  In specific types of partially rectangular billiards we estimate the mass of
an eigenfunction of energy $E$ in the region outside the rectangular set in the
high-energy limit. We use the adiabatic ansatz to compare the Dirichlet energy
form with a second quadratic form for which separation of variables applies.
This allows us to use sharp one-dimensional control estimates and to derive the
bound assuming that $E$ is not resonating with the Dirichlet spectrum of the
rectangular part.
