On Second-Order $L^{\infty }$ Variational Problems with Lower-Order Terms

Abstract

In this paper we study $2$nd order $L^{\infty }$ variational problems, through seeking to minimise a supremal functional involving the Hessian of admissible functions as well as lower-order terms. Specifically, given a bounded domain $\hat{I}\text{c◯}\subseteq {\mathbb{R}}^n$ and $\mathrm{H}:\hat{I}\text{c◯}\times (\mathbb{R}\times {\mathbb{R}}^n\times {\mathbb{R}}_s^{n^{\otimes 2}})\to \mathbb{R}$, we consider the functional \[ \mathrm{E}_\infty(u, \mathcal{O}) :=\underset{ \mathcal{O}}{\mathrm{ess}\sup}\hspace{1mm}\mathrm H (\cdot,u,\mathrm D u,\mathrm D^2u ) , \ \ u\in W^{2,\infty}(Ω), \ \mathcal{O} \subseteq Ω\text{ measurable}. \] We establish the existence of minimisers subject to (first-order) Dirichlet data on $\partial \hat{I}\text{c◯}$ under natural assumptions, and, when $n=1$, we also show the existence of absolute minimisers. We further derive a necessary fully nonlinear PDE of third-order which arises as the analogue of the Euler-Lagrange equation for absolute minimisers, and is given by $${\mathrm{H}}_{\mathrm{X}}(\cdot ,u,\mathrm{D}u,{\mathrm{D}}^{2}u):\mathrm{D}(\mathrm{H}(\cdot ,u,\mathrm{D}u,{\mathrm{D}}^{2}u))\otimes \mathrm{D}(\mathrm{H}(\cdot ,u,\mathrm{D}u,{\mathrm{D}}^{2}u))=0\text{ in }\hat{I}\text{c◯}.$$ We then rigorously derive this PDE from smooth absolute minimisers, and prove the existence of generalised D-solutions to the (first-order) Dirichlet problem. Our work generalises the key results obtained in [26] which first studied problems of this type with pure Hessian dependence only, providing at the same time considerably simpler streamlined proofs.

21 pages