Solutions of Vectorial Hamilton-Jacobi Equations are Rank-One Absolute Minimisers in $L^{\mathrm{\infty }}$

Given the supremal functional $E_\infty(u,\Omega’)=ess,\sup_{\Omega’} H(\cdot,D u)$definedon$W^{1,\infty}_{loc}(\Omega,$\mathbb{R}$^N)$,$\Omega’ \Subset \Omega\subseteq $\mathbb{R}$^n$, we identify a class of vectorial rank-one Absolute Minimisers by proving a statement slightly stronger than the next claim: vectorial solutions of the Hamilton-Jacobi equation $H(\cdot,D u)=c$arerank-oneAbsoluteMinimisersiftheyare$C^1$. Our minimality notion is a generalisation of the classical $L^{\mathrm{\infty }}$ variational principle of Aronsson to the vector case and emerged in earlier work of the author. The assumptions are minimal, requiring only continuity and rank-one convexity of the level sets.