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

## Camillo De Lellis (Institute for Advanced Study, Princeton, and University of Zuerich)

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 Müller 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.

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