Contents
Preface
Introduction


1. Scalar wave scattering by one small body of an arbitrary shape
1.1 Impedance bodies
1.2 Acoustically soft bodies (the Dirichlet boundary condition)
1.3 Acoustically hard bodies (the Neumann boundary condition)
1.4 The interface (transmission) boundary condition
1.5 Summary of the results


2. Scalar wave scattering by many small bodies of an arbitrary shape
2.1 Impedance bodies
2.2 The Dirichlet boundary condition
2.3 The Neumann boundary condition
2.4 The transmission boundary condition
2.5 Wave scattering in an inhomogeneous medium
2.6 Summary of the results


3. Creating materials with a desired refraction coefficient
3.1 Scalar wave scattering. Formula for the refraction coefficient
3.2 A recipe for creating materials with a desired refraction coefficient
3.3 A discussion of the practical implementation of the recipe
3.4 Summary of the results


4. Wave-focusing materials
4.1 What is a wave-focusing material?
4.2 Creating wave-focusing materials
4.3 Computational aspects of the problem
4.4 Open problems
4.5 Summary of the results


5. Electromagnetic wave scattering by a single small body of an arbitrary shape
5.1 The impedance boundary condition
5.2 Perfectly conducting bodies
5.3 Formulas for the scattered field in the case of EM wave scattering by one impedance small body of an arbitrary shape
5.4 Summary of the results


6. Many-body scattering problem in the case of small scatterers
6.1 Reduction of the problem to linear algebraic system
6.2 Derivation of the integral equation for the effective field
6.3 Summary of the results


7. Creating materials with a desired refraction coefficient
7.1 A formula for the refraction coefficient
7.2 Formula for the magnetic permeability
7.3 Summary of the results


8. Electromagnetic wave scattering by many nanowires
8.1 Statement of the problem
8.2 Asymptotic solution of the problem
8.3 Many-body scattering problem equation for the effective field
8.4 Physical properties of the limiting medium
8.5 Summary of the results


9. Heat transfer in a medium in which many small bodies are embedded
9.1 Introduction
9.2 Derivation of the equation for the limiting temperature
9.3 Various results
9.4 Summary of the results


10. Quantum-mechanical wave scattering by many potentials with small support
10.1 Problem formulation
10.2 Proofs
10.3 Summary of the results


11. Some results from the potential theory
11.1 Potentials of the simple and double layers
11.2 Replacement of the surface potentials
11.3 Asymptotic behavior of the solution to the Helmholtz equation under the impedance boundary condition
11.4 Some properties of the electrical capacitance
11.5 Summary of the results


12. Collocation method
12.1 Convergence of the collocation method
12.2 Collocation method and homogenization
12.3 Summary of the results


13. Some inverse problems related to small scatterers
13.1 Finding the position and size of a small body from the scattering data
13.2 Finding small subsurface inhomogeneities
13.3 Inverse radio measurements problem
13.4 Summary of the results


Appendix
A1. Banach and Hilbert spaces
A2. A result from perturbation theory
A3. The Fredholm alternative
Bibliographical notes
Bibliography
Index.