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Consider the supremal functional \[ \tag{1} \label{1} E_\infty(u,A) \,:=\,\|L(\cdot,u,D u)\|_{L^\infty(A)},\quad A\subseteq \Omega, \] applied to$W^{1,\infty}$ maps $u:\Omega\subseteq \mathbb{R}\longrightarrow \mathbb{R}^N$,$N\geq 1$. Under certain assumptions on $L$, we prove for any given boundarydata the existence of a map which is:  i) a vectorial Absolute Minimiser of \eqref{1} in the sense of Aronsson,  ii) a generalised solution to the ODE system associated to \eqref{1} as theanalogue of the Euler-Lagrange equations,  iii) a limit of minimisers of the respective $L^p$ functionals as$p\rightarrow\infty$ for any $q\geq 1$ in the strong $W^{1,q}$ topology \&  iv) partially $C^2$ on $\Omega$ off an exceptional compact nowhere dense set.  \noi {Our method is based on $L^p$ approximations and stable a priori partialregularity estimates. For item ii) we utilise the recently proposed by theauthor notion of $\mathcal{D}$-solutions in order to characterise the limit asa generalised solution. Our results are motivated from and apply to DataAssimilation in Meteorology.}

Absolutely Minimising Generalised Solutions to the Equations of Vectorial Calculus of Variations in $L^\infty$