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Differentiability of typical Lipschitz functions

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If you have a question about this talk, please contact Jon Bennett.

This talk is devoted to differentiability properties of Lipschitz functions.

The classical Rademacher Theorem guarantees that every Lipschitz function between finite-dimensional spaces is differentiable almost everywhere. This means that for every set T of positive Lebesgue measure and for every Lipschitz function f defined on the whole space the set of points from T where f is differentiable is non-empty and is ‘much larger’ than the set of points where it is not differentiable.

A major direction in geometric measure theory research of the last two decades has been to explore to what extent this is true for Lebesgue null sets. Even for real-valued Lipschitz functions, there are null subsets S of Rn (with n>1) such that every Lipschitz function on Rn has points of differentiability in S; one says that S is a universal differentiability set (UDS).

Some sets T which are not UDS still have the property that a typical Lipschitz function has points of differentiability in T. We characterise such sets completely in the language of Geometric Measure Theory: these are exactly the sets which cannot be covered by an F-sigma 1-purely unrectifiable set. We also show that for all remaining sets a typical 1-Lipschitz function is nowhere differentiable, even directionally, at each point.

Surprisingly though, no matter how good the set T is, it turns out that a typical 1-Lipschitz function is non-differentiable at a typical point of T. As above, ‘typical’ is used in the sense of Baire category.

The results in the talk are based on two joint papers with Michael Dymond.

This talk is part of the Analysis seminar series.

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