$\mathcal{D}$-solutions to the system of vectorial Calculus of Variations in $L^\infty$ via the singular value problem

TL;DR

This paper introduces $ ext{D}$-solutions for a class of vectorial calculus of variations problems in $L^ty$, using a novel approach based on the singular value problem, without requiring convexity of the integrand.

Contribution

It constructs $ ext{D}$-solutions for the system in $L^ty$ without convexity assumptions, linking solutions to the singular value problem for general dimensions.

Findings

Existence of $ ext{D}$-solutions for the system.

Solutions are $W^{1,ty}$-submersions.

Established a new link between calculus of variations and the singular value problem.

Abstract

For $\mathrm{H} \in C^2(\mathbb{R}^{N \times n})$ and $u : \Omega \subseteq \mathbb{R}^n \to \mathbb{R}^N$, consider the system \[ \label{1}\mathrm{A}\_\infty u\, :=\,\Big(\mathrm{H}\_P \otimes \mathrm{H}\_P + \mathrm{H}[\mathrm{H}\_P]^\bot \mathrm{H}\_{PP}\Big)(\mathrm{D} u): \mathrm{D}^2 u\, =\,0. \tag{1}\]We construct $\mathcal{D}$-solutions to the Dirichlet problem for (1), an apt notion of generalised solutions recently proposed for fully nonlinear systems. Our $\mathcal{D}$-solutions are $W^{1,\infty}$-submersions and are obtained without any convexity hypotheses for $\mathrm{H}$, through a result of independent interest involving existence of strong solutions to the singular value problem for general dimensions $n\neq N$.