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

Abstract

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