The eigenvalue problem for the $$\infty $$-Bilaplacian

Abstract

We consider the problem of finding and describing minimisers of the Rayleigh quotient $$\begin{aligned} \Lambda _\infty \, :=\, \inf _{u\in \mathcal {W}^{2,\infty }(\Omega )\setminus \{0\} }\frac{\Vert \Delta u\Vert _{L^\infty (\Omega )}}{\Vert u\Vert _{L^\infty (\Omega )}}, \end{aligned}$$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 \(u=0\) on \(\partial \Omega \) or \(u=|\mathrm {D}u|=0\) on \(\partial \Omega \). Our first main result, obtained through approximation by \(L^p\)-problems as \(p\rightarrow \infty \), is the existence of a minimiser \(u_\infty \in \mathcal {W}^{2,\infty }(\Omega )\) satisfying $$\begin{aligned} \left\{ \begin{array}{ll} \Delta u_\infty \, \in \, \Lambda _\infty \mathrm {Sgn}(f_\infty ) &{} \text { a.e. in }\Omega , \\ \Delta f_\infty \, =\, \mu _\infty &{} \text { in }\mathcal {D}'(\Omega ), \end{array} \right. \end{aligned}$$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.