A Rademacher type theorem for Hamiltonians $H(x,p)$ and application to absolute minimizers

TL;DR

This paper proves a Rademacher type theorem for Hamiltonians with weak regularity conditions and applies it to establish the existence of absolute minimizers for related $L^ Infty$-functionals, extending previous results in the field.

Contribution

It introduces a Rademacher theorem under minimal assumptions on Hamiltonians and uses it to prove existence of absolute minimizers, broadening the scope of prior work.

Findings

Rademacher theorem holds for Hamiltonians measurable in $x$ and quasiconvex in $p$

Counterexample shows necessity of lower-semicontinuity in $p$

Existence of absolute minimizers for $L^ Infty$-functionals established

Abstract

We establish a Rademacher type theorem involving Hamiltonians $H(x,p)$ under very weak conditions in both of Euclidean and Carnot-Carath\'eodory spaces. In particular,$H(x,p)$ is assumed to be only measurable in the variable $x$, and to be quasiconvex and lower-semicontinuous in the variable $p$. Without the lower-semicontinuity in the variable $p$, we provide a counter example showing the failure of such a Rademacher type theorem. Moreover, by applying such a Rademacher type theorem we build up an existence result of absolute minimizers for the corresponding $L^\infty$-functional. These improve or extend several known results in the literature.