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

TL;DR

The paper addresses the problem of minimizing the first eigenvalue $\Gamma(\Omega)$ of the Dirichlet polyharmonic operator $(-\Delta)^m$ under a fixed volume, proving existence of an optimal shape in the broad class of $1$-quasi-open sets for dimensions $2\le d\le 4m$ and establishing $C^{m-1,\alpha}$ regularity of the corresponding eigenfunction on $\mathbb{R}^d$. The existence result relies on a concentration-compactness analysis applied to first eigenfunctions of a minimizing sequence, yielding a limit function whose derivatives up to order $m-1$ define a $1$-quasi-open optimal set. Regularity is then obtained via quasi-minimizer theory for Dirichlet energy functionals, showing the eigenfunction is a local quasi-minimizer with forcing term $f=\Gamma(\Omega)u$ and, under the dimension constraint, belongs to $C^{m-1,\alpha}(\mathbb{R}^d)$. Consequently, an open optimal set can be constructed from the eigenfunction, thereby achieving existence and partial regularity for the clamped plate problem up to dimension $d=8$ when $m=2$, while acknowledging open questions about boundedness and sharper regularity of the optimal shape.

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 $C^{m-1,α}$ Hölder 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 $8$.

Theorems & Definitions (11)

• proof
• proof
• proof
• Proposition 14
• proof
• proof : Proof of Proposition \ref{['prop:compacité']}
• proof : Proof of Theorem \ref{["thm:existence d'une forme optimale"]}
• proof
• proof
• proof : Proof of Theorem \ref{['thm:regularite fonction propre']}