The Eigenvalue Problem for the $\infty$-Bilaplacian

Abstract

We consider the problem of finding and describing minimisers of the Rayleigh quotient \[ \Lambda_\infty \, :=\, \inf_{u\in \mathcal{W}^{2,\infty}(\Omega)\setminus\{0\} }\frac{\|\Delta u\|_{L^\infty(\Omega)}}{\|u\|_{L^\infty(\Omega)}}, \] where $\Omega \subseteq \mathbb{R}^n$ is a bounded $C^{1,1}$ domain and $\mathcal{W}^{2,\infty}(\Omega)$ is a class of weakly twice differentiable functions satisfying either $u=0$ or $u=|\mathrm{D} u|=0$ on $\partial \Omega$. Our first main result, obtained through approximation by $L^p$-problems as $p\to \infty$, is the existence of a minimiser $u_\infty \in \mathcal{W}^{2,\infty}(\Omega)$ satisfying \[ \left\{ \begin{array}{ll} \Delta u_\infty \, \in \, \Lambda_\infty \mathrm{Sgn}(f_\infty) & \text{ a.e. in }\Omega, \\ \Delta f_\infty \, =\, \mu_\infty & \text{ in }\mathcal{D}'(\Omega), \end{array} \right. \] for some $f_\infty\in L^1(\Omega)\cap BV_{\text{loc}}(\Omega)$ and a measure $\mu_\infty \in \mathcal{M}(\Omega)$, for either choice of boundary conditions. Here Sgn is the multi-valued sign function. We also study the dependence of the eigenvalue $\Lambda_\infty$ on the domain, establishing the validity of a Faber-Krahn type inequality: among all $C^{1,1}$ domains with fixed measure, the ball is a strict minimiser of $\Omega \mapsto \Lambda_\infty(\Omega)$. This result is shown to hold true for either choice of boundary conditions and in every dimension.