# A unifying perspective on linear continuum equations prevalent in physics. Part VII: Boundary value and scattering problems

@inproceedings{Milton2021AUP, title={A unifying perspective on linear continuum equations prevalent in physics. Part VII: Boundary value and scattering problems}, author={Graeme W. Milton}, year={2021} }

We consider simply connected bodies or regions of finite extent in space or space-time and write conservation laws associated with the equations in Parts I-IV. We review earlier work where, for elliptic equations, the boundary value problem is reformulated as a problem in the abstract theory of composites and the associated effective operator is equated with the Dirichlet-to-Neumann map that governs the response of the body. The dielectric polarizability problem and acoustic and electromagnetic… Expand

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A unifying perspective on linear continuum equations prevalent in physics. Part II: Canonical forms for time-harmonic equations

- Physics, Mathematics
- 2020

Following some past advances, we reformulate a large class of linear continuum science equations in the format of the extended abstract theory of composites so that we can apply this theory to better… Expand

#### References

SHOWING 1-10 OF 31 REFERENCES

A unifying perspective on linear continuum equations prevalent in physics. Part I: Canonical forms for static and quasistatic equations

- Physics
- 2020

Following some past advances, we reformulate a large class of linear continuum physical equations in the format of the extended abstract theory of composites so that we can apply this theory to… Expand

On cloaking for elasticity and physical equations with a transformation invariant form

- Physics
- 2006

In this paper, we investigate how the form of the conventional elastodynamic equations changes under curvilinear transformations. The equations get mapped to a more general form in which the density… Expand

Variational principles for complex conductivity, viscoelasticity, and similar problems in media with complex moduli

- Physics
- 1994

Linear processes in media with dissipation arising in conductivity, optics, viscoelasticity, etc. are considered. Time‐periodic fields in such media are described by linear differential equations for… Expand

Propagation and localization of elastic waves in highly anisotropic periodic composites via two-scale homogenization

- Physics
- 2009

Abstract Wave propagation in periodic elastic composites whose phases may have not only highly contrasting but possibly also (in particular) highly anisotropic stiffnesses and moderately contrasting… Expand

Bounds on complex polarizabilities and a new perspective on scattering by a lossy inclusion

- Physics, Mathematics
- 2017

Here we obtain explicit formulae for bounds on the complex electrical polarizability at a given frequency of an inclusion with known volume that follow directly from the quasistatic bounds of Bergman… Expand

On modifications of Newton's second law and linear continuum elastodynamics

- Mathematics
- Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2007

In this paper, we suggest a new perspective, where Newton's second law of motion is replaced by a more general law which is a better approximation for describing the motion of seemingly rigid… Expand

A unifying perspective on linear continuum equations prevalent in science. Part VI: rapidly converging series expansions for their solution

- Physics, Mathematics
- 2020

We obtain rapidly convergent series expansions of resolvents of operators taking the form ${\bf A}=\Gamma_1{\bf B}\Gamma_1$ where $\Gamma_1({\bf k})$ is a projection that acts locally in Fourier… Expand

The Theory of Composites

- Mathematics
- 2002

Einstein, have contributed to the theory of composite materials. Mathematically, it is the study of partial differential equations with rapid oscillations in their coefficients. Although extensively… Expand

Anisotropic mass density by two-dimensional acoustic metamaterials

- Physics
- 2008

We show that specially designed two-dimensional arrangements of full elastic cylinders embedded in a nonviscous fluid or gas define (in the homogenization limit) a new class of acoustic metamaterials… Expand

A unifying perspective on linear continuum equations prevalent in physics. Part V: resolvents; bounds on their spectrum; and their Stieltjes integral representations when the operator is not selfadjoint

- Physics, Mathematics
- 2020

We consider resolvents of operators taking the form ${\bf A}=\Gamma_1{\bf B}\Gamma_1$ where $\Gamma_1({\bf k})$ is a projection that acts locally in Fourier space and ${\bf B}({\bf x})$ is an… Expand