Calculus of Variations and Geometric Measure Theory

Rademacher's theorem for Euclidean measures

Andrea Marchese (Università di Trento)

created by magnani on 02 May 2012

16 may 2012 -- 17:00   [open in google calendar]

Sala Seminari, Department of Mathematics, Pisa University


For every Euclidean Radon measure $\mu$ we state an adapted version of Rademacher's theorem, which is, in a certain sense, the best possible for the measure $\mu$. We define a sort of fibre bundle (actually a map $S$ that at each point $x\in\mathbb{R}^n$ associates a vector subspace $S(x)$ of $T_x\mathbb{R}^n$, possibly with non-costant dimension $k(x)$) such that every Lipschitz function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is differentiable at $x$, along $S(x)$, for $\mu$-a.e. $x$. We prove that $S$ is maximal in the following sense: there exists a Lipschitz function $g:\mathbb{R}^n\rightarrow\mathbb{R}$ which doesn't admit derivative at $\mu$-a.e. $x$, along any direction not belonging to $S(x)$. Joint work with Giovanni Alberti.