# Meso-Scale Approximations of Fields Around Clusters of Defects

## Abstract

We present a review of recent results on asymptotic approximations of solutions to boundary value problems in domains with large clusters of small defects. There are no assumptions which require periodicity within the cluster. The asymptotic approximations we discuss are uniform and include the boundary layers occurring in neighborhoods of singularly perturbed boundaries of the domains concerned. The term “meso-scale” is used to describe these approximations since they go beyond the conventional constraints of the homogenization theory.

- Authors:

- Linköping University (Sweden)
- University of Liverpool (United Kingdom)

- Publication Date:

- OSTI Identifier:
- 22773952

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Mathematical Sciences

- Additional Journal Information:
- Journal Volume: 232; Journal Issue: 4; Other Information: Copyright (c) 2018 Springer Science+Business Media, LLC, part of Springer Nature; http://www.springer-ny.com; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1072-3374

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; APPROXIMATIONS; ASYMPTOTIC SOLUTIONS; BOUNDARY LAYERS; BOUNDARY-VALUE PROBLEMS; LIMITING VALUES; PERIODICITY; REVIEWS

### Citation Formats

```
Maz’ya, V. G., E-mail: vladimir.mazya@liu.se, and Movchan, A. B., E-mail: abm@liverpool.ac.uk.
```*Meso-Scale Approximations of Fields Around Clusters of Defects*. United States: N. p., 2018.
Web. doi:10.1007/S10958-018-3883-0.

```
Maz’ya, V. G., E-mail: vladimir.mazya@liu.se, & Movchan, A. B., E-mail: abm@liverpool.ac.uk.
```*Meso-Scale Approximations of Fields Around Clusters of Defects*. United States. doi:10.1007/S10958-018-3883-0.

```
Maz’ya, V. G., E-mail: vladimir.mazya@liu.se, and Movchan, A. B., E-mail: abm@liverpool.ac.uk. Sun .
"Meso-Scale Approximations of Fields Around Clusters of Defects". United States. doi:10.1007/S10958-018-3883-0.
```

```
@article{osti_22773952,
```

title = {Meso-Scale Approximations of Fields Around Clusters of Defects},

author = {Maz’ya, V. G., E-mail: vladimir.mazya@liu.se and Movchan, A. B., E-mail: abm@liverpool.ac.uk},

abstractNote = {We present a review of recent results on asymptotic approximations of solutions to boundary value problems in domains with large clusters of small defects. There are no assumptions which require periodicity within the cluster. The asymptotic approximations we discuss are uniform and include the boundary layers occurring in neighborhoods of singularly perturbed boundaries of the domains concerned. The term “meso-scale” is used to describe these approximations since they go beyond the conventional constraints of the homogenization theory.},

doi = {10.1007/S10958-018-3883-0},

journal = {Journal of Mathematical Sciences},

issn = {1072-3374},

number = 4,

volume = 232,

place = {United States},

year = {2018},

month = {7}

}

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