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, we prove that, if u ∈ SBD ( Ω ) , ( u h ) ⊂ SBD ( Ω ) converges to u strongly in L 1 ( Ω , R n ) and the measures | E j u h | converge weakly * to a measure ν singular with respect to the Lebesgue measure, then ∫ Ω f ( x , E u ) d x ⩽ lim inf h → ∞ ∫ Ω f ( x , E u h ) d x provided the integrand f satisfies a weak convexity property and standard growth assumptions of order p > 1 .