Convergence of solutions and eigenelements of Steklov type boundary value problems with boundary conditions of rapidly varying type

Abstract

We consider boundary value problems for the Laplace operator in a domain with boundary conditions of rapidly varying type: the Dirichlet homogeneous condition and the third (Fourier) boundary condition or a Steklov type condition. We construct the limit (homogenized) problem and prove that solutions, eigenvalues, and eigenfunctions of the original problem converge respectively to solutions, eigenvalues, and eigenfunctions of the limit problem. Bibliography: 47 titles. Illustrations: 2 figures.

Boundary homogenization with large reaction terms on a strainer-type wall

“…We mention some of the first works in which keywords such as critical sizes and critical relations between parameters have been introduced [7,29,30] and [35], also [8] for nonhomogeneous boundary conditions. Let us refer to [5,6] and references therein for rapidly alternating Dirichlet-Steklov boundary conditions and [11,18,28] for further references and possible applications in the framework of Geophysics and Winkler beds (foundations). See [9][10][11][12][13][14][15] and [32] for an extensive and updated bibliography on different boundary homogenization problems with Robin-type boundary conditions.…”

On the asymptotic behaviour of eigenvalues of a boundary-value problem in a planar domain of Steklov sieve type

“…Later, many papers were published in which the homogenization of diverse elliptic spectral problems was considered. In particular, problems with a spectral condition of Steklov type were studied; see, for example, [3]- [6]. In these papers, homogenization problems of a spectral problem of Steklov type with a fast change of the type of the boundary condition were investigated (for the scalar equation, see [3], and for the system of elasticity theory, see [4]).…”

“…In particular, problems with a spectral condition of Steklov type were studied; see, for example, [3]- [6]. In these papers, homogenization problems of a spectral problem of Steklov type with a fast change of the type of the boundary condition were investigated (for the scalar equation, see [3], and for the system of elasticity theory, see [4]). In [5], the leading terms of the asymptotic expansion of the eigenvalue in a dense cascade connection were constructed.…”

Asymptotics of a spectral-sieve problem

“…There is a vast literature devoted to homogenization of spectral problems including Steklov-type problems, see for instance [22], [16], [21]. Some results on homogenization of Steklov problems can be found in [4], [13], [15], [19].…”