D-Solutions to the System of Vectorial Calculus of Variations in L-infinity via the Baire Category Method for the Singular Values

Gisella Croce, Nikos Katzourakis, Giovanni Pisante

Published 2016-04-15Version 1

For $\mathrm{H}\in C^2(\R^{N\by n})$ and $u :\Om \sub \R^n \larrow \R^N$, consider the system \[ \label{1}\mathrm{A}\_\infty u\, :=\,\Big(\mathrm{H}\_P \ot \mathrm{H}\_P + \mathrm{H}[\mathrm{H}\_P]^\bot \mathrm{H}\_{PP}\Big)(\D u):\D^2u\, =\,0. \tag{1}\]The PDE system \eqref{1} is associated to the supremal functional\[ \label{2}\ \ \ \ \mathrm{E}\_\infty(u,\Omega')\, =\, \big\| \mathrm{H}(\D u)\|\_{L^\infty(\Om')}, \ \ \ u\in W^{1,\infty}\_{\text{loc}}(\Om,\R^N),\ \Omega'\Subset \Omega, \tag{2} \]and first arose in recent work of the 2nd author as the analogue of the Euler-Lagrange equation. Herein we employ the Dacorogna-Marcellini Baire Category method to construct $\mD$-solutions to the Dirichlet problem for \eqref{1}, an apt notion of generalised solutions recently proposed for fully nonlinear systems. Our $\mD$-solutions to \eqref{1} are $W^{1,\infty}$-submersions corresponding to "critical points" of \eqref{2} and are obtained without any convexity hypotheses. Along the way we establish a result of independent interest by proving existence of strong solutions to the singular value problem for general dimensions $n\neq N$.