Abstract
We prove a differentiability result for monotone Sobolev functions on doubling metric measure spaces supporting a Poincaré inequality. This generalizes a result used by Rickman in proving the differentiability of quasiregular mappings. Our main tools are a Stepanov differentiability theorem in doubling metric measure spaces supporting a Poincaré inequality, proved in 2004 by Balogh, Rogovin and Zürcher and a Sobolev embedding theorem on spheres proved by Hajlasz and Koskela. As an application, it is shown that continuous quasiminimizers for the p- energy integral with p>Q-1 are almost everywhere Cheeger differentiable, where Q is the doubling expo-nent of the underlying metric measure space.
Cuvinte cheie
doubling metric measure space
Poincaré inequality
function monotone in the sense of Lebesgue
Sobolev function
Cheeger differentiability
p-energy integral
quasiminimizer.