arXiv research

Study compares eigenvalues on spherically symmetric manifolds to Euclidean balls.

The paper finds universal inequalities for eigenvalues on hyperbolic spaces.

The paper compares eigenvalues of Dirichlet, Neumann, and Laplacian on graphs.

Universal inequalities found for Laplacian eigenvalues on convex domains.

Estimates for eigenvalues on Riemannian manifolds using classical inequalities.

Estimates eigenvalues of poly-Laplace operator on lattice subgraphs.

Geodesic balls with non-negative Ricci curvature have a sharp lower bound on their first Dirichlet eigenvalue.

Paper proves new inequalities for hyperbolic space Laplacian eigenvalues.

Paper calculates eigenvalues of a specific triangle on a sphere.

In this paper, we mainly study eigenvalue problems of p-Laplacian on domains with an interior hole. Firstly we prove Faber-Krahn-type inequalities, and Cheng-type eigenvalue comparison theorems on manifolds. Secondly, we prove a comparison theorem for eigenvalues with inner Dirichlet and outer Neumann boundary in minim…

Eigenvalues of manifolds with cylindrical boundaries approximated by graph Laplacians.

Introduces differential forms to study inequalities between eigenvalues.

Inequalities between the Dirichlet and Neumann eigenvalues of the Laplacian have received much attention in the literature, but open problems abound. Here, we study the number of Neumann eigenvalues no greater than the first Dirichlet eigenvalue. Based on a combination of analytical and numerical results, we conjecture…

Paper derives inequalities for eigenvalues of Witten-Laplacian under fixed volume constraint.

Study estimates eigenvalues for concave Hessian operators on convex domains.

The paper explores inequalities between eigenvalues on Riemannian manifolds.

The paper examines how the first Steklov-Dirichlet eigenvalue changes with the distance between two concentric circles.

Sharp lower bound for $p$-Laplacian eigenvalue on non-compact manifolds.

Liu's paper contains an error regarding eigenvalues.

The paper connects eigenvalue problems for various operators and establishes inequalities and asymptotic formulas for heat traces.

In this paper, we derive the CR Reilly's formula and its applications to studying of the first eigenvalue estimate for CR Dirichlet eigenvalue problem and embedded p-minimal hypersurfaces. In particular, we obtain the first Dirichlet eigenvalue estimate in a compact pseudohermitian (2n+1)-manifold with boundary and the…

Sharp lower bound found for geodesic ball eigenvalues.

Study on eigenvalue rate of geodesic balls in asymptotically hyperbolic Einstein manifolds.

Researchers found the first and second eigenvalues are Courant-sharp on a Möbius strip.

In this paper we consider a domain in a space of negative constant sectional curvature. Such assumption about the sectional curvature let us develop a new technique and improve existing lower bounds of eigenvalues from Dirichlet eigenvalue problem, obtained by Alessandro Savo in 2009.

In this paper, we first introduce higher order Dirichlet-to-Neumann maps on graphs which can be viewed as a discrete analogue of the corresponding Dirichlet-to-Neumann maps on compact Riemannian manifolds with boundary and a higher order generalization of the Dirichlet-to-Neumann map on graphs introduced by Hua-Huang-W…

We consider a shape optimization problem for the first mixed Steklov-Dirichlet eigenvalues of domains bounded by two balls in two-point homogeneous space. We give a geometric proof which is motivated by Newton's shell theorem

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 Ste…

We give a new estimate on the lower bound of the first Dirichlet eigenvalue of a compact Riemannian manifold with negative lower bound of Ricci curvature and provide a solution for a conjecture of H. C. Yang.

Estimates eigenvalues and spectrum for graph substructures using isocapacitary constants.

We compute the first Dirichlet eigenvalue of a geodesic ball in a rotationally symmetric model space in terms of the moment spectrum for the Brownian motion exit times from the ball. This expression implies an estimate as exact as you want for the first Dirichlet eigenvalue of a geodesic ball in these rotationally symm…

The paper finds metrics for surfaces with boundaries that match specific eigenvalues and areas.

In this paper, we prove some analogues of Payne-Polya-Weinberger, Hile-Protter and Yang's inequalities for Dirichlet (discrete) Laplace eigenvalues on any subset in the integer lattice $\Z^n.$ This partially answers a question posed by Chung and Oden.

The paper compares Dirichlet and Neumann eigenvalues on various curved surfaces.

We study a Dirichlet-to-Neumann eigenvalue problem for differential forms on a compact Riemannian manifold with smooth boundary. This problem is a natural generalization of the classical Steklov problem on functions. We derive a number of upper and lower bounds for the first eigenvalue in several contexts: many of thes…

We give a new estimate on the lower bound for the first Dirichlet eigenvalue for a compact manifold with positive Ricci curvature in terms of the in-diameter and the lower bound of the Ricci curvature. The result improves the previous estimates.

The paper studies eigenvalue problems on manifolds and recovers known inequalities.

The paper finds lower bounds for the first eigenvalue of p-Laplacian in specific manifolds.

Study on biharmonic Steklov problems with Neumann boundary conditions and eigenvalue estimates.

Universal inequalities for Laplacian eigenvalues on discrete groups.

The paper finds metrics for manifolds with prescribed volumes and eigenvalues.

The Dirichlet eigenvalues of the Laplace-Beltrami operator are larger on an annulus than on any other surface of revolution in $\mathbb{R}^3$ with the same boundary. This is established by defining a sequence of shrinking cylinders about the axis of symmetry and proving that flattening a surface outside of each cylinde…

We introduce higher-order Poincar'e constants for compact weighted manifolds and estimate them from above in terms of subsets. These estimates imply upper bounds for eigenvalues of the weighted Laplacian and the first nontrivial eigenvalue of the $p$-Laplacian. In the case of the closed eigenvalue problem and the Neuma…

In this paper we study spectral properties of Dirichlet-to-Neumann map on differential forms obtained by a slight modification of the definition due to Belishev and Sharafutdinov. The resulting operator $Λ$ is shown to be self-adjoint on the subspace of coclosed forms and to have purely discrete spectrum there.We inves…

New upper bound found for nodal sets of Laplace eigenfunctions.

We obtain upper bounds for the first Dirichlet eigenvalue of a tube around a complex submanifold $P$ of $CP^n$ which depends only on the radius of the tube, the degrees of the polynomials defining $P$ and the first eigenvalue of some model centers of the tube. The bounds are sharp on these models. Moreover, when the mo…

Estimates gaps between eigenvalues for elliptic operators on manifolds.

In this paper, we consider eigenvalues of the Dirichlet biharmonic operator on a bounded domain in a hyperbolic space. We obtain universal bounds on the $(k+1)$th eigenvalue in terms of the first $k$th eigenvalue independent of the domains.