On Compensated Compactness for Nonlinear Elliptic Problems in Perforated Domains - Skrypnik, I. V. (Ukrains’kyi Matematychnyi Zhurnal, 2000)
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: \(_s → ∞ Ar_s ,z_s = 0\) , 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 \(W_m^1 ^ ° (Ω _s)\) whose weak limit is equal to zero.