Supersolid edge and bulk phases of a dipolar quantum gas in a box.

We investigate the novel density distributions acquired by a dipolar Bose-Einstein condensed gas confined in a box potential. Differently from the case of harmonic trapping, the ground state density reveals a strong depletion at the center of the trap and an accumulation near the walls, as a consequence of the competition between the attractive {\it vs} repulsive nature of the long range dipolar force and of the presence of the box potential (\textit{meniscus}-like effect). In a quasi one-dimensional configuration, the ground state density shows pronounced rotonic oscillations when the superfluid to supersolid phase transition is approached. In a quasi two-dimensional geometry we observe the coexistence of a low-density superfluid phase in the bulk with one-dimensional supersolid edges or, by increasing the atom number, the coexistence of a supersolid bulk with a high density superfluid phase at edges. Remarkably, the geometry of the supersolid bulk reflects the symmetry of the confining potential even for large systems.