  We consider Dirichlet spaces with superharmonic weights. This class contains
both the harmonic weights and the power weights. Our main result is a
characterization of the Dirichlet spaces with superharmonic weights that can be
identified as de Branges-Rovnyak spaces. As an application, we obtain the
dilation inequality \[ {\cal D}_\omega(f_r)\le \frac{2r}{1+r}{\cal D}_\omega(f)
\qquad(0\le r<1), \] where ${\cal D}_\omega$ denotes the Dirichlet integral
with superharmonic weight $\omega$, and $f_r(z):=f(rz)$ is the $r$-dilation of
the holomorphic function $f$.
