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

  • ‡ Corresponding author

    ‡ Corresponding author 

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

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  • 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 $n≠ N$ .

    Mathematics Subject Classification: 35D99, 35F05.


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