Solution of boundary value problems in cylinders separated by a three-layer film into two semicylinders

Boundary value problems are considered for the class of equations $\partial_x^2u+L[u]=0$ in cylinders $D=(x\in R,\,y\in Q\subseteq R^m)$ with an infinitely thin film at $x=0$ consisting of three sublayers with alternating high and low permeability ($L$-linear differential operator with respect to $y_i$). The solutions of the problems are expressed in terms of those of the corresponding classical boundary value problems in homogeneous cylinders $D$ with no film. The resulting formulas have the form of simple quadrature rules, which are amenable to numerical computations.