Rustem R. Gadyl'shin

On regular and singular perturbations of acoustic and quantum waveguides

Abstract We consider regular and singular perturbations of the Dirichlet and Neumann boundary value problems for the Helmholtz equation in n -dimensional cylinders. The existence of eigenvalues and their asymptotics are studied. To cite this article: R.R. Gadylshin, C. R. Mecanique 332 (2004).

Asymptotics for eigenelements of Laplacian in domain with oscillating boundary: multiple eigenvalues

We study the asymptotic behavior of the solutions of a spectral problem for the Laplacian in a domain with rapidly oscillating boundary. We consider the case where the eigenvalue of the limit problem is multiple. We construct the leading terms of the asymptotic expansions for the eigenelements and verify the asymptotics.

Effective membrane permeability: estimates and low concentration asymptotics

The paper deals with a mathematical model of a steady-state diffusion process through a periodic membrane. For a wide class of periodic membranes, we define the effective permeability and obtain upper and lower estimates of the effective permeability. For periodic membranes made from two materials with different absorbing properties, we study the asymptotic behavior of the effective permeability when the fraction of one material tends to zero (low concentration asymptotics). When the low fraction material forms homothetically vanishing disperse periodic inclusions in the host material, low concentration approximations are built by the method of matched asymptotic expansions. We also show that our results are consistent with those which can be obtained by a boundary homogenization. Finally, we analyze formulas used in physical, chemical, and biological investigations to describe effective membrane properties.

On the scattering of H-polarized electromagnetic field by an ideally conductive cylindrical body of a trapping type

Abstract We consider a two-dimensional analog of Helmholtz resonator with walls of finite thickness in the critical case when there exists an eigenfrequency which is the limit of poles generated by both the bounded component of the resonator and the narrow connecting channel. Under the assumption that the limit eigenfrequency is a simple eigenfrequency of the bounded component, the asymptotics of two poles converging to this eigenfrequency are constructed by using the method of matching asymptotic expansions. Explicit formulas for the leading terms of the asymptotics of poles and of the solution of the scattering problem are obtained.

On resonances in a homogenization problem

Abstract We consider a two-dimensional boundary value problem for the Helmholtz equation with Neumann boundary condition on a set of arcs. This set is obtained from a closed curve by cutting out small holes situated closely each to other and having locally periodic structure. We construct asymptotics of scattering frequencies (poles of analytic continuation of solutions) with small imaginary parts and show that these scattering frequencies imply resonances. To cite this article: R.R.xa0Gadylshin, C. R. Mecanique 331 (2003).

Eigenvalues and scattering frequencies for domains with narrow appendixes and tubes

Abstract The eigenvalue problem of the Dirichlet Laplacian in a singularly perturbed region, which is described as a bounded domain with a thin appendix is considered. The expansion of eigenvalues in the power series with respect to a small parameter (radius of the cross-section of the appendix) is constructed. This result is extended to the Helmholtz resonator with the finite thickness of a shell.