Sergey A. Nazarov

A gap in the essential spectrum of a cylindrical waveguide with a periodic aperturbation of the surface

It is proved that small periodic singular perturbation of a cylindrical waveguide surface may open a gap in the continuous spectrum of the Dirichlet problem for the Laplace operator. If the perturbation period is long and the caverns in the cylinder are small, the gap certainly opens.

Essential spectrum of a periodic elastic waveguide may contain arbitrarily many gaps

We construct a family of periodic elastic waveguides Π h , depending on a small geometrical parameter, with the following property: as h → +0, the number of gaps in the essential spectrum of the elasticity system on Π h grows unboundedly.

Spectra of open waveguides in periodic media

Abstract We study the essential spectra of formally self-adjoint elliptic systems on doubly periodic planar domains perturbed by a semi-infinite periodic row of foreign inclusions. We show that the essential spectrum of the problem consists of the essential spectrum of the purely periodic problem and another component, which is the union of the discrete spectra of model problems in the infinite perturbation strip; these model problems arise by an application of the partial Floquet–Bloch–Gelfand transform.

The Navier―Stokes Problem in a Two-Dimensional Domain with Angular Outlets to Infinity

The Navier―Stokes problem in a plane domain with two angular outlets to infinity is provided, as usual, either with the flux condition or with the pressure drop one. For small data it is proved that there exists a solution with the decay O(|x|-1) of the velocity field as |x| → ∞ (if one of the angles is equal to or greater than π, then additional symmetry assumptions are needed). Since the nonlinear and linear terms are asymptotically of the same power, the results are based on a complete investigation of the linearized Stokes problem in weighted spaces with a detached asymptotics (the angular parts in the representations are not fixed). Bibliography: 16 titles.

THE LOCALIZATION FOR EIGENFUNCTIONS OF THE DIRICHLET PROBLEM IN THIN POLYHEDRA NEAR THE VERTICES

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.

Gap in the essential spectrum of an elliptic formally self-adjoint system of differential equations

For a formally self-adjoint elliptic system of partial differential equations with periodic coefficients in the space ℝn, we show that, by conferring contrast properties to the coefficients of differential operators, one can open a gap in the essential spectrum of the system. We suggest a method based on the derivation of an asymptotically sharp generalized Korn inequality and the use of the maximin principle; this method applies to perforated media as well as to periodic layered and quasi-cylindrical waveguides.

A one-dimensional model of viscous blood flow in an elastic vessel

In this paper we present a one-dimensional model of blood flow in a vessel segment with an elastic wall consisting of several anisotropic layers. The model involves two variables: the radial displacement of the vessels wall and the pressure, and consists of two coupled equations of parabolic and hyperbolic type. Numerical simulations on a straight segment of a blood vessel demonstrate that the model can produce realistic flow fields that may appear under normal conditions in healthy blood vessels; as well as flow that could appear during abnormal conditions. In particular we show that weakening of the elastic properties of the wall may provoke a reverse blood flow in the vessel.

On the Hadamard Formula for Second Order Systems in Non-Smooth Domains

Perturbations of eigenvalues of the Dirichlet problem for a second order elliptic system in a bounded domain Ω in ℝ n are studied under variations of the domain Ω. We investigate the case when the perturbed domain is located in a d-neighborhood of the reference Lipschitz domain. A new asymptotic formula is derived; it contains terms that are absent in the classical formula of Hadamard. The latter is valid only for smooth domains and smooth perturbations. We give conditions that guarantee the validity of the Hadamard formula. The general asymptotic formula is applied when Ω is perturbed by small curvilinear and circular cuts. Most of the results are new even for the Laplace operator.

Boundary Layers and Boundary Hinge-Support Conditions for Thin Plates

The boundary-layer phenomenon is investigated for a thin three-dimensional plate. An arbitrary anisotropy of elastic properties and nonhomogeneity in longitudinal and transversal directions are assumed. Several initial asymptotic terms are constructed and a way of continuing the asymptotic procedure is described. Precise estimates of the difference between the exact and approximate solutions are obtained for the energy norm. An amalgamated problem is formed that gives the two-term asymptotics of the fields of displacements and stresses at a distance from the lateral side of the plate, whereas the edge effects are simulated by conditions of elastic fastening containing integral characteristics of the boundary-layer problem in the semistrip. Hinge-support conditions due to a sufficiently narrow clamped zone at the edge of the plate are derived and justified as well. Bibliography: 48 titles.

The spectrum, radiation conditions and the Fredholm property for the Dirichlet Laplacian in a perforated plane with semi-infinite inclusions

Abstract We consider the spectral Dirichlet problem for the Laplace operator in the plane Ω ∘ with double-periodic perforation but also in the domain Ω • with a semi-infinite foreign inclusion so that the Floquet–Bloch technique and the Gelfand transform do not apply directly. We describe waves which are localized near the inclusion and propagate along it. We give a formulation of the problem with radiation conditions that provides a Fredholm operator of index zero. The main conclusion concerns the spectra σ ∘ and σ • of the problems in Ω ∘ and Ω • , namely we present a concrete geometry which supports the relation σ ∘ ⫋ σ • due to a new non-empty spectral band caused by the semi-infinite inclusion called an open waveguide in the double-periodic medium.