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Let Ω ⊂ C be a bounded domain. In this note, we use complex variable methods to study the number of critical points of the function v = v Ω that solves the elliptic problem Δ v = - 2 in Ω , with boundary values v = 0 on ∂ Ω. This problem has a classical flavor but is especially motivated by recent studies on localization of eigenfunctions. We provide an upper bound on the number of critical points of v when Ω belongs to a special class of domains in the plane, namely, domains for which the boundary ∂ Ω is contained in { z : | z | 2 = f (z) + f (z) ¯ } , where f ′ (z) is a rational function. We furnish examples of domains where this bound is attained. We also prove a bound on the number of critical points in the case when Ω is a quadrature domain. The note concludes with the statement of some open problems and conjectures. [ABSTRACT FROM AUTHOR]

The localization of eigenfunctions of the Laplace operator in a domain divided by a perforated barrier is proven. The localization takes place with sufficiently small holes in the barrier. In this case, the measure of the barrier can be arbitrarily small. [ABSTRACT FROM AUTHOR]

We consider the homogenization of a singularly perturbed self-adjoint fourth order elliptic operator with locally periodic coefficients, stated in a bounded domain. We impose Dirichlet boundary conditions on the boundary of the domain. The presence of large parameters in the lower order terms and the dependence of the coefficients on the slow variable lead to localization of the eigenfunctions. We show that the jth eigenfunction can be approximated by a rescaled function that is constructed in terms of the jth eigenfunction of fourth or second order effective operators with constant coefficients. [ABSTRACT FROM AUTHOR]

We consider the homogenization of an elliptic spectral problem with a large potential stated in a thin cylinder with a locally periodic perforation. The size of the perforation gradually varies from point to point. We impose homogeneous Neumann boundary conditions on the boundary of perforation and on the lateral boundary of the cylinder. The presence of a large parameter 1/εin front of the potential and the dependence of the perforation on the slow variable give rise to the effect of localization of the eigenfunctions. We show that thejth eigenfunction can be approximated by a scaled exponentially decaying function that is constructed in terms of thejth eigenfunction of a one-dimensional harmonic oscillator operator. [ABSTRACT FROM PUBLISHER]

Under some geometric assumptions, we show that eigenfunctions of the Dirichlet problem for the Laplace operator in an n-dimensional thin polyhedron localize near one of its vertices. We construct and justify asymptotics for the eigenvalues and eigenfunctions. For waveguides, which are thin layers between periodic polyhedral surfaces, we establish the presence of gaps and find asymptotics for their geometric characteristics. [ABSTRACT FROM AUTHOR]

We establish that the principal eigenfunction of the Dirichlet problem in a domain with a thin heavy edging admits localization near the corner point of opening angle α > π. The edging amounts to a boundary strip of small width ɛ with the density function ɛ, m > 0, while it is O(1) in the remaining part of the domain. We derive the result by analyzing the essential and discrete spectra of an auxiliary problem in an infinite angle without the small parameter. We state several open questions about the structure of spectra of both problems. [ABSTRACT FROM AUTHOR]

A simple sufficient condition on a curved end of a straight cylinder is found that provides a localization of the principal eigenfunction of the mixed boundary value for the Laplace operator with the Dirichlet conditions on the lateral side. Namely, when the small parameter, i.e., the ratio between the diameter and the length of the cylinder, tends to zero, the eigenfunction concentrates in the vicinity of the ends and decays exponentially in the interior. Similar effects are observed in the Dirichlet and Neumann problems, too. [ABSTRACT FROM AUTHOR]