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

Abstract

Given the supremal functional E∞⁢(u,Ω′)=ess⁢supΩ′⁡H⁢(⋅,D⁢u){E_{\infty}(u,\Omega^{\prime})=\operatornamewithlimits{ess\,sup}_{\Omega^{% \prime}}H(\,\cdot\,,\mathrm{D}u)}, defined on Wloc1,∞⁢(Ω,ℝN){W^{1,\infty}_{\mathrm{loc}}(\Omega,\mathbb{R}^{N})}, with Ω′⋐Ω⊆ℝn{\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 C1{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.

Keywords

• rank-one absolute minimisers
• vectorial hamilton–jacobi equation
• viscosity solutions.
• 35d99
• 35d40
• 35j47
• 35j92
• 35j70
• 35j99