arXiv research

New upper bound for Neumann Laplacian eigenvalues on convex domains.

Universal inequalities for Laplacian eigenvalues on convex domains.

New inequalities for planar convex domains' Laplacian eigenvalues.

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

The paper defines new constants for $p$-Laplacian on manifolds.

Researchers calculate spectral invariants from Dirichlet-to-Neumann map for Witten-Laplacian with potential.

Study connects boundary geometry to symbol of Dirichlet-to-Neumann operator.

The paper finds a family of 2D surfaces where Laplacian and Dirichlet-to-Neumann commute.

The paper classifies solutions to a Liouville equation on a half-space with a specific boundary condition.

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

Sharp isoperimetric inequalities for Neumann eigenvalues in symmetric spaces.

Introduces differential forms to study inequalities between eigenvalues.

For $1\leq q \leq n-2$, we provide explicit examples to demonstrate non-compactness of the Neumann operator for the Kohn Laplacian acting on $L^2$ $(0,q)$-forms on the unit ball in $(2n+1)$-dimensional Heisenberg space.

Recent research connects Hörmander's old work to modern boundary Laplacian analysis.

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

Proven isoperimetric inequality for Witten-Laplacian eigenvalues.

Let $Ω$ be an open, bounded domain in the plane with connected and smooth boundary, and $ω$ an eigenfunction of the Neumann Laplacian corresponding to some Neumann eigenvalue $μ> 0$. If the boundary value of $ω$ is a nonzero constant along the boundary, denoting $0 = μ_1(Ω) < μ_2(Ω) <= ...$ the set of all Neumann eigen…

The paper improves eigenvalue estimates for manifolds with Ricci curvature conditions.

We consider the Laplacian in a domain squeezed between two parallel hypersurfaces in Euclidean spaces of any dimension, subject to Dirichlet boundary conditions on one of the hypersurfaces and Neumann boundary conditions on the other. We derive two-term asymptotics for eigenvalues in the limit when the distance between…

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…

Generalizes Cheeger inequality to Carnot-Carathéodory spaces.

The paper finds a lower bound for a Neumann eigenvalue of surfaces with flat ends.

Two geodesic balls maximize the third Neumann eigenvalue in hyperbolic space.

Paper derives a formula for the determinant of Dirichlet-to-Neumann operator on Riemann surfaces.

Study proves inequalities for eigenvalues of symmetric domains in space forms.

Abstract: Expresses zeta-determinant of Dirichlet-to-Neumann operator on forms.

New contractible domains on half-sphere with constant boundary Laplacian eigenfunctions.

Study sharp lower bounds on negative eigenvalues of magnetic Pauli operator.

For a given bounded domain $Ω\subset {\Bbb R}^n$ with $C^1$-smooth boundary, we prove the Pólya conjecture for the Neumann eigenvalues. In other words, we prove that \begin{eqnarray*} μ_{k+1}\le \frac{(2π)^2k^{2/n}}{(ω_n \cdot \mbox{vol}\, (Ω))^{2/n}} \quad \;\; \mbox{for all} \;\; k=0,1,2,3,\cdots,\end{eqnarray*} wher…

In this paper we study some new von Neumann spectral invariants associated to the Laplacian acting on L^2 differential forms on the universal cover of a closed manifold. These invariants coincide with the Novikov-Shubin invariants whenever there is no spectral gap in the spectrum of the Laplacian, and are homotopy inva…

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…

New algorithms for clustering and dimension reduction using relative von Neumann entropy.

We apply Gromov's ham sandwich method to get (1) domain monotonicity (up to a multiplicative constant factor); (2) reverse domain monotonicity (up to a multiplicative constant factor); and (3) universal inequalities for Neumann eigenvalues of the Laplacian on bounded convex domains in a Euclidean space.

Study eigenvalues of p-Laplacian on quaternionic Kähler manifolds.

We prove Li-Yau-Kröger type bounds for Neumann-type eigenvalues of the poly-harmonic operator and of the biharmonic operator on bounded domains in a Euclidean space. We also prove sharp estimates for lower order eigenvalues of a biharmonic Steklov problem and of the Laplacian, which directly implies two sharp Reilly-ty…

We establish sharp regularity and Fredholm theorems for the \bar{\partial}_b-Neumann problem on domains satisfying some non-generic geometric conditions. We use these domains to construct explicit examples of bad behaviour of the Kohn Laplacian: it is not always hypoelliptic up to the boundary, its partial inverse is n…

We prove the sharp estimate on the first nonzero eigenvalue of the p-laplacian on a compact Riemannian manifold with nonnegative Ricci curvature and possibly with convex boundary (in this case we assume Neumann b.c. on the p-laplacian). The proof is based on a gradient comparison theorem. We will also charachterize the…

Estimates the mass gap for domains with integral Ricci curvature bounds.

Neumann eigenmaps improve landmark-based diffusion map embeddings.

I prove that the spectrum of the Laplace-Beltrami operator with the Neumann boundary condition on a compact Riemannian manifold with boundary admits a fast approximation by the spectra of suitable graph Laplacians on proximity graphs on the manifold, and similar graph approximation works for metric-measure spaces glued…

Study sets lower bounds for Kähler manifolds' Laplacian eigenvalues.

We adapt the results of Part 1 to include the unit ball in the Heisenberg group, the model domain with characteristic boundary points. In particular, we construct function spaces on which the Kohn Laplacian with the \bar{\partial}_b-Neumann boundary conditions is an isomorphism. As an application, we establish sharp re…

In this paper, we derive from the supersymmetry of the Witten Laplacian Brascamp-Lieb's type inequalities for general differential forms on compact Riemannian manifolds with boundary. In addition to the supersymmetry, our results essentially follow from suitable decompositions of the quadratic forms associated with the…

The paper studies stability of commutativity properties of the Dirichlet-to-Neumann map.

Dirichlet-Neumann duality for Riemannian submersions

In this paper, we give a sharp lower bound for the first (nonzero) Neumann eigenvalue of Finsler-Laplacian in Finsler manifolds in terms of diameter, dimension, weighted Ricci curvature.

In this note, we investigate upper bounds of the Neumann eigenvalue problem for the Laplacian of a bounded domain (with smooth boundary) in a given complete (not compact a priori) Riemannian manifold with Ricci bounded below . For this, we use test functions for the Rayleigh quotient subordinated to a family of open se…

The study finds no extremal metrics for eigenvalues on compact manifolds but constructs examples for annuli.