Download The general theory of homogenization: A personalized introduction PDF

TitleThe general theory of homogenization: A personalized introduction
LanguageEnglish
File Size2.8 MB
Total Pages493
Table of Contents
                            Cover
Lecture Notes of
the Unione Matematica Italiana, 7
	The General Theory
of Homogenization
		ISBN 9783642051944
		Dedication
		Preface
		Contents
1 Why Do I Write?
2 A Personalized Overview of Homogenization I
3 A Personalized Overview of Homogenization II
4 An Academic Question of Jacques-Louis Lions
5 A Useful Generalization by François Murat
6 Homogenization of an Elliptic Equation
7 The Div–Curl Lemma
8 Physical Implications of Homogenization
9 A Framework with Differential Forms
10 Properties of H-Convergence
11 Homogenization of Monotone Operators
12 Homogenization of Laminated Materials
13 Correctors in Linear Homogenization
14 Correctors in Nonlinear Homogenization
15 Holes with Dirichlet Conditions
16 Holes with Neumann Conditions
17 Compensated Compactness
18 A Lemma for Studying Boundary Layers
19 A Model in Hydrodynamics
20 Problems in Dimension N = 2
21 Bounds on Effective Coefficients
22 Functions Attached to Geometries
23 Memory Effects
24 Other Nonlocal Effects
25 The Hashin–Shtrikman Construction
26 Confocal Ellipsoids and Spheres
27 Laminations Again, and Again
28 Wave Front Sets, H-Measures
29 Small-Amplitude Homogenization
30 H-Measures and Bounds on Effective Coefficients
31 H-Measures and Propagation Effects
32 Variants of H-Measures
33 Relations Between Young Measuresand H-Measures
34 Conclusion
35 Biographical Information
36 Abbreviations and Mathematical Notation
References
                        
Document Text Contents
Page 2

Lecture Notes of 7
the Unione Matematica Italiana

Page 246

Chapter 21

Bounds on Effective Coefficients

I mentioned the use of symmetries for showing that the effective conductiv-
ity of a checkerboard is isotropic, or simply diagonal in the Mortola–Steffé
conjecture; it follows from using mirror symmetries in Lemma 21.1.

Lemma 21.1. If An ∈ M(α, β;Ω) H-converges to Aeff , if ϕ is a diffeomor-
phism from Ω onto ϕ(Ω), and Bn is defined in ϕ(Ω) by

Bn
(
ϕ(x)

)
=

1
det

(
∇ϕ(x)

)∇ϕ(x)An(x)∇ϕT (x) a.e. x ∈ Ω, (21.1)

then Bn ∈ M
(
α′, β′;ϕ(Ω)

)
H-converges to Beff , defined in ϕ(Ω) by

Beff
(
ϕ(x)

)
=

1
det

(
∇ϕ(x)

)∇ϕ(x)Aeff (x)∇ϕT (x) a.e. x ∈ Ω. (21.2)

Proof. If −div
(
An grad(un)

)
= f in Ω, one defines vn in ϕ(Ω) by

vn = un ◦ ϕ−1 in ϕ(Ω), i.e., un = vn ◦ ϕ in Ω, (21.3)
grad(un) = ∇ϕT grad(vn) ◦ ϕ in Ω. (21.4)

In the case f ∈ L2(Ω),1 one writes the equation in variational form


Ω

(
An grad(un), grad(w)

)
dx =



Ω

f w dx for all w ∈ C1c (Ω), (21.5)

and, after making the change of variables x = ϕ(y), one deduces that

− div
(
Bn grad(vn)

)
= g in ϕ(Ω), g

(
ϕ(x)

)
=

1
det

(
∇ϕ(x)

)f(x) a.e. x ∈ Ω.

(21.6)

1 The generalization to the case f ∈ H−1(Ω) is straightforward.

L. Tartar, The General Theory of Homogenization, Lecture Notes
of the Unione Matematica Italiana 7, DOI 10.1007/978-3-642-05195-1 21,
c© Springer-Verlag Berlin Heidelberg 2009

223

Page 247

N:\Springer\Lenz\Titelei\lnp\lnpe.dvi


224 21 Bounds on Effective Coefficients

The L2(ϕ(Ω); RN ) weak limits of grad(vn) and Bn grad(vn) are deduced
from grad(u∞) and Aeff grad(v∞), and it gives (21.2) for Beff .


One often uses Lemma 21.1 when ϕ is a rotation (modulo a translation), for
example in proving that sets like K(θ) in (20.2) are defined by constraints on
the eigenvalues of Aeff . My initial method with François MURAT for finding
bounds, and the generalization that I made, are not restricted to symmetric
An, so they apply to H-convergence, but I lack physical intuition for the non-
symmetric case,2 for which I do not know what question to ask concerning
rotations. For a similar reason, I chose not to work on questions of homog-
enization in linearized elasticity, a theory which is not frame-indifferent, so
that unrealistic effects deprive the mathematical results of much of their
value!

Sergio SPAGNOLO proved a generalization of Lemma 21.1, by using a se-
quence ϕn, with uniform bounds for the partial derivatives of the components
of ϕn and those of its inverse ψn, and using the Reshetnyak theorem for pass-
ing to the limit in the Jacobian determinant appearing in (21.6) for gn.

Definition 21.2. For θ ∈ (0, 1), 0 < α ≤ β <∞,

λ−(θ) =
( θ
α

+
1 − θ
β

)−1
, λ+(θ) = θ α+ (1 − θ)β, (21.7)

B(θ) = {A ∈ Lsym(RN ; RN ) | λ−(θ)I ≤ A ≤ λ+(θ)I}, (21.8)
H(θ) = {A ∈ B(θ) | det(A) = λ−(θ)λN−1+ (θ)}, (21.9)

K(θ) = {Aeff | for mixtures using proportions θ, 1 − θ}. (21.10)

The “definition” (21.10) is only an intuitive idea, and must be explained.
In the early 1970s, François MURAT and myself used the intuition that if one
mixes materials which were obtained as mixtures of some initial materials,
then the result can be obtained by mixing directly the initial materials in
an adapted way, and from the mathematical point of view, it is here that
the metrizability property of H-convergence is important: one is looking at
the closure of a set containing the tensors of the form

(
αχ + β (1 − χ)

)
I

with χ being the characteristic function of an arbitrary measurable set (or
an open set); one identifies some first-generation sets contained in the se-
quential closure of the initial set, then one identifies some second-generation
sets contained in the sequential closure of some first-generation sets, and one
repeats the process finitely many times, and since the topology is metrizable
every set constructed is included in the sequential closure of the initial set.

2 The only instance that I have heard of non-symmetric tensors occurring in a real-
istic situation is the Hall effect, which Graeme MILTON studied, but it concerns an
electrical current in a thin ribbon, so that the macroscopic direction of the current is
imposed, and the situation is not subject to a complete frame indifference!

Page 492

470 Index

Rayleigh–Bénard instability, 51
Rellich–Kondrašov theorem, 206
Reshetnyak theorem, 224
Reynolds number, 44, 203
Riccati equation, 291, 306, 308
Riesz operators, 94, 336, 339, 353
Riesz theorem, 77

Saint-Venant principle, 200
Schrödinger equation, 402, 404, 434
second commutation lemma, 370, 373
second principle, 7, 8, 251, 303
semi-classical measures, 329, 332, 385,

387, 389, 390, 392, 395, 405
semi-group, 35, 255, 261
singular support, 325
Sobolev embedding theorem, 150, 206
Sobolev space, 42, 94, 178, 182, 213, 372
Sommerfeld radiation condition, 41
speed of light, 8, 44, 98, 274, 406
speed of sound, 9
stationary-phase principle, 326
Stewartson triple deck, 439
Stokes equation, 26, 30, 32, 72, 175, 436
Stone–Čech compactification, 67
strong ellipticity condition, 143
symbols, 330, 333, 342–344, 353, 354,

371, 372, 374, 376, 379

Taylor expansion, 193, 257–259, 271,
273, 285, 329, 350, 355, 357,
371, 415

trace theorem, 92, 178
transport, 47, 51, 54, 106, 111, 262, 303,

327, 369, 370, 373, 377–381,
383, 402, 406

V-ellipticity, 79
variational inequality, 120, 132
vector potential, 12, 108, 109, 406
very strong ellipticity condition, 143
viscosity, 44, 45, 202–205, 209, 274
Vitali covering, 283, 317
von Kármán vortices , 203
vorticity, 204

wave equation, 40, 72, 102, 103, 110,
141, 160, 261, 326, 341, 370,
376–382, 434, 437, 439

wave front sets, 325
Weierstrass theorem, 258, 336
Wheatstone bridge, 243
Wigner measures, 390, 392
Wigner transform, 390, 391

Young inequality, 388
Young measures, 52–54, 66, 67, 76,

251–253, 303, 322, 327, 328,
331, 361, 365, 390, 391, 409,
410, 413, 414, 424, 426–429,
435

Yukawa potential, 40

Page 493

Editor in Chief: Franco Brezzi

Editorial Policy

1. The UMI Lecture Notes aim to report new developments in all areas of mathematics and
their applications - quickly, informally and at a high level. Mathematical texts analysing

new developments in modelling and numerical simulation are also welcome.



2. Manuscripts should be submitted to
Redazione Lecture Notes U.M.I.

[email protected]

and possibly to one of the editors of the Board informing, in this case, the Redazione

about the submission. In general, manuscripts will be sent out to external referees for

evaluation. If a decision cannot yet be reached on the basis of the first 2 reports, further

referees may be contacted. The author will be informed of this. A final decision to publish

can be made only on the basis of the complete manuscript, however a refereeing process

leading to a preliminary decision can be based on a pre-final or incomplete manuscript.

The strict minimum amount of material that will be considered should include a detailed

outline describing the planned contents of each chapter, a bibliography and several sample

chapters.



3. Manuscripts should in general be submitted in English. Final manuscripts should contain
at least 100 pages of mathematical text and should always include

- a table of contents;
- an informative introduction, with adequate motivation and perhaps some historical

remarks: it should be accessible to a reader not intimately familiar with the topic

treated;

- a subject index: as a rule this is genuinely helpful for the reader.


4. For evaluation purposes, please submit manuscripts in electronic form, preferably as pdf-
or zipped ps- files. Authors are asked, if their manuscript is accepted for publication, to

use the LaTeX2e style files available from Springer's web-server at

ftp://ftp.springer.de/pub/tex/latex/svmonot1/ for monographs

and at

ftp://ftp.springer.de/pub/tex/latex/svmultt1/ for multi-authored volumes



5. Authors receive a total of 50 free copies of their volume, but no royalties. They are
entitled to a discount of 33.3 % on the price of Springer books purchased for their personal

use, if ordering directly from Springer.



6. Commitment to publish is made by letter of intent rather than by signing a formal contract.
Springer-Verlag secures the copyright for each volume. Authors are free to reuse material

contained in their LNM volumes in later publications: A brief written (or e-mail) request

for formal permission is sufficient.

Similer Documents