# Some remarks on the theory of elasticity for compressible eohookean materials

##
Camillo De Lellis
(Institute for Advanced Study, Princeton)

created by tomarelli on 03 Dec 2001

6 dec 2001

**Abstract.**

Giovedi' 6 dicembre alle 18 i seminario di Calcolo delle Variazioni sara'
tenuto da Camillo De Lellis, ecco titolo ed abstract:

Some remarks on the theory of elasticity for compressible eohookean materials

In this talk I will present some results contained in a joint work
with Sergio Conti.
In Neohookean elasticity one minimizes functionals which depend
on the $L^2$ norm of the deformation gradient, plus a nonlinear
function of the determinant, with some notion of invertibility to
represent non-interpenetrability of matter.
An existence theory which includes a precise notion of invertibility
and allows for cavitation was formulated by MÃ¼ller and Spector,
however only for the case where some $L^p$-norm of the gradient with
$p>2$ is controlled (in three dimensions).
We first characterize their class of functions
in terms of properties of the associated rectifiable current. Then
we address the physically relevant $p=2$ case, and show how their
notion of invertibility can be extended to $p=2$. The class of functions
so obtained is however not closed. We prove this by giving an explicit
construction, which has interesting consequences even in other
frameworks.