Submitted Paper

Inserted: 13 jun 2003
Last Updated: 2 dec 2003

Year: 2003

Abstract:

The purpose of this paper is to study the lower semicontinuity with respect to the strong $L^1$-convergence, of some integral functionals defined in the space $SBD$ of special functions with bounded deformation. Precisely, let $\Omega\subset R^n$ be an open set. We prove that, if $u\in SBD(\Omega)$, $(u_h)\subset SBD(\Omega)$ converges to $u$ strongly in $L^1(\Omega,R^n)$ and the measures $
E^ju_h
$ converge weakly $*$ to a measure $\nu$ singular with respect to the Lebesgue measure, then $$\int_\Omega f(x,\,\mathcal Eu)dx\leq\liminf_{{h\to\infty}\int\Omega} f(x,\,\mathcal Eu_h)dx$$ provided the integrand $f$ satisfies a weak convexity property and standard growth assumptions of order $p>1$.

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