Homogenization

Andrii Khrabustovskyi works at our faculty in the group Nonlinear
Partial Differential Equations and is a member of the CRC Wave
phenomena: analysis and numerics. He was born in Kharkiv in the
Ukraine and finished his studies as well as his PhD at the
Kharkiv National University and the Institute for Low Temperature
Physics and Engineering of the National Academy of Sciences of
Ukraine. He joined our faculty in 2012 as postdoc in the former
Research Training Group 1294 Analysis, Simulation and Design of
Nanotechnological Processes, which was active until 2014. Gudrun
Thäter talked with him about one of his research interests
Asymptotic analysis and homogenization of PDEs.


Photonic crystals are periodic dielectric media in which
electromagnetic waves from certain frequency ranges cannot
propagate. Mathematically speaking this is due to gaps in the
spectrum of the related differential operators. For that an
interesting question is if there are gaps inbetween bands of the
spectrum of operators related to wave propagation, especially on
periodic geometries and with periodic coeffecicients in the
operator. It is known that the spectrum of periodic selfadjoint
operators has bandstructure. This means the spectrum is a locally
finite union of compact intervals called bands. In general, the
bands may overlap and the existence of gaps is therefore not
guaranteed. A simple example for that is the spectrum of the
Laplacian in L2(Rn) which is the half axis [0,∞).
The classic approach to such problems in the whole space case is
the Floquet–Bloch theory.


Homogenization is a collection of mathematical tools which are
applied to media with strongly inhomogeneous parameters or highly
oscillating geometry. Roughly spoken the aim is to replace the
complicated inhomogeneous by a simpler homogeneous medium with
similar properties and characteristics. In our case we deal with
PDEs with periodic coefficients in a periodic geometry which is
considered to be infinite. In the limit of a characteristic small
parameter going to zero it behaves like a corresponding
homogeneous medium. To make this a bit more mathematically
rigorous one can consider a sequence of operators with a small
parameter (e.g. concerning cell size or material properties) and
has to prove some properties in the limit as the parameter goes
to zero. The optimal result is that it converges to some operator
which is the right homogeneous one. If this limit operator has
gaps in its spectrum then the gaps are present in the spectra of
pre-limit operators (for small enough parameter).


The advantages of the homogenization approach compared to the
classical one with Floquet Bloch theory are:
The knowledge of the limit operator is helpful and only
available through homogenization. For finite domains Floquet Bloch
does not work well. Though we always have a discrete spectrum we
might want to have the gaps in fixed position independent of the
size of our domain. Here the homogenization theory works in
principle also for the bounded case (it is just a bit technical).

An interesting geometry in this context is a domain with
periodically distributed holes. The question arises: what happens
if the sizes of holes and the period simultaneously go to zero?
The easiest operator which we can study is the Laplace operator
subject to the Dirichlet boundary conditions. There are three
possible regimes:
For holes of the same order as the period (even slightly
smaller) the Dirichelet conditions on the boundary of holes
dominate -- the solution for the corresponding Poisson equation
tends to zero. For significantly smaller holes the influence on the
holes is so small that the problem "forgets" about the influence of
the holes as the parameter goes to zero. There is a borderline case
which lies between cases 1 and 2. It represents some interesting
effects and can explain the occurance of so-called strange terms.

A traditional ansatz in homogenization works with the concept of
so-called slow and fast variables. The name comes from the
following observation. If we consider an infinite layer in
cylindrical coordinates, then the variable r measures the
distance from the origin when going "along the layer", ϕ the
angle in that plane, and z is the variable which goes into the
finite direction perpendicular to that plane. When we have
functions y(x)=rkf(z,ϕ) then the derivative with respect to r
changes the power to rk−1 while the other derivatives leave that
power unchanged. In the interesting case k is negative and the
r-derivate makes it decreasing even faster. This leads to the
name fast variable. The properties in this simple example
translate as follows. For any function u we will think of having
a set of slow and fast variables (characteristic to the problem)
and a small parameter eps and try to find u as
u=u0(x)+εu1(x,y)+ε2u2(x,y)+... where in our applications
typically y=x/ε. One can formally sort through the ε-levels using
the properties of the differential operator. The really hard part
then is to prove that this formal result is indeed true by
finding error estimates in the right (complicated) spaces.


There are many more tools available like the technique of
Tartar/Murat, who use a weak formulation with special test
functions depending on the small parameter. The weak point of
that theory is that we first have to know the resulat as the
parameter goes to zero before we can to construct the test
function. Also the concept of Gamma convergence or the unfolding
trick of Cioranescu are helpful.


An interesting and new application to the mathematical results is
the construction of wave guides. The corresponding domain in
which we place a waveguide is bounded in two directions and
unbounded in one (e.g. an unbounded cylinder).
Serguei Nazarov proposed to make holes in order to make gaps into
the line of the spectrum for a specified wave guide. Andrii
Khrabustovskyi suggests to distribute finitely many traps, which
do not influence the essential spectrum but add eigenvalues. One
interesting effect is that in this way one can find terms which
are nonlocal in time or space and thus stand for memory effects
of the material.
References

P. Exner and A. Khrabustovskyi: On the spectrum of narrow
Neumann waveguide with periodically distributed δ′ traps, Journal
of Physics A: Mathematical and Theoretical, 48 (31) (2015),
315301.

A. Khrabustovskyi: Opening up and control of spectral gaps of
the Laplacian in periodic domains, Journal of Mathematical
Physics, 55 (12) (2014), 121502.

A. Khrabustovskyi: Periodic elliptic operators with
asymptotically preassigned spectrum, Asymptotic Analysis, 82
(1-2) (2013), 1-37.

S.A. Nazarov, G. Thäter: Asymptotics at infinity of solutions
to the Neumann problem in a sieve-type layer, Comptes Rendus
Mecanique 331(1) (2003) 85-90.

S.A. Nazarov: Asymptotic Theory of Thin Plates and Rods:
Vol.1. Dimension Reduction and Integral Estimates. Nauchnaya
Kniga: Novosibirsk, 2002.