Related papers: The fundamental Laplacian eigenvalue of the regula…

We numerically compute the lowest Laplacian eigenvalues of several two-dimensional shapes with dihedral symmetry at arbitrary precision arithmetic. Our approach is based on the method of particular solutions with domain decomposition. We…

In this project, I examine the lowest Dirichlet eigenvalue of the Laplacian within the ellipse as a function of eccentricity. Two existing analytic expansions of the eigenvalue are extended: Close to the circle (eccentricity near zero) nine…

In this article, we study the asymptotics of Dirichlet eigenvalues and eigenfunctions of the fractional Laplacian $(-\Delta)^s$ in bounded open Lipschitz sets in the small order limit $s \to 0^+$. While it is easy to see that all…

We investigate the asymptotic behavior of the eigenvalues of the Laplacian with homogeneous Robin boundary conditions, when the (positive) Robin parameter is diverging. In this framework, since the convergence of the Robin eigenvalues to…

We consider Dirichlet Laplacian in a thin curved three-dimensional rod. The rod is finite. Its cross-section is constant and small, and rotates along the reference curve in an arbitrary way. We find a two-parametric set of the eigenvalues…

This paper investigates the first Dirichlet eigenvalue for the $p$-Laplacian in Riemannian manifolds. Firstly, we establish a lower bound for this eigenvalue under the condition that the domain includes a specific function which fulfills…

We consider the Laplace operator with Dirichlet boundary conditions on a domain in R^d and study the effect that performing a scaling in one direction has on the eigenvalues and corresponding eigenfunctions as a function of the scaling…

The spectrum of the Dirichlet Laplacian on conical layers is analysed through two aspects: the infiniteness of the discrete eigenvalues and their expansions in the small aperture limit. On the one hand, we prove that, for any aperture, the…

We consider the Laplace operator with Dirichlet boundary conditions on a planar domain and study the effect that performing a scaling in one direction has on the spectrum. We derive the asymptotic expansion for the eigenvalues and…

The main difficulty in solving the Helmholtz equation within polygons is due to non-analytic vertices. By using a method nearly identical to that used by Fox, Henrici, and Moler in their 1967 paper; it is demonstrated that such eigenvalue…

In this work we consider the homogeneous Neumann eigenvalue problem for the Laplacian on a bounded Lipschitz domain and a singular perturbation of it, which consists in prescribing zero Dirichlet boundary conditions on a small subset of the…

Motivated by relativistic materials, we develop a numerical scheme to support existing or state new conjectures in the spectral optimisation of eigenvalues of the Dirac operator, subject to infinite-mass boundary conditions. We study the…

In this paper, we investigate the Dirchlet eigenvalue problems of poly-Laplacian with any order and quadratic polynomial operator of the Laplacian. We give some estimates for lower bounds of the sums of their first $k$ eigenvalues which…

We prove that the first Dirichlet eigenvalue of a regular $N$-gon of area $\pi$ has an asymptotic expansion of the form $\lambda_1(1+\sum_{n\ge3}C_n(\lambda_1)N^{-n})$ as $N\to\infty$, where $\lambda_1$ is the first Dirichlet eigenvalue of…

Consider the Laplacian in a bounded domain in R^d with general (mixed) homogeneous boundary conditions. We prove that its eigenfunctions are `quasi-orthogonal' on the boundary with respect to a certain norm. Boundary orthogonality is proved…

We consider the Laplacian in a domain squeezed between two parallel curves in the plane, subject to Dirichlet boundary conditions on one of the curves and Neumann boundary conditions on the other. We derive two-term asymptotics for…

For a bounded domain $\Omega$ with a piecewise smooth boundary in a complete Riemannian manifold $M$, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. By making use of a fact that eigenfunctions form an orthonormal…

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…

In this paper, our main goal is to study the evolution problem associated with the Laplacian operator with Dirichlet boundary conditions on a regular tree. To this end, we place special emphasis on the associated first eigenvalue problem,…

For an $n$-dimensional polytope $\Omega$ in $\mathbb{R}^{n}$, we study lower bounds for eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. In the asymptotic formula on the average of the first $k$ eigenvalues, Li and Yau…