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

Abstract

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 $8$.