The Eigenvalue Problem for the $\infty$-Bilaplacian

Abstract

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 $\Omega \subseteq {\mathbb{R}}^n$ 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 \[ \left\{ \begin{array}{ll} Δu_\infty \, \in \, Λ_\infty \mathrm{Sgn}(f_\infty) & \text{ a.e. in }Ω, \\ Δf_\infty \, =\, μ_\infty & \text{ in }\mathcal{D}'(Ω), \end{array} \right. \] for some $f_{\infty }\in L^1(\Omega )\cap B{V}_{\text{loc}}(\Omega )$ 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.

24 pages; accepted; Journal: Nonlinear Differential Equations and Applications

References in corpus (2)

• Absolutely Minimising Generalised Solutions to the Equations of Vectorial Calculus of Variations in $L^{\infty }$
• Second Order $L^{\infty }$ Variational Problems and the $\infty$-Polylaplacian