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

Gisella Croce; Nikos Katzourakis; Giovanni Pisante

doi:10.3934/dcds.2017266

Article Contents

2017, Volume 37, Issue 12: 6165-6181. Doi: 10.3934/dcds.2017266

This issue Previous Article On the extension and smoothing of the Calabi flow on complex tori Next Article A generalization of Douady's formula

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

1.
Normandie Univ, UNIHAVRE, LMAH, 76600 Le Havre, France
2.
Department of Mathematics and Statistics, University of Reading, Whiteknights, PO Box 220, Reading RG6 6AX, UK
3.
Universitá degli Studi della Campania "Luigi Vanvitelli", Scuola Politecnica e delle Scienze di Base, Dipartimento di Matematica e Fisica, Viale Lincoln, 5, 81100 Caserta, Italy

‡ Corresponding author
‡ Corresponding author

† N.K. has been partially financially supported by the EPSRC grant EP/N017412/1

Received: November 30, 2016

Revised: June 30, 2017

Published: October 2017

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Abstract

For $\mathrm{H}∈ C^2(\mathbb{R}^{N\times n})$ and $u :Ω \subseteq \mathbb{R}^n \longrightarrow \mathbb{R}^N$ , consider the system

$ \label{1}\mathrm{A}_∞ u\, :=\,\Big(\mathrm{H}_P \otimes \mathrm{H}_P + \mathrm{H}[\mathrm{H}_P]^\bot \mathrm{H}_{PP}\Big)(\text{D} u):\text{D}^2u\, =\,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 $W^{1,∞}$ -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 $n≠ N$ .

Keywords:

Vectorial Calculus of Variations in $L^\infty$,
generalised solutions,
fully nonlinear systems,
$\infty$-Laplacian,
young measures,
singular value problem,
Baire Category method,
convex integration.

Mathematics Subject Classification: 35D99, 35F05.

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