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

Abstract

Given the supremal functional $E_{\infty }(u,{\Omega }^{\prime })=ess{\sup }_{{\Omega }^{\prime }}H(\cdot ,Du)$ defined on $W_{loc}^{1,\infty }(\Omega ,{\mathbb{R}}^N)$, ${\Omega }^{\prime }⋐\Omega \subseteq {\mathbb{R}}^n$, 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 $H(\cdot ,Du)=c$ are rank-one Absolute Minimisers if they are $C^1$. Our minimality notion is a generalisation of the classical $L^{\infty }$ 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.

12 pages, 2 figures, Journal: Advances in Nonlinear Analysis

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