Chaire de la Vallée Poussin 2020-21
Prize lectures by
Guido De Philippis (Courant Institute New York University)
General title:
The interplay between differential constraints and pointwise inclusions: old and new results and applications
Inaugural lecture, Tuesday 14 December 2021 at 4.15 pm
PDE constrained inclusions: A general overview
Several problems in analysis and geometry can be reduced to the study of a pointwise inclusion
coupled with a linear differential constraint. This point of view, starting with the seminal work
of Murat and Tartar, has revealed itself to be extremely fruitful. I will present a general overview
on these ideas with a series of old and new results which can be obtained.
Lecture 1: Wednesday 15 December 2021 at 4.15 pm
PDE constrained inclusions: Oscillation an compensated compactness
Differential constraints can strongly restrict possible oscillation of weakly convergent sequences.
Once this is coupled with a pointwise constraint, it is often possible to recover strong convergence
of the sequence. This is the main idea behind the Murat and Tartar theory of compensated
compactness. I will give some details on these ideas and on their applications.
Lecture 2: Thursday 16 December 2021 at 4.15 pm
PDE constrained inclusions: concentration and fine properties of PDE
While the interplay between a differential constraint and oscillation has been understood for
years, much less is known concerning possible concentration . I will present some recent works
in collaboration with F. Rindler in which we describe the structure of singular measures satisfying
a differential constraint.
Lecture 3: Friday 17 December 2021 at 3.30 pm
An application: The converse of Rademachers theorem
I will show some surprising application of the structural results for measures satisfying a differ-
ential constraint to the study of Lipschitz functions. In particular I will show how combining
the ideas developed by Alberti, Csorneyi, Marchese and Preiss with the result presented in the
previous lecture, one can characterize those measures for which Rademachers theorem holds
true.
