The finite element method (FEM) is applied to bound leading eigenvalues of the Laplace operator over polygonal domains. Compared with classical numerical methods, most of which can only give concrete eigenvalue bounds over special domains of symmetry, our proposed algorithm can provide concrete eigenvalue bounds for domains of arbitrary shape, even when the eigenfunction has a singularity. The ...

We consider the principal eigenvalue problem for Laplace-Beltrami operator on upper half of a topological torus under Dirichlet boundary condition. present construction that admits eigenfunction having exact numbers critical points. Furthermore, we manage to identify locations all points explicitly.

In the present paper, we study properties of second Dirichlet eigenvalue fractional Laplacian annuli-like domains and corresponding eigenfunctions. first part, consider an annulus with inner radius $ R outer R+1 $. We show that for sufficiently large any eigenfunction this is nonradial. investigate in form B_1(0)\setminus \overline{B_{\tau}(a)} $, where a unitary ball 0<\tau<1-|a| value m...

In this paper, we consider the Dirichlet Laplacian operator on a curved quantum guide in Rn (n = 2, 3). If the reference curve is not straight but asymptotically straight, we prove that the data of one eigenfunction associated with its eigenvalue determines the curvature.

Hill’s method is a means to numerically approximate spectra of linear differential operators with periodic coefficients. In this paper, we address different issues related to the convergence of Hill’s method. We show the method does not produce any spurious approximations, and that for selfadjoint operators, the method converges in a restricted sense. Furthermore, assuming convergence of an eig...

EI ρA Wn(x) eigenfunction corresponding to the n th eigenvalue Tn(t) generalized function corresponding to the n th eigenvalue Fn(t) forcing function in terms of the n th eigenmode z the deflection of the beam in the complex plane x nondimensional length scale by chord length of the NACA0012 airfoil κ Reduced plunging frequency, ωc U∞ h non-dimensional amplitude,y/c ha maximum nondimensional am...

(1.2) { −∆pu = λa(x)|u|p−2u, u ∈ D 0 (Ω), has the least eigenvalue λ1 > 0 with a positive eigenfunction e1 and λ1 is the only eigenvalue having this property (cf. Proposition 3.1). This gives us a possibility to study the existence of an unbounded branch of positive solutions bifurcating from (λ1, 0). When Ω is bounded, the result is well-known, we refer to the survey article of Amann [2] and t...

In quantum systems, the presence of a disordered potential may induce the appearance of strongly localized quantum states (called Anderson localization), i.e., eigenfunctions that essentially “live” in a very restricted subregion of the entire domain. We show here that solving a simple Dirichlet problem reveals a network of interconnected lines which are the boundaries of the localization subre...

We propose and analyze a new spectral collocation method to solve eigenvalue problems of compact integral operators, particularly, piecewise smooth operator kernels and weakly singular operator kernels of the form 1 |t−s|μ , 0 < μ < 1. We prove that the convergence rate of eigenvalue approximation depends upon the smoothness of the corresponding eigenfunctions for piecewise smooth kernels. On t...

We consider the spectral problem for mixed local and nonlocal p-Laplace operator. discuss existence regularity of eigenfunction associated Dirichlet (p, q)-eigenvalue in a bounded domain Ω ⊂ ℝN under assumption that 1 < p ∞ q p∗ where = Np/(N − p) if N ⩾ N.