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

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