# Localized Modes Due to Defects in High Contrast Periodic Media Via Two-Scale Homogenization

## Abstract

The spectral problem for an infinite periodic medium perturbed by a compact defect is considered. For a high contrast small ε-size periodicity and a finite size defect we consider the critical ε{sup 2}-scaling for the contrast. We employ two-scale homogenization for deriving asymptotically explicit limit equations for the localized modes and associated eigenvalues. Those are expressed in terms of the eigenvalues and eigenfunctions of a perturbed version of a two-scale limit operator introduced by V. V. Zhikov with an emergent explicit nonlinear dependence on the spectral parameter for the spectral problem at the macroscale. Using the method of asymptotic expansions supplemented by a high contrast boundary layer analysis, we establish the existence of the actual eigenvalues near the eigenvalues of the limit operator, with “ε square root” error bounds. An example for circular or spherical defects in a periodic medium with isotropic homogenized properties is given.

- Authors:

- University College London (United Kingdom)

- Publication Date:

- OSTI Identifier:
- 22773998

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Mathematical Sciences

- Additional Journal Information:
- Journal Volume: 232; Journal Issue: 3; 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; ASYMPTOTIC SOLUTIONS; BOUNDARY LAYERS; DISTURBANCES; EIGENFUNCTIONS; EIGENVALUES; EQUATIONS; ERRORS; NONLINEAR PROBLEMS; PERIODICITY; SCALE MODELS; SPHERICAL CONFIGURATION

### Citation Formats

```
Kamotski, I. V., E-mail: i.kamotski@ucl.ac.uk, and Smyshlyaev, V. P., E-mail: v.smyshlyaev@ucl.ac.uk.
```*Localized Modes Due to Defects in High Contrast Periodic Media Via Two-Scale Homogenization*. United States: N. p., 2018.
Web. doi:10.1007/S10958-018-3877-Y.

```
Kamotski, I. V., E-mail: i.kamotski@ucl.ac.uk, & Smyshlyaev, V. P., E-mail: v.smyshlyaev@ucl.ac.uk.
```*Localized Modes Due to Defects in High Contrast Periodic Media Via Two-Scale Homogenization*. United States. doi:10.1007/S10958-018-3877-Y.

```
Kamotski, I. V., E-mail: i.kamotski@ucl.ac.uk, and Smyshlyaev, V. P., E-mail: v.smyshlyaev@ucl.ac.uk. Sun .
"Localized Modes Due to Defects in High Contrast Periodic Media Via Two-Scale Homogenization". United States. doi:10.1007/S10958-018-3877-Y.
```

```
@article{osti_22773998,
```

title = {Localized Modes Due to Defects in High Contrast Periodic Media Via Two-Scale Homogenization},

author = {Kamotski, I. V., E-mail: i.kamotski@ucl.ac.uk and Smyshlyaev, V. P., E-mail: v.smyshlyaev@ucl.ac.uk},

abstractNote = {The spectral problem for an infinite periodic medium perturbed by a compact defect is considered. For a high contrast small ε-size periodicity and a finite size defect we consider the critical ε{sup 2}-scaling for the contrast. We employ two-scale homogenization for deriving asymptotically explicit limit equations for the localized modes and associated eigenvalues. Those are expressed in terms of the eigenvalues and eigenfunctions of a perturbed version of a two-scale limit operator introduced by V. V. Zhikov with an emergent explicit nonlinear dependence on the spectral parameter for the spectral problem at the macroscale. Using the method of asymptotic expansions supplemented by a high contrast boundary layer analysis, we establish the existence of the actual eigenvalues near the eigenvalues of the limit operator, with “ε square root” error bounds. An example for circular or spherical defects in a periodic medium with isotropic homogenized properties is given.},

doi = {10.1007/S10958-018-3877-Y},

journal = {Journal of Mathematical Sciences},

issn = {1072-3374},

number = 3,

volume = 232,

place = {United States},

year = {2018},

month = {7}

}