds which do not depend on $ϕ$ when the Bakry-Emery Ricci curvature is non-negative.

math.DG↗