2019
Том 71
№ 11

All Issues

On Compensated Compactness for Nonlinear Elliptic Problems in Perforated Domains

Skrypnik I. V.

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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.

English version (Springer): Ukrainian Mathematical Journal 52 (2000), no. 11, pp 1749-1767.

Citation Example: Skrypnik I. V. On Compensated Compactness for Nonlinear Elliptic Problems in Perforated Domains // Ukr. Mat. Zh. - 2000. - 52, № 11. - pp. 1534-1549.

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