SEMINAR @ SISSSA, Prof. C. De Lellis course - July 2007

[Barbara Corzani]

Course announcement

Prof. Camillo De Lellis, Department of Mathematics ETH, Zurich, will give a course in SISSA during the first two weeks of July. The first Lecture will be July 4, 11.00, in room B.

Title: "Differential inclusions in the calculus of variations, in
geometry and in fluid dynamics"

In these lectures I will address a theme which is common
to apparently unrelated problems coming from several different areas:
the existence of surprisingly irregular solutions to certain nonlinear partial differential equations.
The examples that I would like to touch in these lectures are:

The exotic $C^1$ isometric embeddings of Nash;

The existence of Lipschitz but nowhere differentiable solutions
to elliptic systems of partial differential equations, due to M\"uller
and Sverak;

The existence of Lipschitz maps taking values in finite
sets of incompatible gradients, due to Kirchheim;

The existence of energy--decreasing solutions to the Euler
equations.

The proofs that we will see in these lectures rely on techniques
which exploit similar ideas in different contexts: Gromov's convex
integration, its ``rougher'' variant by M\"uller and Sverak
and Baire Category arguments (the latter being first pioneered by
Cellina and Perrotta in the early eighties and then rediscovered by
several authors in the last years). We will also address Tartar's
wave analysis: a general method to investigate whether for a given PDE
one can use similar constructions to generate very irregular solutions.