  We consider attractive particle systems in $\Z^d$ with product invariant
measures. We prove that when particles are restricted to a subset of $\Z^d$,
with birth and death dynamics at the boundaries, the hydrodynamic limit is
given by the unique entropy solution of a conservation law, with boundary
conditions in the sense of Bardos, Leroux and N\'ed\'elec. For the hydrostatic
limit between parallel hyperplanes, we prove a multidimensional version of the
phase diagram conjectured in \cite{ps}, and show that it is robust with respect
to perturbations of the boundaries.
