  For a transmission problem in a truncated two-dimensional cylinder located
beneath the graph of a function u, the shape derivative of the Dirichlet energy
(with respect to u) is shown to be well-defined and is computed. The main
difficulties in this context arise from the weak regularity of the domain and
the possible non-empty intersection of the graph of u and the transmission
interface. The result is applied to establish the existence of a solution to a
free boundary transmission problem for an electrostatic actuator.
