DIFFERENTIABILITY OF MONOTONE SOBOLEV FUNCTIONS ON METRIC SPACES
MARCELINA MOCANU âVasile Alecsandri" University of BacÄu, Faculty of Sciences Department of Mathematics, Informatics and Education Sciences Calea MÄrÄĆeĆti 157, BacÄu 600115, ROMANIA
Corresponding author: mmocanu@ub.ro
Issue:
SSRSMI, Number 1, Volume XXIV
Section:
Volume 24, Number 1
Abstract:
We prove a differentiability result for monotone Sobolev functions on doubling metric measure spaces supporting a PoincarĂ© 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 PoincarĂ© inequality, proved in 2004 by Balogh, Rogovin and ZĂŒrcher 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.
Keywords:
doubling metric measure space , Poincaré inequality , function monotone in the sense of Lebesgue , Sobolev function , Cheeger differentiability , p-energy integral , quasiminimizer .
Code [ID]:
SSRSMI20140124V24S01A0005 [0004062]
DOI:
Full paper:
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