On Compensated Compactness for Nonlinear Elliptic Problems in Perforated Domains

Abstract

We consider a sequence of Dirichlet problems for a nonlinear divergent operator A: W _m ¹(Ω _s ) → [W _m ¹(Ω _s )]^* in a sequence of perforated domains Ω _s ⊂ Ω. Under a certain condition imposed on the local capacity of the set Ω \ Ω _s , we prove the following principle of compensated compactness: ${\mathop {\lim }\limits_{s \to \infty }} \left\langle {Ar_s ,z_s } \right\rangle = 0$ , where r _s(x) and z _s(x) are sequences weakly convergent in W _m ¹(Ω) and such that r _s(x) is an analog of a corrector for a homogenization problem and z _s(x) is an arbitrary sequence from ${\mathop {W_m^1 }\limits^ \circ} (\Omega _s)$ whose weak limit is equal to zero.