The buffer boundary layer in a mixed boundary value problem with contrasting coefficients

We construct asymptotics for the solution to a mixed boundary value problem in a thin compound cylinder $\Omega_h^0\cup\Omega_h^1\cup\dots\cup\Omega_h^J$ for a scalar second-order differential equation with piecewise linear coefficients of orders $h^m$ in $\Omega_h^0$ and $h^0$ in $\Omega_h^j$, $j=1,\dots,J$, where $h>0$ is a small parameter, and $m=0,1,2$ (the typical cases). For $m=1$ and $J>1$ we discover the new phenomenon of a one-dimensional buffer boundary layer, which enables us to combine smooth-type expansion to a higher-dimensional boundary layer concentrated in the immediate vicinity of the bases of the cylinder.