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

Nikos Katzourakis, Tristan Pryer

Published 2016-05-25Version 1

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. Our analysis relies heavily on the recently proposed by the first author theory of $\mathcal{D}$-solutions.