$\mathcal{D}$-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 $n\ne N$.

References in corpus (1)

• Weak versus $\mathcal{D}$-solutions to linear hyperbolic first order systems with constant coefficients

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• Vectorial variational principles in $L^{\infty }$ and their characterisation through PDE systems