Solutions of vectorial Hamilton–Jacobi equations are rank-one absolute minimisers in L ∞ L^{\infty}

Abstract

Abstract Given the supremal functional E ∞ ⁢ ( u , Ω ′ ) = ess ⁢ sup Ω ′ ⁡ H ⁢ ( ⋅ , D ⁢ u ) {E_{\infty}(u,\Omega^{\prime})=\operatornamewithlimits{ess\,sup}_{\Omega^{% \prime}}H(\,\cdot\,,\mathrm{D}u)} , defined on W loc 1 , ∞ ⁢ ( Ω , ℝ N ) {W^{1,\infty}_{\mathrm{loc}}(\Omega,\mathbb{R}^{N})} , with Ω ′ ⋐ Ω ⊆ ℝ n {\Omega^{\prime}\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 ⁢ ( ⋅ , D ⁢ u ) = c {H(\,\cdot\,,\mathrm{D}u)=c} are rank-one absolute minimisers if they are C 1 {C^{1}} . Our minimality notion is a generalisation of the classical L ∞ {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.