The system of partial differential equations governing the dynamics and, in
the stationary case, the electro-elastic equilibrium of a piezoelectric
crystal when a given electric potential is applied on the surface, is studied
in connection with the aspects of existence, uniqueness and regularity of
solutions. We prove that the corresponding semigroup is in fact a group of
isometries. When the data are periodic functions we also provide a condition
for the existence of forced periodic vibrations in both the damped and undamped case.