A lower semicontinuity result for some integral functionals in the space SBD

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 $U$ be a bounded open subset of $R^n$. If $u\in $SBD$(U)$, $(u_h)\subset $SBD$(U)$ converges to $u$ strongly in $L^1(U,R^n)$ and the measures $|E^ju_h|$ converge weakly * to a measure $ν$ singular with respect to the Lebesgue measure, then $$\int_Uf(x,{\mathcal E}u)dx\leq\liminf_{h\to\infty} \int_Uf(x,{\mathcal E}u_h)dx$$ provided $f$ satisfies some weak convexity property and the standard growth assumptions of order $p>1$.