Seminar Analysis: Filip Rindler (University of Warwick)

Rademacher’s Theorem, Cheeger’s Conjecture and PDEs for measures

The classical Rademacher Theorem asserts that every Lipschitz function is differentiablea lmost everywhere with respect to Lebesgue measure. On the other hand, Preiss (’90) gave a surprising example of a nullset in the plane such that every Lipschitz function is differentiable at at least one point of this set. Thus, it is a natural question to ask whether there exists a singular measure such that all Lipschitz functions are differentiable with respect to this singular measure. It turns out that this question has an intricate connection to the geometric structure of normal one-currents. In this talk I will present a converse to Rademacher’s Theorem, which settles the question in the negative in all dimensions: if a positive measure μ has the property that all Lipschitz functions are μ-a.e. differentiable, then μ is absolutely continuous with respect to Lebesgue measure (in the plane, this question was already solved by Alberti, Csornyei and Preiss in ’05). In a geometric context, Cheeger conjectured in ’99 that in all Lipschitz differentiability spaces (which are essentially Lipschitz manifolds in which Rademacher’s Theorem holds) likewise there is a “functional converse” to Rademacher’s Theorem. As the second main result, I will present a recent solution to this conjecture.Technically, the proofs of both of these theorems are based on a recent structure result for the singular parts of PDE-constrained measures, its corollary on the structure of normalone-currents, and the powerful theory of Alberti representations.

This is a joint work with A. Marchese and G. De Philippis

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