A Rademacher type theorem for Hamiltonians $H(x,p)$ and 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; the lower-simicontinuity in the variable p is sharp. Moreover, applying such Rademacher type theorem we build up an existence result of absolute minimizers for corresponding L ∞ -functional. These improve or extend several known results in the literature.