Calculus of Variations and Geometric Measure Theory

G. Alberti - A. Marchese

On the differentiability of Lipschitz functions with respect to measures in the Euclidean space

created by marchese on 11 Aug 2014
modified by alberti on 28 May 2017


Published Paper

Inserted: 11 aug 2014
Last Updated: 28 may 2017

Journal: Geom. Funct. Anal. (GAFA)
Volume: 26
Number: 1
Pages: 1-66
Year: 2016
Doi: 10.1007/s00039-016-0354-y


For every finite measure $\mu$ on $\mathbb{R}^n$ we define a decomposability bundle $V(\mu,\cdot)$ related to the decompositions of $\mu$ in terms of rectifiable one-dimensional measures. We then show that every Lipschitz function on $\mathbb{R}^n$ is differentiable at $\mu$-a.e. $x$ with respect to the subspace $V(\mu,x)$, and prove that this differentiability result is optimal, in the sense that, following $[4]$, we can construct Lipschitz functions which are not differentiable at $\mu$-a.e. $x$ in any direction which is not in $V(\mu,x)$. As a consequence we obtain a differentiability result for Lipschitz functions with respect to (measures associated to) $k$-dimensional normal currents, which we use to extend certain basic formulas involving normal currents and maps of class $C^1$ to Lipschitz maps.

Keywords: Rademacher theorem, differentiability, Lipschitz functions, normal currents