  We prove the existence of an open set minimizing the first eigenvalue of the
Dirichlet polylaplacian of order $m\geq1$ under volume constraint. Moreover,
the corresponding eigenfunction is shown to enjoy $C^{m-1,\alpha}$ H\"older
regularity. This is performed for dimension $2\leq d\leq 4m$. In particular,
our analysis answers the question of the existence of an optimal shape for the
clamped plate up to dimension $8$.
