Calculus of Variations and Geometric Measure Theory
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Non-differentiability sets of Lipschitz functions

Andrea Merlo

created by delnin on 25 May 2018

29 may 2018 -- 15:00   [open in google calendar]

Aula Magna, Dip. di Matematica, Univ. di Pisa

Abstract.

The study of differentiability properties of Lipschitz functions has a long story. It started with H. Lebesgue who proved the almost everywhere differentiability of Lipschitz functions on the real line. In 1919 H. Rademacher understood that this almost everywhere differentiability was not just a property of the line itself, but a property of Lipschitz maps themselves.

Such a beautiful result left a lot of questions open one of which is: does a viceversa hold for the Rademacher theorem, i.e., is it possible to give a characterization of non-differentiability sets of Lipschitz functions?

In 1990 D. Preiss built a dense set in euclidean spaces on which every real valued Lipschitz function has a differentiability point. This amazing and counterintuitive result made it clear that a converse for the Rademacher's Theorem was not a straightforward problem at all.

In the talk I will present the main ideas and techniques that have been originally introduced by G. Alberti, M. Csörnyei and D. Preiss in their (still) unpublished paper to study this problem. In particular, I will describe: how the width function is defined; some of its applications in the plane; how to build a lot of non differentiable functions along every line, at any point of a given compact purely unrectifiable set.

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