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 $n\neq N$.
Categories: math.AP