Asymptotic expansion for a solution of an elliptic boundary-value problem in a thin cascade domain

TL;DR

This paper develops and justifies an asymptotic expansion for the solution of a Neumann boundary-value problem for the Poisson equation in a thin cascade domain, providing precise estimates for the approximation's accuracy.

Contribution

It introduces a new asymptotic expansion method for elliptic problems in thin cascade domains, with rigorous justification and error estimates.

Findings

Asymptotic expansion accurately approximates the solution.

Energetic and uniform pointwise estimates are established.

The approach extends to nonuniform boundary conditions.

Abstract

Asymptotic expansion is constructed and justified for the solution to a nonuniform Neumann boundary-value problem for the Poisson equation with the right-hand side that depends both on longitudinal and transversal variables in a thin cascade domain. Asymptotic energetic and uniform pointwise estimates for the difference between the solution of the initial problem and the solution of the corresponding limiting problem are proved.