The second eigenfunction of the Laplace-Beltrami operator follows the pattern of the overall shape of an object. This geometric property is well known and used for various applications including mesh processing, feature extraction, manifold learning, data embedding and the minimum linear arrangement problem. Surprisingly, this geometric property has not been mathematically formulated yet. This ...

We study the asymptotic behavior of the rst eigenvalue and eigenfunction of a one-dimensional periodic elliptic operator with Neumann boundary conditions. The second order elliptic equation is not self-adjoint and is singularly perturbed since, denoting by ε the period, each derivative is scaled by an ε factor. The main di culty is that the domain size is not an integer multiple of the period. ...

For almost Mathieu operators, it is shown that the occurrence of Cantor spectrum and the existence, for every point in the spectrum and suitable phase parameters, of at least one localized eigenfunction which decays exponentially are inconsistent properties.

An alternative method for deriving water wave dispersion relations and evolution equations is to use a weak formulation. The free-surface displacement η is written as an eigenfunction expansion,

In this work we show that, generically in the set of C2 bounded regions of Rn, n ≥ 2, the inequality R Ω φ 3 6= 0 holds for any eigenfunction of the Laplacian with either Dirichlet or Neumann boundary conditions.