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

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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.

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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

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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 =∞

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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|.

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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)

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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

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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

. . .

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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) .

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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)).

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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)

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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.

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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

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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

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