Second Order $L^\infty$ Variational Problems and the $\infty$-Polylaplacian

Abstract

In this paper we initiate the study of $2$nd order variational problems in $L^\infty$, seeking to minimise the $L^\infty$ norm of a function of the hessian. We also derive and study the respective PDE arising as the analogue of the Euler-Lagrange equation. Given $\mathrm{H}\in C^1(\mathbb{R}^{n\times n}_s)$, for the functional \[ \label{1} \mathrm{E}_\infty(u,\mathcal{O})\, =\, \big\| \mathrm{H}\big(\mathrm{D}^2 u\big) \big\|_{L^\infty(\mathcal{O})}, \ \ \ u\in W^{2,\infty}(\Omega),\ \mathcal{O}\subseteq \Omega, \tag{1} \] the associated equation is the fully nonlinear 3rd order PDE \[ \label{2} \mathrm{A}^2_\infty u\, :=\,\big(\mathrm{H}_X\big(\mathrm{D}^2u\big)\big)^{\otimes 3}:\big(\mathrm{D}^3u\big)^{\otimes 2}\, =\,0. \tag{2} \] Special cases arise when $\mathrm{H}$ is the Euclidean length of either the full hessian or of the Laplacian, leading to the $\infty$-Polylaplacian and the $\infty$-Bilaplacian respectively. We establish several results for \eqref{1} and \eqref{2}, including existence of minimisers, of absolute minimisers and of "critical point" generalised solutions, proving also variational characterisations and uniqueness. We also construct explicit generalised solutions and perform numerical experiments.