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

TL;DR

This paper investigates second-order $L^ abla$ variational problems involving the Hessian and lower-order terms, establishing existence of minimisers, deriving a third-order nonlinear PDE, and extending previous results with simpler proofs.

Contribution

It introduces a new framework for second-order $L^ abla$ variational problems with lower-order terms, including existence results and a novel PDE for absolute minimisers.

Findings

Existence of minimisers under natural assumptions.

Derivation of a third-order fully nonlinear PDE.

Existence of generalized D-solutions for the Dirichlet problem.

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 $\Omega\subseteq \mathbb R^n$ and $\mathrm H : \Omega\times\big(\mathbb R \times\mathbb R^n \times \mathbb R^{n^{\otimes2}}_s \big) \to \mathbb R$, we consider the functional \[ \mathrm{E}_\infty(u, \mathcal{O}) :=\underset{ \mathcal{O}}{\mathrm{ess}\sup}\hspace{1mm}\mathrm H (\cdot,u,\mathrm D u,\mathrm D^2u ) , \ \ u\in W^{2,\infty}(\Omega), \ \mathcal{O} \subseteq \Omega \text{ measurable}. \] We establish the existence of minimisers subject to (first-order) Dirichlet data on $\partial \Omega$ 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 $$ \ \ \mathrm H_{\mathrm X}(\cdot,u,\mathrm D u,\mathrm D^2u): \mathrm D\big(\mathrm H(\cdot,u,\mathrm D u,\mathrm D^2u)\big)\otimes \mathrm D\big(\mathrm H(\cdot,u,\mathrm D u,\mathrm D^2u)\big)=0\ \ \text{ in }\Omega. $$ 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.