Asymptotic analysis of a boundary-value problem in a thin cascade domain with a local joint

TL;DR

This paper develops a rigorous asymptotic analysis for a boundary-value problem in a thin, irregular domain with a small joint, providing estimates that reveal the joint's influence as the domain's thickness tends to zero.

Contribution

The paper introduces a complete asymptotic expansion for the solution of a Poisson problem in a thin domain with a joint, including energetic and uniform estimates that account for geometric irregularities.

Findings

Asymptotic expansion accurately approximates the solution as  

Geometric irregularity of the joint significantly affects the solution

Uniform estimates quantify the joint's influence in the limit 

Abstract

A nonuniform Neumann boundary-value problem is considered for the Poisson equation in a thin domain $\Omega_\varepsilon$ coinciding with two thin rectangles connected through a joint of diameter ${\cal O}(\varepsilon)$. A rigorous procedure is developed to construct the complete asymptotic expansion for the solution as the small parameter $\varepsilon \to 0.$ Energetic and uniform pointwise estimates for the difference between the solution of the starting problem $(\varepsilon >0)$ and the solution of the corresponding limit problem $(\varepsilon =0)$ are proved, from which the influence of the geometric irregularity of the joint is observed.