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≥1 under volume constraint. Moreover, the corresponding eigenfunction is shown to enjoy Cm-1,α H\"older regularity. This is performed for dimension 2≤ d≤ 4m. In particular, our analysis answers the question of the existence of an optimal shape for the clamped plate up to dimension $8$.