Asymptotics and Estimates of Degrees of Convergence in Three-Dimensional Boundary Value Problem with Frequent Interchange of Boundary Conditions

TL;DR

This paper investigates the asymptotic behavior and convergence rates of eigenvalues in a three-dimensional boundary value problem with rapidly alternating boundary conditions on a cylinder, using homogenization techniques.

Contribution

It develops asymptotic expansions for eigenvalues and eigenfunctions in a perturbed problem with alternating boundary conditions, and estimates convergence rates under rapid variation.

Findings

Constructed leading terms of eigenvalue asymptotics.

Estimated convergence degree for eigenvalues with rapidly varying strips.

Analyzed the homogenized problem with Dirichlet boundary condition.

Abstract

We consider a singular perturbed eigenvalue problem for Laplace operator in a cylinder with frequent interchange of type of boundary condition on a lateral surface. These boundary conditions are prescribed by partition of lateral surface in a great number of narrow strips on those the Dirichlet and Neumann conditions are imposed by turns. We study the case of the homogenized problem containing Dirichlet condition on the lateral surface. When the width of strips varies slowly, we construct the leading terms of eigenelements' asymptotics expansions. We also estimate the degree of convergence for eigenvalues if the strips' width varies rapidly.