On Compensated Compactness for Nonlinear Elliptic Problems in Perforated Domains

  • I. V. Skrypnik

Abstract

We consider a sequence of Dirichlet problems for a nonlinear divergent operator A: W m 1 s ) → [W m 1 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.
Published
25.11.2000
How to Cite
SkrypnikI. V. “On Compensated Compactness for Nonlinear Elliptic Problems in Perforated Domains”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 52, no. 11, Nov. 2000, pp. 1534-49, https://umj.imath.kiev.ua/index.php/umj/article/view/4559.
Section
Research articles