# 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.

Published

25.11.2000

How to Cite

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 52, no. 11, Nov. 2000, pp. 1534-49, https://umj.imath.kiev.ua/index.php/umj/article/view/4559.

Issue

Section

Research articles