ABSTRACT

A classical theorem due to Liouville says that if the gradient of 
a map is a rotation at every point, then it is a constant rotation.

The rigidity estimate of Friesecke, James and Mueller (2002) quantifies 
this: for maps $v: U\subset \mathbb{R}^n \to \mathbb{R}^n$ 
the $L^2$-distance of $\beta=\nabla u$ from a single rotation is bounded 
by its $L^2$-distance from the group of all rotations.

Stefan Mueller, Caterina Zeppieri and myself recently proved a generalised 
rigidity estimate for matrix-valued fields $\beta$ that are 
not necessarily gradients, in dimension 2. In this case it is still 
possible to bound the $L^2$-distance of $\beta$ from a single rotation 
in terms of its $L^2$-distance from the rotations, modulo an error depending 
on the total variation of $\textrm{Curl} \beta$. One of the key 
ingredients in our proof is a fine regularity result for two-dimensional 
$L^1$-vector fields with divergence in $H^{-2}$ proved by Brezis 
and van Schaftingen.

In this talk I will discuss generalised rigidity estimates and their role 
in dislocation theory.