arXiv2017
We consider the problem of finding and describing minimisers of the Rayleigh quotient \[ Λ_\infty \, :=\, \inf_{u\in \mathcal{W}^{2,\infty}(Ω)\setminus\{0\} }\frac{\|Δu\|_{L^\infty(Ω)}}{\|u\|_{L^\infty(Ω)}}, \] where $Ω\subseteq \mathbb{R}^n$ is a bounded $C^{1,1}$ domain and $\mathcal{W}^{2,\infty}(Ω)$ is a class of weakly twice differentiable functions satisfying either $u=0$ or $u=|\mathrm{D} u|=0$ on $\partial Ω$. Our first main result, obtained through approximation by $L^p$-problems as $p\to \infty$, is the existence of a minimiser $u_\infty \in \mathcal{W}^{2,\infty}(Ω)$ satisfying \[ \left\{ \begin{array}{ll} Δu_\infty \, \in \, Λ_\infty \mathrm{Sgn}(f_\infty) & \text{ a.e. in }Ω, \\ Δf_\infty \, =\, μ_\infty & \text{ in }\mathcal{D}'(Ω), \end{array} \right. \] for some $f_\infty\in L^1(Ω)\cap BV_{\text{loc}}(Ω)$ and a measure $μ_\infty \in \mathcal{M}(Ω)$, for either choice of boundary conditions. Here Sgn is the multi-valued sign function. We also study the dependence of the eigenvalue $Λ_\infty$ 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 $Ω\mapsto Λ_\infty(Ω)$. This result is shown to hold true for either choice of boundary conditions and in every dimension.