MERCOLEDI' 16 MAGGIO 2012
17:00-18:00, Sala Seminari (Dip. Matematica)
SEMINARI DI CALCOLO DELLE VARIAZIONI E ANALISI GEOMETRICA
Rademacher's theorem for Euclidean measuresAndrea Marchese (Universita` di Pisa)
For every Euclidean Radon measure
μ we state an adapted version of Rademacher's theorem, which is, in a certain sense, the best possible for the measure
μ. We define a sort of fibre bundle (actually a map S that at each point x of R
n associates a vector subspace
S(x) of
T
xR
n, possibly with non-costant dimension
k(x)) such that every Lipschitz functio
n f:R
n→R is differentiable at
x, along
S(x), for
μ-a.e.
x. We prove tha
t S is maximal in the following sense: there exists a Lipschitz function
g:R
n→R which doesn't admit derivative at
μ-a.e.
x, along any direction not belonging to
S(x). Joint work with Giovanni Alberti.