Related papers: On the Effectless Cut Method for Laplacian Eigenva…

The celebrated conjecture by Payne, P\'{o}lya and Weinberger (1956) states that for the fixed membrane problem, the ratio of the first two eigenvalues, $\lambda_2/\lambda_1$, is maximized by a disk. A more general dimensional version of…

The lowest eigenmode of thin axisymmetric shells is investigated for two physical models (acoustics and elasticity) as the shell thickness (2$\epsilon$) tends to zero. Using a novel asymptotic expansion we determine the behavior of the…

The purpose of this article is to approximately compute the eigenvalues of the symmetric Dirichlet Laplacian within an interval $(0,\Lambda)$. A novel domain decomposition Ritz method, partition of unity condensed pole interpolation method,…

We study a stochastic primal-dual method for constrained optimization over Riemannian manifolds with bounded sectional curvature. We prove non-asymptotic convergence to the optimal objective value. More precisely, for the class of…

We study the Laplacian in deformed thin (bounded or unbounded) tubes in ?$\R^3$, i.e., tubular regions along a curve $r(s)$ whose cross sections are multiplied by an appropriate deformation function $h(s)> 0$. One the main requirements on…

In this paper, we investigate the Dirichlet problem of Laplacian on complete Riemannian manifolds. By constructing new trial functions, we obtain a sharp upper bound of the gap of the consecutive eigenvalues in the sense of the order, which…

We study asymptotic shape optimization for Riesz means of Robin Laplacian eigenvalues among cuboids of fixed measure. Our focus is the regime where the Robin parameter is proportional to the square root of the spectral parameter defining…

The third eigenvalue of the Robin Laplacian on a simply-connected planar domain of given area is bounded above by the third eigenvalue of a disjoint union of two disks, provided the Robin parameter lies in a certain range and is scaled in…

We consider Robin Laplace operators on a class of two-dimensional domains with cusps. Our main results include the formula for the asymptotic distribution of the eigenvalues of such operators. In particular, we show how the eigenvalue…

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…

The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators $\Theta$ which give rise to self-adjoint Laplacians $-\Delta_{\Theta, \Omega}$ in $L^2(\Omega; d^n x)$ with (nonlocal and local)…

We develop a computational method for extremal Steklov eigenvalue problems and apply it to study the problem of maximizing the $p$-th Steklov eigenvalue as a function of the domain with a volume constraint. In contrast to the optimal…

Spherical regression explores relationships between variables on spherical domains. We develop a nonparametric model that uses a diffeomorphic map from a sphere to itself. The restriction of this mapping to diffeomorphisms is natural in…

We consider singular perturbed eigenvalue problem for Laplace operator in a two-dimensional domain. In the boundary we select a set depending on a character small parameter and consisting of a great number of small disjoint parts. On this…

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 study Riesz means of the eigenvalues of the Heisenberg Laplacian with Dirichlet boundary conditions on bounded domains. We obtain an inequality with a sharp leading term and an additional lower order term, improving the result of Hanson…

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 bi-Laplacian eigenvalue problem for the modes of vibration of a thin elastic plate with a discrete set of clamped points. A high-order boundary integral equation method is developed for efficient numerical determination of…

The purpose of this paper is three-fold. Firstly we attack a nonlinear interface problem on an unbounded domain with nonmonotone set-valued transmission conditions. The investigated problem involves a nonlinear monotone partial differential…

We prove that Klein quartic maximizes the multiplicity of the first positive eigenvalue of the Laplacian among all closed hyperbolic surfaces of genus $3$, with multiplicity equal to $8$. We also obtain partial results in genus $2$, where…