Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Composite Structure with Highly Contrasting
Conductivities:
Homogenization of Hyperbolic Equation
A. K. Nandakumaran
Department of Mathematics
Indian Institute of Science, Bangalore 560012.
Email: [email protected]
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Outline
1
A Brief Introduction to Homogenization
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Outline
1
A Brief Introduction to Homogenization
2
High Contrasting Composite
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Outline
1
A Brief Introduction to Homogenization
2
High Contrasting Composite
3
Apriori Estimates and Extension Operators
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Outline
1
A Brief Introduction to Homogenization
2
High Contrasting Composite
3
Apriori Estimates and Extension Operators
4
Two-scale System
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
A Brief Introduction to
Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
What is Homogenization (Mathematical)?
♣ Let Ω be a bounded domain in Rn . Define, for α, β > 0, the class of
matrix functions:
E(Ω) = E(α, β, Ω) = {A = [aij (x)] : A is symmetric and satisfies (1)}.
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
What is Homogenization (Mathematical)?
♣ Let Ω be a bounded domain in Rn . Define, for α, β > 0, the class of
matrix functions:
E(Ω) = E(α, β, Ω) = {A = [aij (x)] : A is symmetric and satisfies (1)}.
♣ We assume the matrix A satisfies
α|ξ|2 ≤ hA(x)ξ, ξi = aij ξi ξj ≤ β|ξ|2 , ∀ξ ∈ Rn .
(1)
The first inequality is nothing but the uniform ellipticity.
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
What is Homogenization (Mathematical)?
♣ Let Ω be a bounded domain in Rn . Define, for α, β > 0, the class of
matrix functions:
E(Ω) = E(α, β, Ω) = {A = [aij (x)] : A is symmetric and satisfies (1)}.
♣ We assume the matrix A satisfies
α|ξ|2 ≤ hA(x)ξ, ξi = aij ξi ξj ≤ β|ξ|2 , ∀ξ ∈ Rn .
(1)
The first inequality is nothing but the uniform ellipticity.
♣ Given an element A ∈ E(Ω), associate the PDE operator
∂
∂
(aij
) and introduce the elliptic boundary value problem
A=−
∂xi
∂xj
Au = f in Ω
u
Composite Structure: Homogenization
(2)
= 0 on ∂Ω.
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
What is Homogenization (Mathematical)?, Conti..
♣ The aim is to introduce certain convergence in the above class of
matrix functions relevant to the homogenization theory.
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
What is Homogenization (Mathematical)?, Conti..
♣ The aim is to introduce certain convergence in the above class of
matrix functions relevant to the homogenization theory.
♣ (G- convergence or H - convergence): We say a family
{[aεij ]}ε>0 , H-converges to [a∗ij ] as ε → 0 if
i)
uε * u in H01 (Ω) weak
ii) aεij (x)
∂uε
∂u
* a∗ij (x)
in L2 (Ω) weak.
∂xj
∂xj
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
What is Homogenization (Mathematical)?, Conti..
♣ The aim is to introduce certain convergence in the above class of
matrix functions relevant to the homogenization theory.
♣ (G- convergence or H - convergence): We say a family
{[aεij ]}ε>0 , H-converges to [a∗ij ] as ε → 0 if
i)
uε * u in H01 (Ω) weak
ii) aεij (x)
∂uε
∂u
* a∗ij (x)
in L2 (Ω) weak.
∂xj
∂xj
♣ Here uε , u are, respectively, the solution of (2) corresponding to the
operators Aε , A∗ and we write
H
H
[aεij ] * [a∗ij ]or simply Aε * A∗ .
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
What is Homogenization (Mathematical)?, Conti..
♣ So in a nutshell, one has a family of differential operators and we
would like to obtain the limit operator. In some sense the limit
behaviour of the differential operators.
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
What is Homogenization (Mathematical)?, Conti..
♣ So in a nutshell, one has a family of differential operators and we
would like to obtain the limit operator. In some sense the limit
behaviour of the differential operators.
♣ The differential operators need not be elliptic, it can come from
many other situations, like parabolic, hyperbolic based on applications.
But studying for elliptic operators is more fundamental even in other
applications.
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
What is Homogenization (Mathematical)?, Conti..
♣ So in a nutshell, one has a family of differential operators and we
would like to obtain the limit operator. In some sense the limit
behaviour of the differential operators.
♣ The differential operators need not be elliptic, it can come from
many other situations, like parabolic, hyperbolic based on applications.
But studying for elliptic operators is more fundamental even in other
applications.
♣ Why do we need to study such a special convergence?
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
What is Homogenization (Mathematical)?, Conti..
♣ So in a nutshell, one has a family of differential operators and we
would like to obtain the limit operator. In some sense the limit
behaviour of the differential operators.
♣ The differential operators need not be elliptic, it can come from
many other situations, like parabolic, hyperbolic based on applications.
But studying for elliptic operators is more fundamental even in other
applications.
♣ Why do we need to study such a special convergence?
Why it is different from other asymptotic analysis and a new name
homogenization?
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
What is Homogenization (Mathematical)?, Conti..
♣ So in a nutshell, one has a family of differential operators and we
would like to obtain the limit operator. In some sense the limit
behaviour of the differential operators.
♣ The differential operators need not be elliptic, it can come from
many other situations, like parabolic, hyperbolic based on applications.
But studying for elliptic operators is more fundamental even in other
applications.
♣ Why do we need to study such a special convergence?
Why it is different from other asymptotic analysis and a new name
homogenization?
For this one need to understand the physical applications of
homogenization.
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
What is Homogenization (Physical)?
♣ It is the study of macroscopic and/or bulk behavior of solutions to
partial differential or other type of equations posed on a heterogeneous
domain/media, where the heterogeneities are present at microscopic
scale ε.
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
What is Homogenization (Physical)?
♣ It is the study of macroscopic and/or bulk behavior of solutions to
partial differential or other type of equations posed on a heterogeneous
domain/media, where the heterogeneities are present at microscopic
scale ε.
♣ The heterogeneities can be in the form of fine mixing of two or more
materials with different physical properties in the domain or due to
singularities in the domain in the form of pores, granules etc.
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Homogenization has many applications in :
• the study of properties (structural, electro-magnetic, thermal etc.) of
composites.
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Homogenization has many applications in :
• the study of properties (structural, electro-magnetic, thermal etc.) of
composites.
• study of flow in porous media (flow of oil, water through subsurface,
pollution of ground water, flow of resins and polymers in moulds etc.)
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Homogenization has many applications in :
• the study of properties (structural, electro-magnetic, thermal etc.) of
composites.
• study of flow in porous media (flow of oil, water through subsurface,
pollution of ground water, flow of resins and polymers in moulds etc.)
• analysis of vibrations of thin structures.
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Homogenization has many applications in :
• the study of properties (structural, electro-magnetic, thermal etc.) of
composites.
• study of flow in porous media (flow of oil, water through subsurface,
pollution of ground water, flow of resins and polymers in moulds etc.)
• analysis of vibrations of thin structures.
• homogenization with oscillating (rough) boundary.
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Few Sample Composites
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Few Sample Composites; conti...
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Few Sample Composites; conti...
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Few Sample Composites; conti...
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Sample Composites; Conti..
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Sample Composites, Conti..
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Periodically Perforated Domain
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Oscillating Boundary: Sample Domains
γ
ε
x2
M‘
M
−
M−
Ω+
ε
Ω−
Γu
ω
Γs
Γ
s
Γb
0
Composite Structure: Homogenization
x1
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Oscillating Boundary: Sample Domains
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Nature of a typical problem in homogenization
♣ The problem posed on the heterogeneous domain has a unique
solution.
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Nature of a typical problem in homogenization
♣ The problem posed on the heterogeneous domain has a unique
solution.
♣ Heterogeneities cause the solution to develop high frequency
oscillations.
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Nature of a typical problem in homogenization
♣ The problem posed on the heterogeneous domain has a unique
solution.
♣ Heterogeneities cause the solution to develop high frequency
oscillations.
♣ So a direct numerical study will not be able to capture either the
bulk behavior or the oscillations present in the solutions which are at a
microscopic level.
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Periodic Oscillations in Heterogeneities
1.0
0.5
0.2
0.4
0.6
0.8
1.0
-0.5
-1.0
1
Ε =
1.0
0.5
0.2
0.4
0.6
0.8
1.0
-0.5
-1.0
Ε =
1  5
1.0
0.5
0.2
0.4
0.6
0.8
1.0
-0.5
-1.0
Ε = 1  10
1.0
0.5
0.2
0.4
0.6
0.8
1.0
-0.5
-1.0
Ε = 1  20
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
What is the way out?
Do an asymptotic analysis as ε −→ 0, that is we try to approximate
the solution uε corresponding to the heterogeneous domain, for small
ε > 0, by the solution of a homogenized problem in which the small
parameter do not appear.
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
What is the way out?
Do an asymptotic analysis as ε −→ 0, that is we try to approximate
the solution uε corresponding to the heterogeneous domain, for small
ε > 0, by the solution of a homogenized problem in which the small
parameter do not appear.
The mathematical problem is:
• to identify the homogenized equation
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
What is the way out?
Do an asymptotic analysis as ε −→ 0, that is we try to approximate
the solution uε corresponding to the heterogeneous domain, for small
ε > 0, by the solution of a homogenized problem in which the small
parameter do not appear.
The mathematical problem is:
• to identify the homogenized equation
• to prove convergence of uε corresponding to the heterogeneous
equation to that of the homogeneous equations as ε −→ 0.
Composite Structure: Homogenization
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52
Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
What is the way out?
Do an asymptotic analysis as ε −→ 0, that is we try to approximate
the solution uε corresponding to the heterogeneous domain, for small
ε > 0, by the solution of a homogenized problem in which the small
parameter do not appear.
The mathematical problem is:
• to identify the homogenized equation
• to prove convergence of uε corresponding to the heterogeneous
equation to that of the homogeneous equations as ε −→ 0.
• to capture the oscillations in uε (known as correctors in the
literature)
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52
Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
What is the way out?
Do an asymptotic analysis as ε −→ 0, that is we try to approximate
the solution uε corresponding to the heterogeneous domain, for small
ε > 0, by the solution of a homogenized problem in which the small
parameter do not appear.
The mathematical problem is:
• to identify the homogenized equation
• to prove convergence of uε corresponding to the heterogeneous
equation to that of the homogeneous equations as ε −→ 0.
• to capture the oscillations in uε (known as correctors in the
literature)
• the solution together with the correctors can be used (and also can
be computed numerically) to get good approximation to the original
solution.
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Various Methods
• Formal asymptotic expansion
• Energy method via test functions
• Compensated Compactness
• Gamma Convergence
• Two Scale (Multi-scale) Convergence
• Fourier (Bloch wave) method
• Unfolding method
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Composite Structure with Two High Contrasting Materials
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Hyperbolic Equation

Lε uε := u00ε − div aε (x)A x, xε ∇uε + uε = fε



in ΩT = (0, T ) × Ω,
(Pε )



uε = 0 on ∂ΩT = (0, T ) × Γ, uε (0) = u0ε , u0ε (0) = u1ε .
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Hyperbolic Equation

Lε uε := u00ε − div aε (x)A x, xε ∇uε + uε = fε



in ΩT = (0, T ) × Ω,
(Pε )



uε = 0 on ∂ΩT = (0, T ) × Γ, uε (0) = u0ε , u0ε (0) = u1ε .
• The coefficients aε takes the form aε (x) = αε2 χBε + χMε and A is
uniformly elliptic.
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Hyperbolic Equation

Lε uε := u00ε − div aε (x)A x, xε ∇uε + uε = fε



in ΩT = (0, T ) × Ω,
(Pε )



uε = 0 on ∂ΩT = (0, T ) × Γ, uε (0) = u0ε , u0ε (0) = u1ε .
• The coefficients aε takes the form aε (x) = αε2 χBε + χMε and A is
uniformly elliptic.
αε
• Here αε is a small parameter which goes to zero. Let α = lim .
ε→0 ε
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Hyperbolic Equation

Lε uε := u00ε − div aε (x)A x, xε ∇uε + uε = fε



in ΩT = (0, T ) × Ω,
(Pε )



uε = 0 on ∂ΩT = (0, T ) × Γ, uε (0) = u0ε , u0ε (0) = u1ε .
• The coefficients aε takes the form aε (x) = αε2 χBε + χMε and A is
uniformly elliptic.
αε
• Here αε is a small parameter which goes to zero. Let α = lim .
ε→0 ε
• 3 cases; α = 0, α = +∞ and 0 < α < ∞ which is the critical case.
We mainly concentrate on critical case and take αε = ε.
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Hyperbolic Equation

Lε uε := u00ε − div aε (x)A x, xε ∇uε + uε = fε



in ΩT = (0, T ) × Ω,
(Pε )



uε = 0 on ∂ΩT = (0, T ) × Γ, uε (0) = u0ε , u0ε (0) = u1ε .
• The coefficients aε takes the form aε (x) = αε2 χBε + χMε and A is
uniformly elliptic.
αε
• Here αε is a small parameter which goes to zero. Let α = lim .
ε→0 ε
• 3 cases; α = 0, α = +∞ and 0 < α < ∞ which is the critical case.
We mainly concentrate on critical case and take αε = ε.
• Hence Bε is the soft inclusions and Mε is stiff material part.
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Hyperbolic Equation

Lε uε := u00ε − div aε (x)A x, xε ∇uε + uε = fε



in ΩT = (0, T ) × Ω,
(Pε )



uε = 0 on ∂ΩT = (0, T ) × Γ, uε (0) = u0ε , u0ε (0) = u1ε .
• The coefficients aε takes the form aε (x) = αε2 χBε + χMε and A is
uniformly elliptic.
αε
• Here αε is a small parameter which goes to zero. Let α = lim .
ε→0 ε
• 3 cases; α = 0, α = +∞ and 0 < α < ∞ which is the critical case.
We mainly concentrate on critical case and take αε = ε.
• Hence Bε is the soft inclusions and Mε is stiff material part.
• We wish to study the limiting behavior of the solution.
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Assumptions on the Data
Z
(A1)
|u0ε |2
Z
+
Ω
Composite Structure: Homogenization
Bε
αε2 |∇u0ε |2
Z
+
|∇u0ε |2
Z
+
Mε
Cebu, Phillipines: January 15, 2016
|u1ε |2 < ∞
Ω
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Assumptions on the Data
Z
(A1)
|u0ε |2
Z
+
Ω
(A2)
Composite Structure: Homogenization
Bε
αε2 |∇u0ε |2
Z
+
|∇u0ε |2
Z
+
Mε
|u1ε |2 < ∞
Ω
kfε kL2 (ΩT ) ≤ C,
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Assumptions on the Data
Z
(A1)
|u0ε |2
Z
+
Ω
αε2 |∇u0ε |2
Bε
Z
+
|∇u0ε |2
Z
|u1ε |2 < ∞
+
Mε
Ω
kfε kL2 (ΩT ) ≤ C,
(A2)
• The assumption (A1) is natural based on energy estimates which we
are going to present it shortly. From the second assumption, we get
fε * f weakly in L2 (ΩT )
fε * f0 (t, x, y) in L2 (ΩT ) and f (t, x) =
2−s
Composite Structure: Homogenization
Cebu, Phillipines: January 15, 2016
(3)
Z
f0 (t, x, y)dy (4)
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Assumptions on the Data
Z
(A1)
|u0ε |2
Z
+
Ω
αε2 |∇u0ε |2
Bε
Z
+
|∇u0ε |2
Z
|u1ε |2 < ∞
+
Mε
Ω
kfε kL2 (ΩT ) ≤ C,
(A2)
• The assumption (A1) is natural based on energy estimates which we
are going to present it shortly. From the second assumption, we get
fε * f weakly in L2 (ΩT )
fε * f0 (t, x, y) in L2 (ΩT ) and f (t, x) =
2−s
(3)
Z
f0 (t, x, y)dy (4)
• The second convergence is known as two-scale convergence which we
use it for our homogenization problem.
23 /
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Two Scale Convergence
Definition (Two-scale convergence)
A sequence of functions {vε } in L2 (ΩT ) is said to two-scale converge to
2s
a limit v ∈ L2 (ΩT × Y ) (denoted as vε ** v) if
Z
Z Z
x
dx dt →
v(t, x, y)φ(t, x, y) dy dx dt
vε φ t, x,
ε
ΩT
ΩT Y
For all φ ∈ L2 ΩT ; C# (Y ) .
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Two Scale Convergence
Definition (Two-scale convergence)
A sequence of functions {vε } in L2 (ΩT ) is said to two-scale converge to
2s
a limit v ∈ L2 (ΩT × Y ) (denoted as vε ** v) if
Z
Z Z
x
dx dt →
v(t, x, y)φ(t, x, y) dy dx dt
vε φ t, x,
ε
ΩT
ΩT Y
For all φ ∈ L2 ΩT ; C# (Y ) .
Further, if v0 is the weak limit of {vε } in L2 (ΩT ), then
Z
v0 (t, x) =
v(t, x, y)dy.
Y
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Two Scale Convergence
We have the following compactness theorem.
Theorem (Compactness)
For any bounded sequence vε in L2 (ΩT ), there exist a subsequence and
v ∈ L2 (ΩT × Y ) such that, vε two-scale converges to v along the
subsequence.
Also, if vε is bounded in L2 (0, T ; H 1 (Ω)), then v is independent of y
1 (Y )) such
and is in L2 (0, T ; H 1 (Ω)), and there exists a v1 ∈ L2 (ΩT ; H#
that, up to a subsequence, ∇vε two-scale converges to ∇v + ∇y v1 .
Composite Structure: Homogenization
Cebu, Phillipines: January 15, 2016
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Energy of the System
Eε (t) =
1
2
Z
|u0ε (t)|2 +
Z
|uε (t)|2
Ω
Z Ω
Z
2
2
2
χMε |∇uε (t)|
+ αε χBε |∇uε (t)| +
Ω
Ω
= Eε1 (t) + Eε2 (t) + Eε3 (t) + Eε4 (t)
Composite Structure: Homogenization
(5)
Cebu, Phillipines: January 15, 2016
(6)
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Energy of the System
Eε (t) =
1
2
Z
|u0ε (t)|2 +
Z
|uε (t)|2
Ω
Z Ω
Z
2
2
2
χMε |∇uε (t)|
+ αε χBε |∇uε (t)| +
Ω
(5)
Ω
= Eε1 (t) + Eε2 (t) + Eε3 (t) + Eε4 (t)
(6)
Proposition
There exists a constant C > 0 independent of ε such that
Eε (t) < C
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Apriori Estimates
• The above proposition will give us the following estimates
Proposition
There exists a constant C > 0 independent of ε such that

kuε kL∞
ku0ε kL∞
2 ≤ C,
2 ≤ C,

t Lx
t Lx
 kα ∇u k ∞ 2
ε
ε Lt Lx (Bε ) ≤ C,
Composite Structure: Homogenization


(7)
k∇uε kL∞
≤ C. 
2
t Lx (Mε )
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Apriori Estimates
• The above proposition will give us the following estimates
Proposition
There exists a constant C > 0 independent of ε such that

kuε kL∞
ku0ε kL∞
2 ≤ C,
2 ≤ C,

t Lx
t Lx
 kα ∇u k ∞ 2
ε
ε Lt Lx (Bε ) ≤ C,


(7)
k∇uε kL∞
≤ C. 
2
t Lx (Mε )
• Thus, we have the correct L2 estimates for the solution and its
time-derivative. Also gradient estimate in the stiff part.
Composite Structure: Homogenization
Cebu, Phillipines: January 15, 2016
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52
Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Apriori Estimates
• The above proposition will give us the following estimates
Proposition
There exists a constant C > 0 independent of ε such that

kuε kL∞
ku0ε kL∞
2 ≤ C,
2 ≤ C,

t Lx
t Lx
 kα ∇u k ∞ 2
ε
ε Lt Lx (Bε ) ≤ C,


(7)
k∇uε kL∞
≤ C. 
2
t Lx (Mε )
• Thus, we have the correct L2 estimates for the solution and its
time-derivative. Also gradient estimate in the stiff part.
• The difficulty in this problem: The gradient estimate in the soft
inclusions is of order ε−1 and hence, in general H 1 estimate is of order
ε−1 .
Composite Structure: Homogenization
Cebu, Phillipines: January 15, 2016
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Apriori Estimates
• The above proposition will give us the following estimates
Proposition
There exists a constant C > 0 independent of ε such that

kuε kL∞
ku0ε kL∞
2 ≤ C,
2 ≤ C,

t Lx
t Lx
 kα ∇u k ∞ 2
ε
ε Lt Lx (Bε ) ≤ C,


(7)
k∇uε kL∞
≤ C. 
2
t Lx (Mε )
• Thus, we have the correct L2 estimates for the solution and its
time-derivative. Also gradient estimate in the stiff part.
• The difficulty in this problem: The gradient estimate in the soft
inclusions is of order ε−1 and hence, in general H 1 estimate is of order
ε−1 .
• This also motivates the assumption (A1) on the initial data.
Composite Structure: Homogenization
Cebu, Phillipines: January 15, 2016
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Extension Operators
• We use the idea from perforated domains, where one extends the
solution from the domain part to the perforations via linear operators.
Composite Structure: Homogenization
Cebu, Phillipines: January 15, 2016
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52
Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Extension Operators
• We use the idea from perforated domains, where one extends the
solution from the domain part to the perforations via linear operators.
• Thus idea is to restrict the solution to the stiff part of the domain
and treating the soft part as perforations, extend the solution in a
bounded way. This is done using the Extension Lemma.
Composite Structure: Homogenization
Cebu, Phillipines: January 15, 2016
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Extension Operators
• We use the idea from perforated domains, where one extends the
solution from the domain part to the perforations via linear operators.
• Thus idea is to restrict the solution to the stiff part of the domain
and treating the soft part as perforations, extend the solution in a
bounded way. This is done using the Extension Lemma.
• Then study the convergence of the actual solution and the extended
function using two-scale convergence and connect them
Composite Structure: Homogenization
Cebu, Phillipines: January 15, 2016
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Extension Lemma
Lemma (Cioranescu-Donato)
There exists a linear continuous operator
P ε ∈ L L∞ (0, T ; H k (Mε )); L∞ (0, T ; H k (Ω)) , k = 0, 1 such that, for
some constant C independent of ε: for any φ ∈ L∞ (0, T ; H k (Mε ));
 ε
P φ = φ in Mε × (0, T)








P ε φ0 = (P ε φ)0 in Ω × (0, T)




kP ε φkL∞ (0,T ;L2 (Ω)) ≤ C kφkL∞ (0,T ;L2 (Mε ))






kP ε φ0 kL∞ (0,T ;L2 (Ω)) ≤ C kφ0 kL∞ (0,T ;L2 (Mε ))






k∇(P ε φ)kL∞ (0,T ;L2 (Ω)) ≤ C k∇φkL∞ (0,T ;L2 (Mε ))
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Estimate and Convergence on extended functions
• Let ũε be the extension of uε restricted to Mε , that is
ũε = P ε (uε |Mε ). Then
kũε kL∞
≤ C,
1
t H0 (Ω)
Composite Structure: Homogenization
kũ0ε kL∞
≤C
2
t Lx (Ω)
Cebu, Phillipines: January 15, 2016
(8)
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Estimate and Convergence on extended functions
• Let ũε be the extension of uε restricted to Mε , that is
ũε = P ε (uε |Mε ). Then
kũε kL∞
≤ C,
1
t H0 (Ω)
kũ0ε kL∞
≤C
2
t Lx (Ω)
(8)
=⇒
1
ũε * ũ weak∗ in L∞
t H0 (Ω)
)
(9)
2
ũ0ε * ũ0 weak∗ in L∞
t Lx (Ω)
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Estimate and Convergence on extended functions
• Let ũε be the extension of uε restricted to Mε , that is
ũε = P ε (uε |Mε ). Then
kũε kL∞
≤ C,
1
t H0 (Ω)
kũ0ε kL∞
≤C
2
t Lx (Ω)
(8)
=⇒
1
ũε * ũ weak∗ in L∞
t H0 (Ω)
)
(9)
2
ũ0ε * ũ0 weak∗ in L∞
t Lx (Ω)
=⇒
ũε → ũ strongly in L2 (ΩT ) and C([0, T], L2 (Ω))
Composite Structure: Homogenization
Cebu, Phillipines: January 15, 2016
A.K.N/IISc
(10)
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52
Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Further Analysis on uε
• We assume that αε = O(ε), i.e.,
Composite Structure: Homogenization
αε
ε
→ α ∈ (0, ∞)
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52
Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Further Analysis on uε
• We assume that αε = O(ε), i.e.,
αε
ε
→ α ∈ (0, ∞)
• Since uε , ε∇uε are bounded in L2 (ΩT ), we apply two-scale
convergence to get v0 = v0 (t, x, y) ∈ L2 (ΩT ; H]1 (Y )) such that

uε * v0 in L2 (ΩT )


2−s



2
ε∇uε * ∇y v0 in L (ΩT )
2−s




0
0
2

uε * v0 in L (ΩT )
(11)
2−s
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Further Analysis on uε ; Conti...
• Let u is the weak limit of uε in L2 (ΩT ), then
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Further Analysis on uε ; Conti...
• Let u is the weak limit of uε in L2 (ΩT ), then
Z
u = v0 (t, x, y)dy.
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Further Analysis on uε ; Conti...
• Let u is the weak limit of uε in L2 (ΩT ), then
Z
u = v0 (t, x, y)dy.
The difficulty is that we do not have convergence in H 1 .
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Further Analysis on uε ; Conti...
• Let u is the weak limit of uε in L2 (ΩT ), then
Z
u = v0 (t, x, y)dy.
The difficulty is that we do not have convergence in H 1 .
• But ∇uε is bounded in L2 (Mε ), we have ∇uε χMε bounded in
L2 (ΩT ), by suitably applying 2-scale convergence, we have
∇y v0 (t, x, y) = 0 in Ω × M a.e.
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Further Analysis on uε ; Conti...
• Let u is the weak limit of uε in L2 (ΩT ), then
Z
u = v0 (t, x, y)dy.
The difficulty is that we do not have convergence in H 1 .
• But ∇uε is bounded in L2 (Mε ), we have ∇uε χMε bounded in
L2 (ΩT ), by suitably applying 2-scale convergence, we have
∇y v0 (t, x, y) = 0 in Ω × M a.e.
• Thus v0 (t, x, y) is constant in y variable in Ω × M . In fact, using the
identity and 2-scale convergence, we get v0 (t, x, y) = ũ(t, x) in Ω × M .
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Further Analysis on uε ; Conti...
• Let u is the weak limit of uε in L2 (ΩT ), then
Z
u = v0 (t, x, y)dy.
The difficulty is that we do not have convergence in H 1 .
• But ∇uε is bounded in L2 (Mε ), we have ∇uε χMε bounded in
L2 (ΩT ), by suitably applying 2-scale convergence, we have
∇y v0 (t, x, y) = 0 in Ω × M a.e.
• Thus v0 (t, x, y) is constant in y variable in Ω × M . In fact, using the
identity and 2-scale convergence, we get v0 (t, x, y) = ũ(t, x) in Ω × M .
• For (t, x, y) ∈ ΩT × Y , define
v(t, x, y) := v0 (t, x, y) − ũ(t, x).
Thus v vanishes outside B.
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Further Analysis on uε ; Conti...
We get the following regularity results:
v ∈ L∞ (0, T ; L2 (Ω; H01 (B))) and v 0 ∈ L∞ (0, T ; L2 (Ω; L2 (B)))
1
0
∞
2
ũ ∈ L∞
t H0 (Ω), ũ ∈ Lt (L (Ω)).
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Further Analysis on uε ; Conti...
We get the following regularity results:
v ∈ L∞ (0, T ; L2 (Ω; H01 (B))) and v 0 ∈ L∞ (0, T ; L2 (Ω; L2 (B)))
1
0
∞
2
ũ ∈ L∞
t H0 (Ω), ũ ∈ Lt (L (Ω)).
2s
uε **v0 (t, x, y) = ũ(t, x) + v(t, x, y).
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Further Analysis on uε ; Conti...
We get the following regularity results:
v ∈ L∞ (0, T ; L2 (Ω; H01 (B))) and v 0 ∈ L∞ (0, T ; L2 (Ω; L2 (B)))
1
0
∞
2
ũ ∈ L∞
t H0 (Ω), ũ ∈ Lt (L (Ω)).
2s
uε **v0 (t, x, y) = ũ(t, x) + v(t, x, y).
• The aim, indeed, is to identify v0 which is the two-scale limit of our
original problem.
Composite Structure: Homogenization
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52
Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Further Analysis on uε ; Conti...
We get the following regularity results:
v ∈ L∞ (0, T ; L2 (Ω; H01 (B))) and v 0 ∈ L∞ (0, T ; L2 (Ω; L2 (B)))
1
0
∞
2
ũ ∈ L∞
t H0 (Ω), ũ ∈ Lt (L (Ω)).
2s
uε **v0 (t, x, y) = ũ(t, x) + v(t, x, y).
• The aim, indeed, is to identify v0 which is the two-scale limit of our
original problem.
• The difficulty is that, in general, we will not have regularity for v0
with respect to the spatial variable x. Indeed, we do get regularity
with respect to the fast variable y. On the other hand, ũ, the limit of
the extended function has H 1 regularity with respect to x.
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Consolidation of all the Discussion
• For the two-scale limit v0 of the given problem, we have the
representation
v0 (t, x, y) = ũ(t, x) + v(t, x, y),
where
ũ ∈ L2 (0, T ; H01 (Ω)), v ∈ L∞ (0, T ; L2 (Ω; H01 (B))).
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Consolidation of all the Discussion
• For the two-scale limit v0 of the given problem, we have the
representation
v0 (t, x, y) = ũ(t, x) + v(t, x, y),
where
ũ ∈ L2 (0, T ; H01 (Ω)), v ∈ L∞ (0, T ; L2 (Ω; H01 (B))).
• Here ũ is the limit of extended function in L∞ (0, T ; H01 (Ω)).
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Consolidation of all the Discussion
• For the two-scale limit v0 of the given problem, we have the
representation
v0 (t, x, y) = ũ(t, x) + v(t, x, y),
where
ũ ∈ L2 (0, T ; H01 (Ω)), v ∈ L∞ (0, T ; L2 (Ω; H01 (B))).
• Here ũ is the limit of extended function in L∞ (0, T ; H01 (Ω)).
• First, we will look for a solution in the above form which satisfies a
weak formulation.
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Limit of ∇uε χMε and ε∇uε χBε
2s
• Clearly εχBε ∇uε **χB (y)∇y v.
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Limit of ∇uε χMε and ε∇uε χBε
2s
• Clearly εχBε ∇uε **χB (y)∇y v.
• We do more analysis on the additional information that ∇uε is
bounded in the stiff part of the material, that is ∇uε χMε is bounded in
L2 (ΩT ).
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Limit of ∇uε χMε and ε∇uε χBε
2s
• Clearly εχBε ∇uε **χB (y)∇y v.
• We do more analysis on the additional information that ∇uε is
bounded in the stiff part of the material, that is ∇uε χMε is bounded in
L2 (ΩT ).
• Let ∇uε χMε * K(t, x, y) in L2 (ΩT ). Then, we have
2−s
Proposition
1 (M )) such that
There exists u1 (t, x, y) ∈ L2 (ΩT ; H#
[K(t, x, y) − ∇x ũ(t, x)] χM (y) = ∇y u1 (t, x, y)χM (y)
Further, u1 vanishes outside M satisfies
u1 ∈ L∞ (0, T ; L2 (Ω; H01 (M ))).
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Remarks
• Thus, with respect to the fast variable y, v vanishes outside soft part
B and u1 vanishes outside the stiff part M taking care of the two
features.
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Remarks
• Thus, with respect to the fast variable y, v vanishes outside soft part
B and u1 vanishes outside the stiff part M taking care of the two
features.
• We now state a weak formulation for the unknowns ũ, v and u1 .
Composite Structure: Homogenization
Cebu, Phillipines: January 15, 2016
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52
Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Remarks
• Thus, with respect to the fast variable y, v vanishes outside soft part
B and u1 vanishes outside the stiff part M taking care of the two
features.
• We now state a weak formulation for the unknowns ũ, v and u1 .
• We then represent u1 in terms of ũ leading to a two-scale
homogenized system for ũ and v.
Composite Structure: Homogenization
Cebu, Phillipines: January 15, 2016
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52
Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Remarks
• Thus, with respect to the fast variable y, v vanishes outside soft part
B and u1 vanishes outside the stiff part M taking care of the two
features.
• We now state a weak formulation for the unknowns ũ, v and u1 .
• We then represent u1 in terms of ũ leading to a two-scale
homogenized system for ũ and v.
• We also derive appropriate initial conditions so that the two-scale
system is well-posed.
Composite Structure: Homogenization
Cebu, Phillipines: January 15, 2016
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52
Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Remarks
• Thus, with respect to the fast variable y, v vanishes outside soft part
B and u1 vanishes outside the stiff part M taking care of the two
features.
• We now state a weak formulation for the unknowns ũ, v and u1 .
• We then represent u1 in terms of ũ leading to a two-scale
homogenized system for ũ and v.
• We also derive appropriate initial conditions so that the two-scale
system is well-posed.
• Deriving a one-scale system (homogenized) remains open, probably,
it is not possible, whereas such a decomposition is possible in an
elliptic system.
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
2-scale system for ũ, v and u1 .
Find ũ, v and u1 in appropriate spaces such that
Z Z
Z Z
00
(ũ + v) (ū + v̄) +
(ũ + v) (ū + v̄)
ΩT Y
ΩT Y
Z Z
+
A(x, y)∇y v · ∇y v̄
ΩT B
Z Z
+
A(x, y)(∇x ũ + ∇y u1 )(∇x ū + ∇y ū1 )
ΩT M
Z
Z Z
=
f ū +
f¯v̄
ΩT
ΩT
Y
for the appropriate test functions ū, v̄ and ū1 . Here f and f¯ are,
respectively, the L2 weak- limit and two-scale limit of the data fε .
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Representation of u1
Proposition
The solution u1 can be given in terms of ũ and solutions to cell problem
as
X ∂ ũ
u1 (t, x, y) =
wi (x, y),
∂xi
where wi satisfies for a.e. x ∈ Ω:

 −divy (A(x, y)(∇y wi (x, ·) + ei )) = 0 in M
 w (x, ·) ∈ H 1 (M )
i
0#
Here {ei , 1 ≤ i ≤ n} is the canonical basis of Rn .
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Representation of u1
Proposition
The solution u1 can be given in terms of ũ and solutions to cell problem
as
X ∂ ũ
u1 (t, x, y) =
wi (x, y),
∂xi
where wi satisfies for a.e. x ∈ Ω:

 −divy (A(x, y)(∇y wi (x, ·) + ei )) = 0 in M
 w (x, ·) ∈ H 1 (M )
i
0#
Here {ei , 1 ≤ i ≤ n} is the canonical basis of Rn .
Introducing the homogenized matrix
Z
∗
A (x) =
A(x, y)(∇y wj + ej )(∇y wi + ei )dy,
M
we may eliminate u1 .
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
2-scale system in terms of ũ and v.
Theorem
The limits ũ and v satisfies
ũ ∈ L∞ (0, T ; H01 (Ω)), ũ0 ∈ L∞ (0, T ; L2 (Ω)),
v ∈ L∞ (0, T ; L2 (Ω; H01 (B))), v 0 ∈ L∞ (0, T ; L2 (Ω; L2 (B)))
and
Z
Z
00
Z
Z
(ũ + v) (ū + v̄) +
(ũ + v) (ū + v̄)
ΩT Y
Z Z
Z
A(x, y)∇y v · ∇y v̄ +
+
A∗ (x)∇ũ · ∇ū
ΩT B
ΩT
Z
Z Z
=
f ū +
f¯v̄
ΩT
Y
ΩT
Composite Structure: Homogenization
ΩT
(12)
Y
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Two-scale Homogenized System (Strong Form)
Find ũ ∈ L∞ (0, T ; H01 (Ω)), v ∈ L∞ (0, T ; L2 (Ω; H01 (B))) s.t.
(ũ + v)00 + (ũ + v) − divy (χB A(x, y)∇y v)
− divx (A∗ (x)∇ũ) = f + f¯
I
I
0
0
ũ(0, x) =
u (x, y) dy ũ (0, x) =
u1 (x, y) dy
M
M
I
v(0, x, y) = u0 (x, y) −
u0 (x, y) dy
M
I
v 0 (0, x, y) = u1 (x, y) −
u1 (x, y) dy
in Ω,
M
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Two-scale Homogenized System (Strong Form)
Find ũ ∈ L∞ (0, T ; H01 (Ω)), v ∈ L∞ (0, T ; L2 (Ω; H01 (B))) s.t.
(ũ + v)00 + (ũ + v) − divy (χB A(x, y)∇y v)
− divx (A∗ (x)∇ũ) = f + f¯
I
I
0
0
ũ(0, x) =
u (x, y) dy ũ (0, x) =
u1 (x, y) dy
M
M
I
v(0, x, y) = u0 (x, y) −
u0 (x, y) dy
M
I
v 0 (0, x, y) = u1 (x, y) −
u1 (x, y) dy
in Ω,
M
• Need to interpret the meaning of ũ(0), v(0) and ũ0 (0), v 0 (0) and
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Two-scale Homogenized System (Strong Form)
Find ũ ∈ L∞ (0, T ; H01 (Ω)), v ∈ L∞ (0, T ; L2 (Ω; H01 (B))) s.t.
(ũ + v)00 + (ũ + v) − divy (χB A(x, y)∇y v)
− divx (A∗ (x)∇ũ) = f + f¯
I
I
0
0
ũ(0, x) =
u (x, y) dy ũ (0, x) =
u1 (x, y) dy
M
M
I
v(0, x, y) = u0 (x, y) −
u0 (x, y) dy
M
I
v 0 (0, x, y) = u1 (x, y) −
u1 (x, y) dy
in Ω,
M
• Need to interpret the meaning of ũ(0), v(0) and ũ0 (0), v 0 (0) and then,
identify them.
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Meaning of Initial Values
• Meaning to v0 (0) = (ũ + v)(0), v00 (0) = (ũ + v)0 (0)
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Meaning of Initial Values
• Meaning to v0 (0) = (ũ + v)(0), v00 (0) = (ũ + v)0 (0)
• This can be interpreted properly using the facts
1
0
∞ 2
00
∞ −1
ũ ∈ L∞
(Ω)
t H0 (Ω), ũ ∈ Lt L (Ω), ũ ∈ Lt H
and
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Meaning of Initial Values
• Meaning to v0 (0) = (ũ + v)(0), v00 (0) = (ũ + v)0 (0)
• This can be interpreted properly using the facts
1
0
∞ 2
00
∞ −1
ũ ∈ L∞
(Ω)
t H0 (Ω), ũ ∈ Lt L (Ω), ũ ∈ Lt H
and
2 1
0
∞ 2 2
00
∞ −1
v ∈ L∞
(Ω × Y ).
t Lx Hy , v ∈ Lt Lx Ly , v ∈ Lt H
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Meaning of Initial Values
• Meaning to v0 (0) = (ũ + v)(0), v00 (0) = (ũ + v)0 (0)
• This can be interpreted properly using the facts
1
0
∞ 2
00
∞ −1
ũ ∈ L∞
(Ω)
t H0 (Ω), ũ ∈ Lt L (Ω), ũ ∈ Lt H
and
2 1
0
∞ 2 2
00
∞ −1
v ∈ L∞
(Ω × Y ).
t Lx Hy , v ∈ Lt Lx Ly , v ∈ Lt H
• The above regularity implies that ũ, v ∈ C([0, T ], L2 (Ω × Y )) and
ũ0 , v 0 ∈ C([0, T ], H −1 (Ω × Y )) and hence
ũ(0), v(0) ∈ L2 (Ω × Y ), ũ0 (0), v 0 (0) ∈ H −1 (Ω × Y )
.
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Identification of Initial Conditions
• Using the boundedness of the initial conditions u0ε and u1ε , let
2−s
2−s
u0ε → u0 (x, y), u1ε → u1 (x, y).
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Identification of Initial Conditions
• Using the boundedness of the initial conditions u0ε and u1ε , let
2−s
2−s
u0ε → u0 (x, y), u1ε → u1 (x, y).
• Using the strong convergence of the extension ũε , the it is easy to
find the initial condition for ũ, ũ0 (0).
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Identification of Initial Conditions
• Using the boundedness of the initial conditions u0ε and u1ε , let
2−s
2−s
u0ε → u0 (x, y), u1ε → u1 (x, y).
• Using the strong convergence of the extension ũε , the it is easy to
find the initial condition for ũ, ũ0 (0).
• However, more work to be done to identify the limit for v(0), v 0 (0).
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Existence
• The above equation defines a hyperbolic system with appropriate
elliptic part. Let X = H01 (Ω), Z = L2 (Ω; H01 (B)). We have
(ũ, v) ∈ L∞ (0, T ; X) × L∞ (0, T ; Z)
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Existence
• The above equation defines a hyperbolic system with appropriate
elliptic part. Let X = H01 (Ω), Z = L2 (Ω; H01 (B)). We have
(ũ, v) ∈ L∞ (0, T ; X) × L∞ (0, T ; Z)
• Elliptic bilinear form: Let A : X × Z → R, U = (u, v) ∈ X × Z
defined by
Z
Z
Z
A(U1 , U2 ) =
u1 u1 +
v1 v1 +
A∗ (x)∇u1 · ∇u2
Ω
Ω×B
Ω
Z
+
A(x, y)∇y v1 · ∇y v2
Ω×B
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Existence; Conti..
• The norm kU k2X×Z = kuk2H 1 (Ω) + k∇y vk2L2 (Ω×B) + kvk2L2 (Ω×B) is
0
equivalent to k∇uk2L2 (Ω) + k∇y vk2L2 (Ω×B) .
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Existence; Conti..
• The norm kU k2X×Z = kuk2H 1 (Ω) + k∇y vk2L2 (Ω×B) + kvk2L2 (Ω×B) is
0
equivalent to k∇uk2L2 (Ω) + k∇y vk2L2 (Ω×B) .
• Clearly A is continuous and
Z
Z
2
A(U, U ) =
|u1 | +
Ω
|v1 |2
Ω×B
Z
Z
∗
A (x)∇u · ∇u +
+
Ω
Z
≥ C
Composite Structure: Homogenization
2
Z
|∇y v|
|∇u| +
Ω
A(x, y)∇y v · ∇y v
Ω×B
2
≥ C1 kU k2X×Z
Ω
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Existence; Conti..
• The norm kU k2X×Z = kuk2H 1 (Ω) + k∇y vk2L2 (Ω×B) + kvk2L2 (Ω×B) is
0
equivalent to k∇uk2L2 (Ω) + k∇y vk2L2 (Ω×B) .
• Clearly A is continuous and
Z
Z
2
A(U, U ) =
|u1 | +
Ω
|v1 |2
Ω×B
Z
Z
∗
A (x)∇u · ∇u +
+
Ω
Z
≥ C
2
Z
|∇y v|
|∇u| +
Ω
A(x, y)∇y v · ∇y v
Ω×B
2
≥ C1 kU k2X×Z
Ω
• The existence of the Hyperbolic system could be worked out.
Composite Structure: Homogenization
Cebu, Phillipines: January 15, 2016
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Remarks
• We do not know how to separate ũ and v so that we have a complete
homogenized equation for the macro variable alone, probably it may
not be possible.
Composite Structure: Homogenization
Cebu, Phillipines: January 15, 2016
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52
Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Remarks
• We do not know how to separate ũ and v so that we have a complete
homogenized equation for the macro variable alone, probably it may
not be possible.
• However, using the unique existence of ũ + v, we have a
representation of ũ + v = û(t, x) + v̂(t, x, y), where û and v̂ are solved
separately.
Composite Structure: Homogenization
Cebu, Phillipines: January 15, 2016
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52
Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Remarks
• We do not know how to separate ũ and v so that we have a complete
homogenized equation for the macro variable alone, probably it may
not be possible.
• However, using the unique existence of ũ + v, we have a
representation of ũ + v = û(t, x) + v̂(t, x, y), where û and v̂ are solved
separately.
• A complete decoupling is possible in an elliptic system.
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Decoupling in elliptic Problem
The elliptic two scale system is given by
Z Z
Z Z
(ũ + v) (ū + v̄) +
A(x, y)∇y v · ∇y v̄
Ω Y
Ω B
Z
Z
∗
+ A (x)∇ũ · ∇ū =
f ū
Ω
Composite Structure: Homogenization
(13)
Ω
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Decoupling in elliptic Problem
The elliptic two scale system is given by
Z Z
Z Z
(ũ + v) (ū + v̄) +
A(x, y)∇y v · ∇y v̄
Ω Y
Ω B
Z
Z
∗
+ A (x)∇ũ · ∇ū =
f ū
Introduce, v̂(x, y) ∈
a.e. x ∈ Ω;
Ω
∞
L (Ω
(13)
Ω
× B) as the solution to the cell problem for
v̂(x, ·) − divy A(x, ·)∇u v̂(x, ·) = 1 in B
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Decoupling in elliptic Problem
The elliptic two scale system is given by
Z Z
Z Z
(ũ + v) (ū + v̄) +
A(x, y)∇y v · ∇y v̄
Ω Y
Ω B
Z
Z
∗
+ A (x)∇ũ · ∇ū =
f ū
Introduce, v̂(x, y) ∈
a.e. x ∈ Ω;
Ω
∞
L (Ω
(13)
Ω
× B) as the solution to the cell problem for
v̂(x, ·) − divy A(x, ·)∇u v̂(x, ·) = 1 in B
Then, v is given by v(x, y) = (f (x) − ũ(x))v̂(x, y) and ũ satisfies the
homogenized equation; ũ ∈ H01 (Ω)
m(x)u(x) − divA∗ (x)∇u(x) = m(x)f (x) in Ω,
Z
v̂(x, y)dy.
where m(x) =
B
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Decoupling in Hyperbolic System Problem
Introduce û(t, x), v̂(t, x, y) as follows:


Find û ∈ L∞ (0, T ; H01 (Ω)), û0 ∈ L∞ (0, T ; L2 (Ω)) satisfying



û00 + û − divx (A∗ (x)∇û) = f in ΩT



 û(0) = 0 = û0 (0).
Composite Structure: Homogenization
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(14)
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Decoupling in Hyperbolic System Problem
Introduce û(t, x), v̂(t, x, y) as follows:


Find û ∈ L∞ (0, T ; H01 (Ω)), û0 ∈ L∞ (0, T ; L2 (Ω)) satisfying



û00 + û − divx (A∗ (x)∇û) = f in ΩT



 û(0) = 0 = û0 (0).
and with x ∈ Ω as parameter

Find v̂ ∈ L∞ (0, T ; H01 (B)), v̂ 0 ∈ L∞ (0, T ; L2 (B)) satisfying






 v̂ 00 + v̂ − div (A(x, y))∇ v̂) = f in (0, T ) × B
y
y


v̂(0, x, y) = u00 (x, y)




 0
v̂ (0, x, y) = u11 (x, y).
Composite Structure: Homogenization
(14)
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(15)
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Decoupling in Hyperbolic System Problem
• Indeed û + v̂ satisfies the two-scale system. By uniqueness, the
two-scale limit of the inhomogenized equation is given by
v0 = ũ + v = û + v̂,
(16)
where û, v̂ are given as in (14) and (15) respectively.
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Decoupling in Hyperbolic System Problem
• Indeed û + v̂ satisfies the two-scale system. By uniqueness, the
two-scale limit of the inhomogenized equation is given by
v0 = ũ + v = û + v̂,
(16)
where û, v̂ are given as in (14) and (15) respectively.
• Though, we have a complete representation of the weak limit via û
and v̂, the equation (14) cannot be treated as a complete homogenized
(one-scale) system as û do not capture the initial values.
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Main Theorem
Theorem (Homogenization)
Let the given data fε , u0ε , u1ε , aε satisfy the assumptions be given. Let
uε be the unique solution to the problem (Pε ). Then,
2s
uε ** ũ(t, x) + v(t, x, y),
where the pair (ũ, v) ∈ L∞ (0, T ; H01 (Ω)) × L2 (ΩT ; H01 (B)) is the unique
solution of the coupled system described earlier.
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
The other two regimes; α := lim
ε
Two-scale System
αε
= 0 or α = +∞.
ε
• The case α = 0; Here the limit problem is
00
∗
Z
(1 − |B|)(u + u) − divx A (x)∇u =
f0 dy,
in ΩT ,
M
with the same initial conditions as earlier.
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
The other two regimes; α := lim
ε
Two-scale System
αε
= 0 or α = +∞.
ε
• The case α = 0; Here the limit problem is
00
∗
Z
(1 − |B|)(u + u) − divx A (x)∇u =
f0 dy,
in ΩT ,
M
with the same initial conditions as earlier.
• The contribution of the soft part Bε in the homogenized equation is
seen through the measure of B in the final macroscopic equation.
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
The case α = ∞
• There is no contribution from the soft material. The homogenized
equation is
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
The case α = ∞
• There is no contribution from the soft material. The homogenized
equation is
Z
00
∗
u + u − divx A (x)∇u =
f0 dy, in ΩT ,
Y
with the same initial conditions as in the critical case.
Composite Structure: Homogenization
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
The case α = ∞
• There is no contribution from the soft material. The homogenized
equation is
Z
00
∗
u + u − divx A (x)∇u =
f0 dy, in ΩT ,
Y
with the same initial conditions as in the critical case.
• Reference: A. K. Nandakumaran and Ali Sili, Homogenization of a
Hyperbolic Equation with Highly Contrasting Diffusivity Coefficients,
Differential and Integral Equations, Volume 29, Numbers 1-2 (2016),
37-54.
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Intro-Homo
High Contrasting Composite
Apriori Estimates and Extension Operators
Two-scale System
Thank You!
Nandakumaran, IISc., Bangalore
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