PICOF 2012 Regularity Estimates in High Conductivity Homogenization - - PowerPoint PPT Presentation

PICOF 2012 Regularity Estimates in High Conductivity Homogenization

Yves Capdeboscq

Oxford Centre for Nonlinear PDE, University of Oxford

With Marc Briane (Rennes) & Luc Nguyen (Princeton)

Outline

Enhanced Resolution for finite contrast (Ammari-Bonnetier-C) The High Contrast case (Briane-C-Nguyen)

Time Reversal Experiment (Physics)

• C. Prada & M. Fink (Wave Motion ’94). . .

For acoustic waves, with random scatterers in a resonant chamber. Step 1 A pulse is emitted at a source location Step 2 The signal is recorded on an array of microphones, behind the scatterers (for a long time T) Step 3 The signal is amplified and played back, backwards, from the array. At time 2T, a peak appears at the location where the pulse was emitted.

Time Reversal Mirror (Maths)

Extracted from Guillaume Bal’s webpage, http://www.columbia.edu/˜gb2030/

• J. P. Fouque, J. Garnier, G. Papanicolaou and K. Sølna, Wave

propagation and time reversal in randomly layered media, Springer, 2007.

• G. Bal and L. Ryzhik Time reversal and refocusing in random

media, SIMA, 2003.

• C. Bardos & M. Fink, Mathematical foundations of the time

reversal mirror, Asymptotic Anal. 2002 http://www.claudebardos.com/pdf/retemp.pdf

Focusing Beyond the Diffraction Limit

• G. Lerosey, J. de Rosny, A. Tourin, A. Derode, M. Fink,Focusing

Beyond the Diffraction Limit with Far-Field Time Reversal Science, 23 February 2007, p.1120-1122 A TRM made of eight commercial dipolar antennas operating at λ = 12 cm. is placed in a 1 m3 reverberating chamber. Ten wavelength away from the TRM is placed a sub-wavelength receiving array consisting of 8 micro-structured antennas λ/30 apart from each other.

Focusing Beyond the Diffraction Limit

Focusing Beyond the Diffraction Limit

Interpretation via homogenization and small volume asymptotics, for passive imaging.

Passive imaging : no source in the medium

Model problem: Helmholtz equation (time harmonic, in TM (transverse magnetic) Polarization). The x3 component of the magnetic field satisfies div(a(x)∇u(x)) + ω2µ(x)u(x) = in BR, u = ϕ on SR with a(x) = 1 ε(x) + iσ(x)/ω, and ε(x): electric permittivity, σ(x) : conductivity, µ(x) : magnetic permeability, ω : 2π λ .

Difference Imaging

We compare the traces of div(a(x)∇u(x)) + ω2µ(x)u(x) = in BR, and div(ad(x)∇ud(x)) + ω2µd(x)ud(x) = in BR, u, ud = ϕ

• n SR

where ud corresponds to (ad, µd) := (a(x), µ(x)) + 1D(x)(aD(x) − a(x), µD(x) − µ(x)). We want to find inclusions from boundary measurements (or the far field). We consider differential measurements, that is, the map Λ : H1/2(SR) → H−1/2(SR) ϕ → a∇ (ud − u) · n|SR

Resolution for passive systems

Example: a homogeneous background (ε0, µ0) and a constant inclusion (ε1, µ1). Suppose D = B(0, r) and r → 0.

Resolution for passive systems

Example: a homogeneous background (ε0, µ0) and a constant inclusion (ε1, µ1). Suppose D = B(0, r) and r → 0. (Λ(φ), φ) = rdR(φ, φ) + o

• rd

The response operator R is the bilinear form given by R(φ, ψ) = 1 rd

• Br

MB∇uφ · ∇uψdy + 1 rd

• Br

ω2mBuφuψdy The term MB is the polarisation tensor, here MB =

2 ε0+ε1 ε1−ε0 ε0

Id, and mB = µ1 − µ0. The function uφ (resp. uψ) satisfies div(ǫ−1

0 ∇uφ) + ω2µ0uφ

= 0 in BR uφ = φ resp. ψ on SR.

Resolution for passive systems

For constant coefficients, R is explicit yielding a representation formula for Λ. The permeability response is, up to o(rd), Λ = C1,BJ1(kr)eiθ J1(kr)eiθ, ·

• +

C0,BJ0(kr)eiθ J0(kr)eiθ, ·

• when d = 2,
• r

Λ = C1,Bj1(kr)eiθ j1(kr)eiθ, ·

• +

C0,Bj0(kr)eiθ j0(kr)eiθ, ·

• when d = 3,

where k2 = µ0ε0ω2

Resolution for passive system

x K 10 K 5 5 10 0.2 0.4 0.6 0.8 1.0 x K 10 K 5 5 10 0.2 0.4 0.6 0.8 1.0

The size of the focal spot, is given by the size first lobe(s) of the (spherical) Bessel Function J0(√µ0ε0ω·) (J1(√µ0ε0ω·) )

Resolution for passive system

x K 10 K 5 5 10 0.2 0.4 0.6 0.8 1.0 x K 10 K 5 5 10 0.2 0.4 0.6 0.8 1.0

The size of the focal spot, is given by the size first lobe(s) of the (spherical) Bessel Function J0(√µ0ε0ω·) (J1(√µ0ε0ω·) ) This size does not depend on the inclusion.

Enhanced resolution in structured media

• lim

|D|→0 ε = 0.

(|D|2, −1/ln|D| . . . ..)

Enhanced resolution Ammari-Bonnetier-C ’09

Introduce scatterers in the medium, on a periodic grid (for example). Bε,j = jε + εB (aε(x), µε(x)) = (as, µs) in ∪j∈Sε Bε,j (a0, µ0)

• therwise.

(1) The true (defective) medium, material parameters are (aε,d(x), µε,d(x)) =    (aD, µD) in D (as, µs) in ∪j Bε,j \ D. (a0, µ0)

• therwise.

Enhanced Resolution in Structured Media

We want to image inclusions (imperfections) from boundary (far field) measurements. We consider difference measurements, namely the map Λ : H1/2(SR) → H−1/2(SR) ϕ → a∇ (uε,D − uε) · n|SR

Enhanced Resolution in Structured Media

We want to image inclusions (imperfections) from boundary (far field) measurements. We consider difference measurements, namely the map Λ : H1/2(SR) → H−1/2(SR) ϕ → a∇ (uε,D − uε) · n|SR Result: if ε → 0 as |D| → 0, the response operator is given by < Λ(ϕ), ϕ >=

• D

M∗∇u∗(x)·∇u∗(x)dx+ω2

• D

m∗u2

∗(x)dx+o (|D|) ,

where M∗ and m∗ are constant polarization terms that depend on the contrast in material constants, where o (|D|) /|D| → 0 uniformly for ϕH1/2 ≤ 1

Enhanced Resolution in Structured Media

We want to image inclusions (imperfections) from boundary (far field) measurements. We consider difference measurements, namely the map Λ : H1/2(SR) → H−1/2(SR) ϕ → a∇ (uε,D − uε) · n|SR Result: if ε → 0 as |D| → 0, the response operator is given by < Λ(ϕ), ϕ >=

• D

M∗∇u∗(x)·∇u∗(x)dx+ω2

• D

m∗u2

∗(x)dx+o (|D|) ,

where M∗ and m∗ are constant polarization terms that depend on the contrast in material constants, where o (|D|) /|D| → 0 uniformly for ϕH1/2 ≤ 1 To be compared with the unscattered response operator, R(ϕ) =

• D
• M∇u0 · ∇u0 + ω2µu2
• dx + o (|D|) .

different asymptotic limits

1 If ε → 0 first , then |D| → 0: correct focal spot, but incorrect

asymptotic formulae.

different asymptotic limits

1 If ε → 0 first , then |D| → 0: correct focal spot, but incorrect

asymptotic formulae.

2 If |D| → 0 first, then ε → 0: the “reference” resolution

calculation.

different asymptotic limits

1 If ε → 0 first , then |D| → 0: correct focal spot, but incorrect

asymptotic formulae.

2 If |D| → 0 first, then ε → 0: the “reference” resolution

calculation.

3 If εd ≪ |D|, then (1) is correct.

Elements of proof

We use the following tools: H-convergence (strong with correctors) uε,d − u∗L2(Ω) , uε − u∗L2(Ω) → 0. (Murat 70s). Local regularity, and smoothness of u∗ The scatterers have a smooth C 1,α boundary (important). Then Li & Vogelius (’01), Li & Nirenberg (’03), and Avellaneda & Lin (’97) give W 1,∞ convergence estimates ||uε − u∗||L∞(D), ||∇uε − Pε∇u∗||L∞(D) → 0, where Pε ∈ L∞(Ω) is the corrector matrix. Similar to Ben Hassen & Bonnetier ’06. Small volume fraction asymptotics This gives an asymptotic formula for 1D∇uε,D/|D| in terms of a polarization tensor, involving Pε, and ∇u∗, not unlike C-Vogelius (’03, ’06).

Elements of proof

We use the following tools: H-convergence (strong with correctors) uε,d − u∗L2(Ω) , uε − u∗L2(Ω) → 0. (Murat 70s). Local regularity, and smoothness of u∗ The scatterers have a smooth C 1,α boundary (important). Then Li & Vogelius (’01), Li & Nirenberg (’03), and Avellaneda & Lin (’97) give W 1,∞ convergence estimates ||uε − u∗||L∞(D), ||∇uε − Pε∇u∗||L∞(D) → 0, where Pε ∈ L∞(Ω) is the corrector matrix. Similar to Ben Hassen & Bonnetier ’06. Small volume fraction asymptotics This gives an asymptotic formula for 1D∇uε,D/|D| in terms of a polarization tensor, involving Pε, and ∇u∗, not unlike C-Vogelius (’03, ’06). That’s it.

High Contrast Materials

Meta-materials can be constructed with high contrast periodic media (Felbacq-Bouchitt´ e ’04, ’09). At well chosen frequencies, infinite resolution would be possible. Collaboration with Marc Briane (Rennes) and Luc Nguyen (Princeton)

High contrast materials ` a la Fenchenko & Khruslov ’81

ω0 ω x3 = −L x3 = L

Consider −div (aε ∇Wε) = 0 in Ω , Wε = Φ

• n ∂Ω .

with aε = 1 except in the rods of radius εrε, where κ− ≤ aεπr2

ε ≤ κ+.

The radius rε satisfies 2π γ+ ≤ −ε2 ln rε ≤ 2π γ− .

High contrast materials

Suppose lim

ε→0 αε πr2 ε = κ ∈ [κ−, κ+],

lim

ε→0

2π ε2 |ln rε| = γ ∈ [γ−, γ+]. and Φb ∈ C 1(¯ Ω). Then, the solution Wε converges weakly in H1(Ω) to the unique solution W∗ of the coupled system            − ∆W∗ + γ (W∗ − V∗) 1Ω0 = 0 in Ω − κ ∂2

33V∗ + γ (V∗ − W∗) = 0

in Ω0 = ω0 × (−L, L) W∗ = Φb

• n ∂Ω

V∗(·, ±L) = Φb(·, ±L) in ω0.

High contrast materials

The function V∗ is a limit 1 |rods|1rodsWε ⇀ V∗ weakly ∗ in M(Ω). Consider the capacity function cσ

ε for some σ ≥ 1, in Ω

cσ

ε (x) :=

       if r = |(x1, x2) − (m, n)ε| ≤ εrε ln r − ln(εrε) ln(εσ/2) − ln(εrε) if r ∈ (εrε, εσ/2), 1 elsewhere. Then ∇Wε · ∇cσ

ε ⇀ γ(U∗ − V∗) weakly ∗ in M(Ω).

∇Wεcσ

ε ⇀ ∇W∗ weakly ∗ in M(Ω).

Strong H convergence with correctors. No extra-regularity

For a C 1 boundary data, lim

ε→0

• Ω

aε

• ∇Wε−∇cσ

ε (W∗ − V∗)−cσ ε ∇W∗−(1−cσ ε ) ∂3V∗ e3

• 2 dx = 0.

The corrector term is just L2. For a bounded contrast, Pε ∈ L∞.

Strong H convergence with correctors. No extra-regularity

For a C 1 boundary data, lim

ε→0

• Ω

aε

• ∇Wε−∇cσ

ε (W∗ − V∗)−cσ ε ∇W∗−(1−cσ ε ) ∂3V∗ e3

• 2 dx = 0.

The corrector term is just L2. For a bounded contrast, Pε ∈ L∞. Let Φ ∈ C 2(¯ Ω) be such that the functions Φ(·, ±L) are both harmonic at most on a set of zero measure. Then, for any Ω′ ⊂ Ω there exists a constant C(Φ, Ω′) > 0 such that for any ρ ∈ (0, 1), lim inf

ε→0

• r2ρ(1−2/p)

ε

∇Wε2

Lp(Ω′)

• ≥ (1 − ρ)C(Φ, Ω′).

Blow-up everywhere at exponential rate (exp(C/ε2)) in W 1,p loc(Ω0).

Proof

Let Oε =

• x ∈ ω0 : 0 < dist (x, rods) < εrρ

ε

• × (−L, L),

Using the definition of cσ

ε , and the continuity of W ∗ − V ∗,

1Oε |∇cσ

ε |2 (V∗−W∗)2 dx ⇀ γ(1−ρ) (V∗−W∗)2

weakly-∗ in M(Ω0). The limit cannot cancel on any Ω′, by analyticity w.r.t. x3. The corrector result shows that 1Oε

• ∇Wε − ∇cσ

ε (W∗ − V∗)

• 2 −

→ 0 strongly in L1

loc(Ω0).

Therefore 1Oε|∇Wε|2 ⇀ γ(1 − ρ)(V∗ − W∗)2 weakly- ∗ in M(Ω0).

Proof

Fix Ω′′ ⋐ Ω′ ⊂ Ω0, and ζ a cut-off function.

• Ω′′ γ(1 − ρ)(V∗ − W∗)2 dx ≤
• Ω0

γ(1 − ρ)(V∗ − W∗)2 ζ dx = lim

ε→0

• Oε

|∇Wε|2 ζ dx ≤ lim inf

ε→0

• Oε∩Ω′ |∇Wε|2 dx

≤ lim inf

ε→0

• |Oε|1−2/p∇Wε2

Lp(Ω′\Qε)

• ≤ C1 lim inf

ε→0

• r2ρ(1−2/p)

ε

∇Wε2

Lp(Ω′\Qε)

• .

And this is valid for all Ω′′ ⋐ Ω′. Also true without touching the rods (ρ′ − ρ....)

Regularity away from the rods

For a fixed ω0 ⋐ ω1 ⋐ ω, define ωε

τ by

ωε

τ = {x ∈ ω1 : dist (x, rods) ≥ ε1+τ}.

(2) Concerning −div(aε∇Uε) = F in H1

0(Ω) we show

Theorem For τ >

1−η 2(1+η) with η ∈

3

4, 1

• , and 0 < ν < 2
• η − 3

4

• , the

solution Uε satisfies Uε − V C 1,ν(ωε

τ×(−L,L)) ≤ C(η, ν)FL2(Ω),

where V ∈ H1

0(Ω) is the solution of

−∆V = F.

Regularity away from the rods

Concerning −div(aε∇Wε) = 0 in H1(Ω), and u = Φb on ∂Ω we show Theorem Let Ωε

τ = ωε τ × (−l, l), with l < L. For τ > 1−η 2(1+η) with η ∈

3

4, 1

• ,

and 0 < ν < 2

• η − 3

4

• , the solution Wε satisfies

WεC 1,ν(Ωε

τ) ≤ C(η, ν, l) ΦbC 1(¯

Ω).

Sketch of Proof

Using the 2 + 1 form of the problem, Ω = ω × (−L, L), we can reduce both problems to −div2(aε ∇2uε) + λ aε uε = f + aε g + div2(aεh) in ω , uε = 0

• n ∂ω ,

with f ∈ L2(ω), √aεg ∈ L2(ω), h ∈ L∞(ω)2 and λ = n2π2 L2 . Compare uε with v, −∆2 + λ v = f + g + div2(h) in ω , v = 0

• n ∂ω ,

Sketch of Proof

Write uε − v = ˜ uε + ˆ uε outside the rods, where ˜ uε ∈ H1(ω \ rods) is the solution to    −∆˜ uε + λ ˜ uε = 0 in ω \ rods , ˜ uε = 0

• n ∂ω ,

˜ uε = um,n

• n ∂rodsm,n .

(3) with um,n = 1 |∂rodsm,n|

• ∂rodsm,n

[uε(x) − v(x)] dσ(x), Constant coefficients, specific geometry, and the b.v. are

• constants. Fine estimates are possible..

˜ uεL∞(ωε

τ) + λβ−1/2∇˜

uεL∞(ωε

τ) ≤ C(β)

λ1/2 sup

m,n |um,n|

for 0 < β < 1

2, and τ > 1−2β 1+2β.

Sketch of Proof

The remainder ˆ uε ∈ H1(ω \ rods) is the solution to −∆ˆ uε + λ ˆ uε = 0 in ω \ rods , ˆ uε = uε − v − um,n,

• n ∂rodsm,n.

(4) Poincar´ e inequality, energy estimates for uε and local estimates for v make this term negligible compared to the first two (v and ˜ uε). So we have a proof if we can bound sup

m,n |um,n| ≤ uεL∞(ω) + vL∞(ω)

appropriately (with the enough decay w.r.t λ).

Sketch of Proof

For any q > 2, and any ω′ ⋐ ω, vL∞(ω′) ≤ C(q, ω′) λ

1 q

• f L2(ω) + gL2(ω) + hL∞(ω)
• .

And if g = 0 and h = 0, we have also uεL∞(ω) ≤ C(q) λ1/q f L2(ω) (5) (de Giorgi-Moser-Nash iteration)

Counter-example

Consider −div2(aε ∇2uε) + λ aε uε = aε f in ω , uε = 0

• n ∂ω ,

Let 1D0,0,ε be the indicator function of the disk D0,0,ε ⊂ R2 centered at the origin and of radius εrε. Set fε := ε−11D0,0,ε and h ≡ 0. Then, we have √aεfεL2(ω) ≤ √κ+, and uεL∞(ω) ≥ 1 ε κ− κ+ κ− γ+ + λ κ+ + o(1), where limε→0 o(1) = 0.

• Proof. Write the energy bound
• ω

aε |∇uε|2 dx +

• ω

λ aε u2

ε dx =

• ω

aε fε uε dx ≤ κ+ ε uεL∞(ω) .

Counter-example

• 2. Set gε := ε−1(1 − cσ

ε ) 1εY ∈ H1 0(ω), for σ = 1, and compute its

energy

• ω

aε |∇gε|2 dx +

• ω

λ aε g2

ε dx ≤ γ+ + λ κ+ + o(1).

• 3. Use Cauchy-Schwarz :
• κ+ ε uεL∞(ω)

1

2 (γ+ + λ κ+ + o(1)) 1 2

≥

• ω

aε ∇uε · ∇gε dx +

• ω

aε uε gε dx =

• ω

aε fε gε dx =

• D0,0,ε

aε ε−2dx ≥ κ−.

Sketch of Proof

However, this is only powers of ε, not of exp(1/ε2). Optimal Sobolev constant:

• ω

aε |v|s dx ≤ C(s) ε2−s

• ω

aε |∇v|2 dx s

2

∀ v ∈ H1

0(ω).

for all s > 2. For ϕε ∈ H1

0 (ω) solution of

−div (aε∇ϕε) + λaεϕε = aεfε + div (aεh) . Then, for any 1 < α < 2, 0 < β < 1 − α/2, we have ϕεL∞(ω) ≤ C(α, β) εαλβ

• √aεf L2(ω) + ε

α+1 2 √aεhL∞(ω)

• .

Can be controlled with the regularity of Φb.

What have we obtained ?

H-convergence (strong with correctors). Uniform Local C 1,α regularity ǫ1+τ away from the rods (everywhere except in a set ∝ ε2τ). Uniform global W 1,p, p > 2 regularity is false (lower bounds) Small volume fraction asymptotics?

Small volume asymptotics for high contrast

Assume that Gε ⊂ Ωε, |Gε| → 0, and |Gε|−11Gε tends to a limit measure µ. There exists a subsequence still denoted ε, and a matrix-valued function M ∈ L2(Ω, µ)3×3 such that the bilinear response form has the following asymptotic form R(Φ) = |Gε|

• Ω

M∇W ∗ · ∇W ∗ dµ + o (|Gε|) for any Φ ∈ C 1(¯ Ω), where W ∗ denotes the solution to the homogenized problem with boundary condition Φ, and lim

ε→0

sup

ΦC1(¯

Ω)≤1

• (|Gε|)

|Gε| = 0. In addition, the matrix-valued function M is symmetric, independent of Φ, and satisfies (γ1 − 1) min

• 1, 1

γ1

• ≤ M(y)ξ · ξ ≤ max
• 1, 1

γ1

• µ-a.e.

Another viewpoint: homogenization theory

In the case of smooth scatterers, with bounded coefficients, homogenization is stable under defect: the effect of the defect is ∝ size × contrast. In the case of rods, with critical coefficients, defects in the substrate not too near the rods are also harmless (in the same way).