A property of Absolute Minimizers in $L^\infty$ Calculus of Variations and of solutions of the Aronsson-Euler equation

We discover a new minimality property of the absolute minimizers supremal functionals, whose variational problems are also known as $L^\infty$ problems. In particular for every minimizer $v$ quasi-convex functional $$ {\rm ess.sup}\,{\Omega}H(x,Dv(x)), we consider set \mathcal{A}(v)=\{x: H (x,Dv(x))={\rm ess.sup}\,{\Omega}H(x,Dv(x))\}, suitably defined. If $u$ is an minimizer, give structure result $\mathcal{A}(u)$ and show that $\mathcal{A}(u)\subset\mathcal{A}(v)$ $v$.