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

Abstract

We establish a Rademacher type theorem involving Hamiltonians H(x, p) under very weak conditions in both of Euclidean and Carnot-Carathéodory 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∞-functional. These improve or extend several known results in the literature.