Existence of an optimal shape for the first eigenvalue of polyharmonic operators

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