  For two-dimensional quantum billiards we derive the partial Weyl law, i.e.
the average density of states, for a subset of eigenstates concentrating on an
invariant region $\Gamma$ of phase space. The leading term is proportional to
the area of the billiard times the phase-space fraction of $\Gamma$. The
boundary term is proportional to the fraction of the boundary where parallel
trajectories belong to $\Gamma$. Our result is numerically confirmed for the
mushroom billiard and the generic cosine billiard, where we count the number of
chaotic and regular states, and for the elliptical billiard, where we consider
rotating and oscillating states.
