##### 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.

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.