On the relaxation of integral functionals depending on the symmetrized gradient

Abstract

We prove results on the relaxation and weak* lower semicontinuity of integral functionals of the form \[ \mathcal{F}[u] := \int_{\Omega} f \bigg( \frac{1}{2} \bigl( \nabla u(x) + \nabla u(x)^T \bigr) \bigg)\,\mathrm{d} x, \qquad u : \Omega \subset \mathbb{R}^d \to \mathbb{R}^d, \] over the space $\mathrm{BD}(\Omega)$ of functions of bounded deformation or over the Temam-Strang space \[ \mathrm{U}(\Omega):=\bigl\{u\in \mathrm{BD}(\Omega): \ \mathrm{div} \ u\in \mathrm{L}^2(\Omega)\bigr\}, \] depending on the growth and shape of the integrand $f$. Such functionals are interesting for example in the study of Hencky plasticity and related models.