Directional oscillations and concentrations and weak↔strong
compactness via microlocal compactness forms
Filip Rindler
Cambridge Centre for Analysis (CCA)
University of Cambridge (DAMTP)
F.Rindler@maths.cam.ac.uk
6 December 2012
Oscillations & concentrations
Prototypical oscillation: uj (x) = sin(jx · n)
,
,
n
, ···
0
In Fourier space: mass wanders out to ∞:
Prototypical concentration in Lp : uj (x) = j d/p h(jx)
,
,
, ···
Why study oscillations / concentrations?
Weak↔strong compactness:


p


 (uj ) ⊂ L bounded,

Vitali’s Convergence Thm
uj → u in measure ⇔ no oscillations,
=⇒
uj → u




(uj ) p-equiintegrable ⇔ no concentrations
Compensated compactness (Tartar’s framework):


uj * u in Lp ,








d


X
∂u
???
j
(k)
−1,p
Auj :=
A
= 0 in W
, =⇒ uj → u


∂x
k


k=1






uj (x) ∈ Z (x) a.e.
Efficient description of microstructure (e.g. for minimizing sequences)
Physical meaning of weak/strong convergence:
Assume: (uj ) ⊂ L2 sequence of “measurements”.
Weak convergence uj * u:
Z
ϕ · (uj − u) dx
Parseval
Z
ϕ̂ · F [uj − u] dξ,
=
Weak convergence is band-limited
Strong convergence uj → u:
Z
|uj − u|2 dx
∀ϕ ∈ S.
Measure with “finite precision”.
Parseval
=
Strong convergence is not band-limited:
Z
|F[uj − u]|2 dξ.
Measure with “infinite precision”.
Conclusion:
Different types of convergence have different physical meanings.
The study of compactness is related to the “physical” properties of
oscillations & concentrations.
Tools to systematically study oscillations & concentrations
Measure compactness / oscillations / concentrations
for a weakly compact generating sequence (uj ) ⊂ Lp :
Location:
Scalar defect measures w*-limj→∞ |uj |p Ld
Location & value-distribution (oscillations):
Young measures (Young ’37,’42)
Location & value-distribution (oscillations & concentrations):
Generalized Young (DiPerna–Majda) measures (DiPerna & Majda ’87)
Location & direction:
H-measures (Tartar ’90, Gérard ’91)
Location & value-distribution & direction:
Microlocal compactness forms (MCFs) ←− in this work (2012).
Features of MCFs
Represent limits of functionals (essentially contains generalized Young
measure).
Preserve directional information of oscillations/concentrations preserved
(contains H-measure).
Easy to “read off” pointwise constraints (u(x) ∈ Z (x)) and differential
constraints (Auj = 0) on generating sequence.
Allow an analogue of the wavefront set from microlocal analysis, but with
respect to weak↔strong convergence (not C∞ -regularity).
Hierarchy of microstructure (e.g. laminates) reflected in MCF.
Allow relaxation of anisotropic functionals.
Can be defined for all Lp -spaces (H-measure only for L2 ).
Have the same compactness as weak convergence.
Definition of MCFs
For h ∈ C(Ω × CN ; CN ):
(S p−1 h)(x, w ) := (1 − |w |)p−1 h x,
w
,
1 − |w |
(x, w ) ∈ Ω × BN .
Space Fp of test functions:
Fp (Ω; CN ) := f ∈ C(Ω × CN × CN ; C) : f (x, z, q) = h(x, z) · q and
S p−1 h ∈ C(Ω × BN ; CN ) .
Fourier multiplier with symbol Ψ ∈ M := C∞ (Sd−1 ; CN×N ):
ξ
û(ξ)
Tψ : Lp (Ω; CN ) → Lp (Ω; CN ),
Tψ [u] := F −1 Ψ
|ξ|
d
Cut-off function η ∈ C∞
c (R ) with η ≡ 1 on B(0, 1), supp η ⊂ B(0, 2).
d
ηR (ξ) := η(ξ/R) for ξ ∈ R and any R > 1.
Microlocal compactness form ω ∈ MCFp (Ω; CN ) is a sesquilinear form
Z
f ⊗ Ψ, ω = ω(f , Ψ) = lim lim
h( q, uj ) · T(1−ηR )Ψ [uj ] dx
R→∞ j→∞
p
N
Ω
for all f ∈ F (Ω; C ), Ψ ∈ M. (Well-defined?)
Existence Theorem
Theorem (R. 2012)
Let (uj ) ⊂ Lp (Ω; CN ) be norm-bounded. Then, after selecting a subsequence,
there exist an MCF ω ∈ MCFp (Ω; CN ) such that
Z
f ⊗ Ψ, ω = lim lim
h( q, uj ) · T(1−ηR )Ψ [uj ] dx
R→∞ j→∞
p
Ω
N
for all f ∈ F (Ω; C ) with f (x, z, q) = h(x, z) · q and Ψ ∈ M. Moreover:
(A) Independence. The above limit (and hence ω) is independent of the
choice of η and the sequence R → ∞.
(B) Basic estimate. For all f , Ψ as above it holds that
f ⊗ Ψ, ω ≤ C · supj k1 + |uj |kpp · kf kFp · kΨkCd ,
where C = Cd max{p, (p − 1)−1 } with a dimensional constant Cd .
Parts of the main definition formula
h( q, uj ) ∈ Lp/(p−1) , T(1−ηR )Ψ [uj ] ∈ Lp .
Product in L1
2
concentrations expressed.
In L , Parseval yields:
Z
Z
ξ
ûj (ξ) dξ.
h( q, uj )·T(1−ηR )Ψ [uj ] dx =
F h( q, uj ) (ξ) · (1 − ηR (ξ))Ψ
|ξ|
Ω
Ω
Representation Theorem
Theorem (R. 2012)
The MCF ω ∈ MCFp (Ω; CN ) from the preceding Existence Theorem can be
considered as a triple ω = (ωx , λω , ωx∞ ) consisting of
(i) a parametrized family (ωx )x∈Ω of continuous (complex-valued) sesquilinear
forms
ωx : C(CN ; CN ) × M → C,
(ii) a positive and finite measure on Ω, and
(iii) a parametrized family (ωx∞ )x∈Ω of continuous (complex-valued)
sesquilinear forms
ωx∞ : C(∂BN ; CN ) × M → C.
Then it holds that
Z
Z
∞
f ⊗ Ψ, ω =
h(x, q) ⊗ Ψ, ωx dx +
h (x, q) ⊗ Ψ, ωx∞ dλω (x).
{z
}
{z
}
Ω|
Ω|
q
q
=ωx∞ (h∞ (x, ),Ψ)
=ωx (h(x, ),Ψ)
“oscillations”
“concentrations”
Recession function: h∞ (x, z) := limx 0 →x
z 0 →z
t→∞
h(x 0 , tz 0 )
(x ∈ Ω, z ∈ CN ).
t p−1
Oscillations
Consider: uj (x) = A1(0,θ) (jx · n0 − bjx · n0 c) + B 1(θ,1) (jx · n0 − bjx · n0 c), x ∈ Ω,
where A, B ∈ CN , n0 ∈ Sd−1 and 1(0,θ) is the indicator function of (0, θ).
What MCF does (uj ) generate?
Lemma (Oscillation Lemma)
Let w ∈ L∞ (R; CN ) be 1-periodic (the “profile function”) and assume that
wj (x) := w (js) generates the homogeneous Young measure ν ∈ M1 (CN ).
Then, the simple oscillation in direction n0 ∈ Sd−1 ,
uj (x) := w (jx · n0 ),
x ∈ Ω,
generates an microlocal compactness form
ω = Ld Ω ⊗ (z − Z0 ) ν(dz) ⊗ δ ±n0
∈ MCF2 (Ω; CN ),
R1
where Z0 := −0 w ds is the average of w over one period cell and
δ ±n0 := δ −n0 + δ +n0 . That is, for all f ∈ Fp (Ω; CN ), Ψ ∈ M,
Z Z
f ⊗ Ψ, ω =
h(x, z) · Ψ(+n0 ) + Ψ(−n0 ) (z − Z0 ) dν(z) dx.
Ω
Oscillations II
Example: Simple oscillation
uj (x) = A1(0,θ) (bjx · n0 c) + B 1(θ,1) (bjx · n0 c),
where A, B ∈ CN , n0 ∈ Sd−1 and
x ∈ Ω,
1(0,θ) is the indicator function of (0, θ).
Young measure: ν = θδA + (1 − θ)δB .
MCF: By the Oscillation Lemma the sequence (uj ) generates
ω = Ld Ω ⊗ θ(A − M)δA + (1 − θ)(B − M)δB ⊗ δ ±n0 ∈ MCF2 (Ω; CN ),
where M := θA + (1 − θ)B.
Proof of Oscillation Lemma:
n0
Oscillations III
Concentrations
Let w ∈ Lp (Rd ) have compact support and w ≥ 0. Further, let Z0 ∈ CN with
|Z0 | = 1. Define
uj (x) := j d/p Z0 w (jx),
x ∈ Rd .
Then, uj generates the MCF
ω = δ0 ⊗ Z0 δ∞Z0 ⊗ µ̄
∈ MCFp (Rd ; CN ),
where µ̄ is the surface measure (acting on Ψ)
Z ∞
µ̄ =
F |w |p−1 (tη) ŵ (tη) t d−1 dt Hd−1
Sd−1 (dη).
0
The above is a shorthand notation for the MCF ω acting on
f (x, z, q) = h(x, z) · q ∈ Fp (Ω; CN ), Ψ ∈ M, as
Z
Ψ(ξ)Z0 dµ̄(ξ).
f ⊗ Ψ = h∞ (0, Z0 ) ·
Sd−1
Remarks
In the Existence Theorem, we can choose for example
(↔ scalar defect measure)
λω = w*-lim |uj |p Ld .
j→∞
(?)
The families (ωx )x , (ωx∞ )x are uniquely determined if we use the canonical
choice (?).
Proofs of existence/representation theorems use techniques from
harmonic analysis, measure theory (disintegration/slicing of measures),
and the theory of generalized Young measures (we use an argument similar
to Kristensen & R. 2010, ARMA).
The representation as before allows us to (in some sense) localize in both
the x- and ξ-variable and so “circumvent the uncertainty principle”!
Reason: We only look at infinite frequencies.
Some properties of MCF
Lemma
Let (uj ) ⊂ Lp (Ω; CN ) generate the MCF ω ∈ MCFp (Ω; CN ) and uj * u in
Lp (Ω; CN ). Then uj → u strongly if and only if ω = 0
The MCF ω measures the difference between weak and strong convergence.
Young measures:
Proposition
Let (uj ) ⊂ Lp (Ω; CN ) generate the MCF ω ∈ MCFp (Ω; CN ) and the
(generalized) Young measure ν ∈ Y(Ω; CN ). Assume further that uj * u = [ν].
Then, the knowledge of ω and of the limit u completely determine ν.
H-measures: They are trivially contained (in the case uj * 0 in L2 ).
Differential constraints
Differential operator A:
A :=
d
X
∂
∂xk
A(k)
k=1
Symbol: A(ξ) :=
Pd
k=1
A(k) ξk .
(Au = 0 ⇔ (2πi)A(ξ)û(ξ) = 0)
Homogeneous symbol: A0 (ξ) :=
Pd
k=1
A(k) ξk /|ξ|.
Murat’s constant-rank property: rank ker A(ξ) = const for all ξ ∈ Sd−1 .
Theorem (R. 2012)
Let (uj ) ⊂ L2 (Rd ; CN ) with uj * u in L2 (Rd ; CN ) generate the MCF
ω ∈ MCF2 (Rd ; CN ). Then,
Auj → 0
in W−1,2 (Rd ; Cl ).
if and only if
2
N
f ⊗ ΨA0 , ω
for all f ∈ F (Ω; C ) and all Ψ ∈ C
bd/2c+1
=0
d−1
(S
; CN×l ).
Higher-order laminates
Homogeneous MCFs: ω ∈ MCFp (Ω; CN ) is called homogeneous if ωx , ωx∞
are constant in x and λω = αLd Ω for a constant α > 0.
Proposition (Laminations)
Let n0 ∈ Sd−1 . Assume that
(i) uj * A = const and vj * B = const in Lp (Rd ; CN ),
(ii) (uj ), (vj ) are p-equiintegrable,
(iii) (uj ), (vj ) generate the homogeneous MCFs ω1 , ω2 ∈ MCFphom (CN ) and
the homogeneous Young measures ν1 , ν2 ∈ M1 (CN ), respectively.
Then, for any θ ∈ (0, 1) there exists a homogeneous microlocal compactness
form ω̄ ∈ MCFphom (CN ) with
ω̄ = θω1 + (1 − θ)ω2 + Ld ⊗ θ(A − X )ν1 + (1 − θ)(B − X )ν2 ⊗ δ ±n0 ,
|
{z
} |
{z
}
faster scales
slower scale
where X := θA + (1 − θ)B.
Proof: Similar argument as in Oscillation Lemma, and averaging.
Hierarchy of scales reflected.
Lamination example
C
n1
A
B A
n2
B A
B A
B A
B A
1 − θ1
M
A
1 − θ2
X
θ2
θ1
B
C
C
A
B A
B A
B A
M = θ1 A + (1 − θ1 )B
X = θ2 M + (1 − θ2 )C
Generated MCF:
ω = θ2 ωA/B + (1 − θ2 )ωC + Ld ⊗ θ2 (M − X )ν1 + (1 − θ2 )(C − X )ν2 ⊗ δ ±n2
n
= Ld Ω ⊗ θ2 θ1 (A − M)δA + θ2 (1 − θ1 )(B − M)δB ⊗ δ ±n1
+ θ2 (M − X )(θ1 δA + (1 − θ1 )δB ) ⊗ δ ±n2
o
+ (1 − θ2 )(C − X )δC ⊗ δ ±n2 .
Wavefront set
Sphere compactification: σCN := CN ] ∞SN−1 .
Wavefront set for ω ∈ MCF(Ω; CN ):
WF(ω) ⊂ Ω × σCN × Sd−1 is the smallest closed subset A of Ω × σCN × Sd−1
with the property that for any f (x, z, q) = ϕ(x)g (z) · q ∈ Fp (Ω; CN ) and any
multiplier Ψ ∈ M,
supp ϕ ⊗ g ⊗ Ψ ⊂ Ac
implies
f ⊗ Ψ, ω = 0.
Examples:
Oscillations: ω = Ld
Ω ⊗ θ(A − M)δA + (1 − θ)(B − M)δB ⊗ δ ±n0 :
WF(ω) = Ω × {A, B} × {+n0 , −n0 }.
Concentrations: ω = δ0 ⊗ Z0 δ∞Z0 ⊗ µ̄:
WF(ω) = (0, ∞Z0 ) × supp µ̄,
“Concentrations lie in ∞SN−1 ”.
Compensated compactness
Consider general framework (Tartar):

uj * u in Lp ,




d

X
∂uj
Auj :=
A(k)
= 0 in W−1,p ,
∂x

k

k=1



uj (x) ∈ Z a.e.,
Z ⊂ CN closed.
Symbol: A(ξ) :=
Pd
k=1
A(k) ξk .
(CC)
(Au = 0 ⇔ (2πi)A(ξ)û(ξ) = 0)
Theorem (R. 2012)
Let (uj ) ⊂ Lp (Rd ; CN ) generate ω ∈ MCFp (Rd ; CN ) and let A be a linear PDE
operator satisfying Murat’s constant-rank property, i.e.
rank ker A(ξ) = const
for all ξ ∈ Sd−1 .
Moreover, assume (CC). Then,
WF(ω) ⊂ Ξ := (x, z, ξ) ∈ Rd × σCN × Sd−1 : spanC Z ∩ ker A(ξ) 6= {0} .
In particular, if Ξ = ∅, then ω = 0 and (uj ) is strongly compact.
Compensated compactness example I: Gradients
Let (uj ) ⊂ Lp (Ω; Rm×d ) generate the MCF ω ∈ MCFp (Ω; Cm×d ) and
(
curl uj = 0
in W−1,p ,
uj (x) ∈ span{M}
a.e.,
M ∈ Rm×d a fixed matrix.
Theorem ⇒
WF(ω) ⊂
(x, z, ξ) ∈ Ω×σRm×d ×Sd−1 : spanC {M}∩ker A(ξ) 6= {0} ,
where ker A(ξ) = { a ⊗ ξ : a ∈ Cm }.
(i) If rank M ≥ 2, then ω = 0 and (uj ) is strongly compact.
(ii) If M = a ⊗ n0 for a ∈ Rm , n0 ∈ Sd−1 , then
WF(ω) ⊂ (x, z, ξ) ∈ Ω × σRm×d × Sd−1 : ξ = ±n0 ,
Oscillations / concentrations have “direction ±n0 ”:
Compensated compactness example II: Linear elasticity
There exists a (2nd order) A acting on Rd×d
sym -valued vector fields with
1
∇v + ∇v T
if and only if
Au = 0,
u=
2
p
d
m×d
Let (uj ) ⊂ Lp (Ω; Rd×d
sym ) generate the MCF ω ∈ MCF (R ; Csym ) with
(
Auj = 0
in W−1,p ,
uj (x) ∈ span{M}
a.e.,
M ∈ Rd×d
sym a fixed matrix.
Theorem ⇒
d−1
(x, z, ξ) ∈ Ω × σRd×d
: spanC {M} ∩ ker A(ξ) 6= {0} ,
sym × S
where ker A(ξ) = a ξ = 21 a ⊗ ξ + ξ ⊗ a : a ∈ Cd .
WF(ω) ⊂
(i) If M ∈
/ {a b}, then ω = 0 and (uj ) is strongly compact.
(ii) If M = γ(a b) for some a, b ∈ Rd with |a| = |b| = 1, γ 6= 0, then
d−1
WF(ω) ⊂ (x, z, ξ) ∈ Ω × σRd×d
: ξ = ±a or ξ = ±b .
sym × S
Compensated compactness example II: Linear elasticity (continued)
The case M = a b (continued):
d−1
WF(ω) ⊂ (x, z, ξ) ∈ Ω × σRd×d
: ξ = ±a or ξ = ±b .
sym × S
Oscillations and concentrations have direction ±a or ±b:
=
+
Weak* lower semicontinuity of integral functionals depending on the
symmetric gradient and in the space BD of functions of bounded deformation
(R. 2011, ARMA)
Summary & Outlook
The theory of MCFs . . .
. . . measures the difference between weak and strong compactness
very weak regularity.
. . . can be considered microlocal analysis for weak↔strong compactness
much more adapted to nonlinear PDEs than classical microlocal
analysis, which measures C∞ -regularity.
. . . allows localizing singularities in x, ξ, and the target space
(“circumvent the uncertainty principle”).
. . . represents differential/pointwise constraints.
. . . enables “geometric” proofs of compensated compactness.
. . . reflects the hierarchy of microstructure.
Outlook:
Relaxation of integral functionals with anisotropy.
Propagation of regularity / singularities for hyperbolic conservation laws.
Finer investigations into shape of microstructure.
Thank you for your attention!