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

TL;DR

The paper studies second-order $L^\infty$ variational problems with Hessian and lower-order terms, introducing the functional $E_\infty$ and proving existence of minimisers under Dirichlet data, with absolute minimisers in 1D. It derives the third-order fully nonlinear PDE $A^2_\infty u=0$ as the Euler–Lagrange analogue and establishes that smooth absolute minimisers satisfy this equation; a $\mathcal{D}$-solutions framework based on Young measures is developed to handle the Dirichlet problem and the link to minimisers. The authors provide existence results for $\mathcal{D}$-solutions to the Dirichlet problem for $A^2_\infty u=0$, using an auxiliary second-order equation and a Baire-category approach, and they show the approach generalises prior Hessian-only results with streamlined proofs.

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

Theorems & Definitions (20)

• Definition 1.1: Global and absolute minimisers
• Definition 2.1: Young measures
• Definition 2.2: Difference quotients and diffuse derivatives
• Definition 2.3: Twice weakly differentiable $\mathcal{D}$-solutions of a third-order PDE
• Remark 2.4
• Theorem 3.1: Existence of global minimisers
• Remark 3.2: More general supremands
• proof : Proof of Theorem \ref{['theorem:existenceofglobalminimisers']}
• Corollary 3.3: Diagonal lower semicontinuity
• proof : Proof of Corollary \ref{['corollary:diagonallowersemicontinuity']}