$\mathcal{D}$-solutions to the system of vectorial Calculus of Variations in $L^\infty$ via 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 $n\neq N$.