Abstract for the talk at 21.02.2012 (11:30 h)Arbeitsgemeinschaft ANGEWANDTE ANALYSIS
Rademacher's teorem for euclidean measures
For every euclidean Radon measure μ we state an adapted version of Rademacher’s theorem, which is, in a certain sense, optimal. We define a sort of fibre bundle (actually just a map S that at each point x ∈ ℝn associates a vector subspace S(x) of Txℝn, possibly with non-costant dimension k(x)) such that every Lipschitz function f : ℝn → ℝ is differentiable at x, along the k(x)-plane S(x), for μ-a.e. x. We prove that S is maximal in the following sense: if T is a vector space such that T(x) ⁄∈ S(x) whenever k(x)≠n, then there exists a Lipschitz function f : ℝn → ℝ which doesn’t admit any derivative in the direction V (x) for μ-a.e. x satisfying k(x)≠n. Joint work with Giovanni Alberti.