# Several versions of the compensated compactness principle

## Abstract

The convergence of the product of a solenoidal vector w{sub {epsilon}} and a gradient {nabla}u{sub {epsilon}} in L{sup 1}({Omega}) (where {Omega} is a region in R{sup d}) is investigated in the case when the factors converge weakly in the spaces L{sup {gamma}({Omega})}{sup d} and L{sup {alpha}({Omega})}{sup d}, respectively, with 1/{gamma}+1/{alpha}>1, which means that the main assumption of the classical div-curl lemma fails. Nevertheless, the same convergence (in the sense of distributions in {Omega}) as in the framework of the div-curl lemma, survives under certain additional assumptions. The new versions of the compensated compactness principle proved in the paper can be used in homogenization and in the theory of G-convergence of monotone operators with non-standard coercivity and growth properties, for instance, some degenerate operators. Bibliography: 20 titles.

- Authors:

- Publication Date:

- OSTI Identifier:
- 21612604

- Resource Type:
- Journal Article

- Journal Name:
- Sbornik. Mathematics

- Additional Journal Information:
- Journal Volume: 202; Journal Issue: 9; Other Information: DOI: 10.1070/SM2011v202n09ABEH004192; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1064-5616

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; COERCIVE FORCE; CONVERGENCE; DISTRIBUTION; MATHEMATICAL SPACE; VECTORS; SPACE; TENSORS

### Citation Formats

```
Pastukhova, Svetlana E, and Khripunova, Anna S.
```*Several versions of the compensated compactness principle*. United States: N. p., 2011.
Web. doi:10.1070/SM2011V202N09ABEH004192; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).

```
Pastukhova, Svetlana E, & Khripunova, Anna S.
```*Several versions of the compensated compactness principle*. United States. doi:10.1070/SM2011V202N09ABEH004192; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).

```
Pastukhova, Svetlana E, and Khripunova, Anna S. Fri .
"Several versions of the compensated compactness principle". United States. doi:10.1070/SM2011V202N09ABEH004192; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).
```

```
@article{osti_21612604,
```

title = {Several versions of the compensated compactness principle},

author = {Pastukhova, Svetlana E and Khripunova, Anna S},

abstractNote = {The convergence of the product of a solenoidal vector w{sub {epsilon}} and a gradient {nabla}u{sub {epsilon}} in L{sup 1}({Omega}) (where {Omega} is a region in R{sup d}) is investigated in the case when the factors converge weakly in the spaces L{sup {gamma}({Omega})}{sup d} and L{sup {alpha}({Omega})}{sup d}, respectively, with 1/{gamma}+1/{alpha}>1, which means that the main assumption of the classical div-curl lemma fails. Nevertheless, the same convergence (in the sense of distributions in {Omega}) as in the framework of the div-curl lemma, survives under certain additional assumptions. The new versions of the compensated compactness principle proved in the paper can be used in homogenization and in the theory of G-convergence of monotone operators with non-standard coercivity and growth properties, for instance, some degenerate operators. Bibliography: 20 titles.},

doi = {10.1070/SM2011V202N09ABEH004192; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA)},

journal = {Sbornik. Mathematics},

issn = {1064-5616},

number = 9,

volume = 202,

place = {United States},

year = {2011},

month = {9}

}