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Localisation principle for 1-scale H-measures
Marko Erceg
Department of Mathematics, Faculty of ScienceUniversity of Zagreb
Joint work with Nenad Antonic and Martin Lazar
Dubrovnik, 29th May, 2014
IntroductionH-measuresSemiclassical measures1-scale H-measuresDefinition
Localisation principleMotivation1-scale H-measures
2 23
H-measures
Ω ⊆ Rd open.
Theorem
If un 0 in L2(Ω;Cr), then there exist a subsequence (un′) andµH ∈Mb(Ω× Sd−1; Mr(C)) such that for every ϕ1, ϕ2 ∈ C0(Ω) andψ ∈ C(Sd−1)
limn′
∫Rd
ϕ1un′(ξ)⊗ ϕ2un′(ξ)ψ( ξ
|ξ|
)dξ = 〈µH , ϕ1ϕ2 ψ〉 .
Measure µH we call the H-measure corresponding to the (sub)sequence (un).
Theorem
unL2loc−→ 0 ⇐⇒ µH = 0 .
[T1] Luc Tartar: H-measures, a new approach for studying homogenisation,oscillations and concentration effects in partial differential equations,Proceedings of the Royal Society of Edinburgh, 115A (1990) 193–230.
3 23
Semiclassical measures
Theorem
If un 0 in L2(Ω;Cr), εn → 0, then there exist a subsequence (un′) andµsc ∈Mb(Ω×Rd; Mr(C)) such that for every ϕ1, ϕ2 ∈ C0(Ω) andψ ∈ S(Rd)
limn′
∫Rd
ϕ1un′(ξ)⊗ ϕ2un′(ξ)ψ(εn′ξ) dξ = 〈µsc, ϕ1ϕ2 ψ〉 .
Measure µsc we call the semiclassical measure with characteristic length εncorresponding to the (sub)sequence (un).
Definition
(un) is (εn)-oscillatory if(∀ϕ ∈ C∞c (Ω)) limR→∞ lim supn
∫|ξ|> R
εn
|ϕun(ξ)|2 dξ = 0 .
Theorem
unL2loc−→ 0 ⇐⇒ µsc = 0 & (un) is (εn)− oscillatory .
[PG] Patrick Gerard: Mesures semi-classiques et ondes de Bloch, Sem. EDP1990–91 (exp. 16), (1991) 4 23
Example 1: Oscillations - one characteristic length
α > 0, k ∈ Zd \ 0,
un(x) := e2πinαk·x L2
loc−− 0 , n→∞
µH = λ(x) δ k|k|
(ξ)
µsc = λ(x)
δ0(ξ) , limn n
αεn = 0δck(ξ) , limn n
αεn = c ∈ 〈0,∞〉0 , limn n
αεn =∞
sin( 4√nπx)
sin(nπx)
sin(n2πx)
n = 2
5 23
Example 2: Oscillations - two characteristic length
0 < α < β, k, s ∈ Zd \ 0,
un(x) := e2πinαk·x L2
loc−− 0 , n→∞
vn(x) := e2πinβ s·x L2
loc−− 0 , n→∞
µH (µsc) is H-measure (semiclassical measure with characteristic length εn,εn → 0) corresponding to (un + vn).
µH = λ(x)(δ k|k|
+ δ s|s|
)(ξ)
µsc = λ(x)
2δ0(ξ) , limn n
βεn = 0(δcs + δ0)(ξ) , limn n
βεn = c ∈ 〈0,∞〉δ0(ξ) , limn n
βεn =∞ & limn nαεn = 0
δck , limn nαεn = c ∈ 〈0,∞〉
0 , limn nαεn =∞
6 23
Compatification of Rd \ 0
Rd
Σ∞
Σ0
Σ0 := 0ξ0 : ξ0 ∈ Sd−1
Σ∞ := ∞ξ0 : ξ0 ∈ Sd−1
K0,∞(Rd) := Rd \ 0 ∪ Σ0 ∪ Σ∞
Corollary
a) C0(Rd) ⊆ C(K0,∞(Rd)).b) ψ ∈ C(Sd−1), ψ π ∈ C(K0,∞(Rd)), where π(ξ) = ξ/|ξ|.
7 23
1-scale H-measures
Theorem
If un 0 in L2(Rd;Cr), εn 0, then there exist a subsequence (un′) andµsc ∈Mb(Rd ×Rd; Mr(C)) such that for every ϕ1, ϕ2 ∈ C0(Rd) andψ ∈ S(Rd)
limn′
∫Rd
(ϕ1un′)(ξ)⊗ (ϕ2un′)(ξ)ψ(εn′ξ) dξ = 〈µsc, ϕ1ϕ2 ψ〉 .
Measure µsc we call the semiclassical measure with characteristic length εncorresponding to the (sub)sequence (un).
[T2] Luc Tartar: The general theory of homogenization: A personalizedintroduction, Springer (2009)
8 23
Some properties
Theorem
ϕ1, ϕ2 ∈ Cc(Ω), ψ ∈ S(Rd), ψ ∈ C(Sd−1).
a) 〈µK0,∞ , ϕ1ϕ2 ψ〉 = 〈µsc, ϕ1ϕ2 ψ〉 ,b) 〈µK0,∞ , ϕ1ϕ2 ψ π〉 = 〈µH , ϕ1ϕ2 ψ〉 ,
where π(ξ) = ξ/|ξ|.
Theorem
a) µ∗K0,∞ = µK0,∞
b) unL2loc−→ 0 ⇐⇒ µK0,∞ = 0
c) µK0,∞(Ω× Σ∞) = 0 ⇐⇒ (un) is (εn)− oscillatory
9 23
Example 1 revisited
un(x) = e2πinαk·x,
µH = λ(x) δ k|k|
(ξ)
µsc = λ(x)
δ0(ξ) , limn n
αεn = 0δck(ξ) , limn n
αεn = c ∈ 〈0,∞〉0 , limn n
αεn =∞
µK0,∞ = λ(x)
δ
0k|k|
(ξ) , limn nαεn = 0
δck(ξ) , limn nαεn = c ∈ 〈0,∞〉
δ∞
k|k|
(ξ) , limn nαεn =∞
10 23
Example 2 revisited
un(x) = e2πinαk·x, vn(x) = e2πin
β s·x,Corresponding measures of (un + vn):
µH = λ(x)(δ k|k|
+ δ s|s|
)(ξ)
µsc = λ(x)
2δ0(ξ) , limn n
βεn = 0(δ0 + δcs)(ξ) , limn n
βεn = c ∈ 〈0,∞〉δ0(ξ) , limn n
βεn =∞ & limn nαεn = 0
δck , limn nαεn = c ∈ 〈0,∞〉
0 , limn nαεn =∞
µK0,∞ = λ(x)
(δ0
k|k|
+ δ0
s|s|
)(ξ) , limn nβεn = 0
(δ0
k|k|
+ δcs)(ξ) , limn nβεn = c ∈ 〈0,∞〉
(δ0
k|k|
+ δ∞
s|s|
)(ξ) , limn nβεn =∞ & limn n
αεn = 0
(δck + δ∞
s|s|
)(ξ) , limn nαεn = c ∈ 〈0,∞〉
(δ∞
k|k|
+ δ∞
s|s|
) , limn nαεn =∞
11 23
Motivation (Localisation principle for H-measures)
Let Ω ⊆ Rd open, m ∈ N, un 0 in L2loc(Ω;Cr), Aα ∈ C(Ω; Mr(C)) and
Pun =∑|α|=m
∂α(Aαun) −→ 0 in H−mloc (Ω;Cr) .
Then we havep(x, ξ)µ>H = 0 ,
where p(x, ξ) =∑|α|=m ξαAα(x) is the principle simbol of P.
Idea: If p is nowhere zero (e.g. elliptic operator of the second order), we knowµH = 0, and that implies un −→ 0 in L2
loc(Ω;Cr).
Applications:• compactness by compensation• small amplitude homogenisation• velocity averaging• averaged control
. . .
12 23
Motivation (localisation principle for semiclassical measures
Let Ω ⊆ Rd open, m ∈ N, un 0 in L2loc(Ω;Cr) and
Pnun =∑|α|6m
ε|α|n ∂α(Aαun) = fn in Ω ,
where• εn → 0, εn > 0• Aα ∈ C(Ω; Mr(C))• fn −→ 0 in L2
loc(Ω;Cr).Then we have
p(x, ξ)µ>sc = 0 ,
where p(x, ξ) =∑|α|6m ξαAα(x), and µsc is semiclassical measure with
characteristic length (εn), corresponding to (un).
Problem: µsc = 0 is not enough for the strong convergence!
13 23
Localisation principle
Let Ω ⊆ Rd open, m ∈ N, un 0 in L2loc(Ω;Cr) and∑
l6|α|6m
ε|α|−ln ∂α(Aαun) = fn in Ω ,
where• l ∈ 0..m• εn → 0, εn > 0• Aα ∈ C(Ω; Mr(C))• fn ∈ H−mloc (Ω;Cr) such that
(∀ϕ ∈ C∞c (Ω))ϕfn
1 +∑ms=l ε
s−ln |ξ|s
−→ 0 in L2(Rd;Cr) (C(εn))
Lemma
a) (C(εn)) is equivalent to
(∀ϕ ∈ C∞c (Ω))ϕfn
1 + |ξ|l + εm−ln |ξ|m−→ 0 in L2(Rd;Cr) .
b) (∃ k ∈ l..m) fn −→ 0 in H−kloc (Ω;Cr) =⇒ (εk−ln fn) satisfies (C(εn)).
14 23
Localisation principle
∑l6|α|6m
ε|α|−ln ∂α(Aαun) = fn in Ω ,
(∀ϕ ∈ C∞c (Ω))ϕfn
1 +∑ms=l ε
s−ln |ξ|s
−→ 0 in L2(Rd;Cr) . (C(εn))
Theorem (Tartar (2009))
Under previous assumptions and l = 1, 1-scale H-measure µK0,∞ with
characteristic length εn corresponding to (un) satisfies
supp (pµ>K0,∞) ⊆ Ω× Σ0 ,
where
p(x, ξ) :=∑
16|α|6m
(2πi)|α|ξα
|ξ|+ |ξ|mAα(x) .
15 23
Proof (Step 1: inserting test function)
∑l6|α|6m
ε|α|−ln ∂α(Aαun) = fn /ϕ ∈ C∞c (Ω)
=⇒∑
l6|α|6m
∑06β6α
(−1)|β|(α
β
)ε|α|−ln ∂α−β
((∂βϕ)Aαun
)= ϕfn
• ∂α−β
((∂βϕ)Aαun
)has compact support
=⇒ ∂α−β
((∂βϕ)Aαun
)−→ x in H−|α|(Ω;Cr) , 0 < β 6 α
=⇒ (−1)|β|(α
β
)ε|α|−ln ∂α−β
((∂βϕ)Aαun
)satisfies (C(εn))
We can rewrite ∑l6|α|6m
ε|α|−ln ∂α(Aαϕun
)= fn
where (fn) satisfies (C(εn)).
16 23
Proof (Step 2: Fourier transform)
After applying Fourier transform and multiplying by 1
1+|ξ|l+εm−ln |ξ|mwe get:
∑l6|α|6m
ε|α|−ln (2πi)|α|ξαAαϕun
1 + |ξ|l + εm−ln |ξ|m=
fn1 + |ξ|l + εm−ln |ξ|m
L2
−→ 0 .
Lemma
(fn) mesurable (vector valued) on Rd, hn > 0 and
(∀ r > 0)(∃ C > 0)(∀n ∈ N)(∀ ξ ∈ Rd \K(0, r)) hn(ξ) > C ,
(un) bounded in L2(Rd;Cr) ∩ L1(Rd;Cr) and fn1+hn
· un −→ 0 in L2(Rd) .
If (h−2n |fn|2) is equiintegrable then
fnhn· un −→ 0 in L2(Rd) .
=⇒∑
l6|α|6m
(2πi)|α|ε|α|−ln ξα
|ξ|l + εm−ln |ξ|mAαϕun −→ 0 in L2(Rd;Cr)
17 23
Proof (Step 3: passing to the limit)
Multiplication by ψ(εn·)ϕ1un, ψ ∈ C(K0,∞(Rd)), ϕ1 ∈ C∞c (Ω), andintegration gives us
0 = limn
∫Rd
ψ(εnξ)
( ∑l6|α|6m
(2πi)|α|(εnξ)α
|εnξ|l + |εnξ|mAαϕun
)⊗(ϕ1un
)dξ
=
⟨ ∑l6|α|6m
(2πi)|α|ξα
|ξ|l + |ξ|mAαµK0,∞ , ϕϕ1 ψ
⟩,
where we have used ξ 7→ ξα
|ξ|l+|ξ|m ∈ C(K0,∞(Rd)), l 6 |α| 6 m.
Taking ϕ1 = 1 on suppϕ and using ¯µK0,∞ = µ>K0,∞ we get the result.Q.E.D.
18 23
Localisation principle - final generalisation
Theorem
εn > 0 bounded un 0 in L2loc(Ω;Cr) and∑
l6|α|6m
ε|α|−ln ∂α(Aαn un) = fn ,
where Aαn ∈ C(Ω; Mr(C)), Aα
n −→ Aα uniformly on compact sets, andfn ∈ H−mloc (Ω;Cr) satisfies (C(εn)).Then for ωn → 0 such that limn
ωnεn
= c ∈ [0,∞], corresponding 1-scaleH-measure µK0,∞ with characteristic length ωn satisfies
pµ>K0,∞ = 0 ,
where
p(x, ξ) :=
∑|α|=l
ξα
|ξ|l+|ξ|mAα(x) , limnωnεn
=∞∑l6|α|6m
(2πic
)|α|ξα
|ξ|l+|ξ|mAα(x) , limnωnεn
= c ∈ 〈0,∞〉∑|α|=m
ξα
|ξ|l+|ξ|mAα(x) , limnωnεn
= 0
19 23
Localisation principle - final generalisation
Theorem (cont.)
Moreover, if there exists ε0 > 0 such that εn > ε0, n ∈ N, we can take
p(x, ξ) :=∑|α|=m
ξα
|ξ|mAα(x) .
Sketch of the proof. Suppose that we have already obtained the result forlimn
ωnεn∈ 〈0,∞〉. Idea is reduce other two cases to this case.
In the case limnωnεn
=∞ we rewrite equations in the form∑l6|α|6m
ω|α|−ln ∂α(Bαn un) = fn ,
for Bαn :=
(εnωn
)|α|−lAαn , and similary for the case limn
ωnεn
= 0 we have∑l6|α|6m
ω|α|−ln ∂α(Bαn un) = gn ,
where Bαn :=
(ωnεn
)m−|α|Aαn , and gn :=
(ωnεn
)m−lfn.
20 23
Localisation principle (H-measures and semiclassical measures)
• Using preceding theorem and µK0,∞ = µH on Ω× Sd−1, we can obtainedknown localisation principle for H-measures.
Theorem
Under the assumptions of the preceding theorem, we have
p(x, ξ)µ>sc = 0 ,
where
p(x, ξ) :=
∑|α|=l ξ
αAα(x) , limnωnεn
=∞∑l6|α|6m
(2πic
)|α|ξαAα(x) , limn
ωnεn
= c ∈ 〈0,∞〉∑|α|=m ξαAα(x) , limn
ωnεn
= 0
21 23
Proof (only case limnωnεn
= c ∈ 〈0,∞〉)
ψ ∈ S(Rd) =⇒ ξ 7→ (|ξ|l + |ξ|)ψ(ξ) ∈ C(K0,∞(Rd))
0 =
⟨ ∑l6|α|6m
(2πi
c
)|α| ξα
|ξ|l + |ξ|mAαµK0,∞ , ϕ (|ξ|l + |ξ|m)ψ
⟩
=
⟨µK0,∞ ,
∑l6|α|6m
(2πi
c
)|α|ϕAα ξαψ
⟩
=
⟨µsc,
∑l6|α|6m
(2πi
c
)|α|ϕAα ξαψ
⟩=
⟨ ∑l6|α|6m
(2πi
c
)|α|ξαAαµsc, ϕ ψ
⟩,
where we have used ξαψ ∈ S(Rd) and that µK0,∞ and µsc coincide on S(Rd).
22 23
Summary
• H-measures do not catch frequency• In some cases, semiclassical measures do not catch direction• 1-scale H-measures are generalisation of H-measures and semiclassical
measures and do not have above anomalies
• Localisation principle for 1-scale H-measures is obtained• Localisation principles for H-measures and semiclassical measures via
localisation principle for 1-scale H-measures
23 23