Characteristic length of sequences via one-scale H-measures
Marko Erceg
Department of Mathematics, Faculty of Science
University of Zagreb
Bressanone, 9th February, 2016
Joint work with Nenad Antonic and Martin Lazar
Introduction
If we have un * 0 in
L2loc (
), Rd open, what we can say about jun j2 ?
2
17
Introduction
L2loc (
), Rd open, what we can say about jun j2 ?
It is bounded in L1loc (
)
If we have un * 0 in
2
17
Introduction
L2loc (
), Rd open, what we can say about jun j2 ?
It is bounded in L1loc (
) ,! M(
), so
If we have un * 0 in
jun j2 * :
is called the defect measure.
2
17
Introduction
L2loc (
), Rd open, what we can say about jun j2 ?
It is bounded in L1loc (
) ,! M(
), so
If we have un * 0 in
jun j2 * :
is called the defect measure.
Of course, we have
un
L2
loc
! 0 () = 0 :
2
17
Introduction
L2loc (
), Rd open, what we can say about jun j2 ?
It is bounded in L1loc (
) ,! M(
), so
If we have un * 0 in
jun j2 * :
is called the defect measure.
Of course, we have
un
L2
loc
! 0 () = 0 :
If the defect measure is not trivial we need another objects to determine all the
properties of the sequence:
H-measures
semiclassical measures
...
2
17
H-measures
Rd open.
Theorem
If un * 0 in L2loc (
; Cr ), then there exist a subsequence (un0 ) and
H 2 M(
Sd 1 ; Mr (C)) such that for any '1 ; '2 2 Cc (
) and
2 C(Sd 1 )
lim
n0
Z Rd
\ \
'1 un0 () '2 un0 ()
jj d = hH ; '1 '2 i :
(Unbounded) Radon measure H we call the H-measure corresponding to the
(sub)sequence (un ).
[T1] Luc Tartar: H-measures, a new approach for studying homogenisation,
oscillations and concentration eects in partial dierential equations,
Proceedings of the Royal Society of Edinburgh, 115A (1990) 193{230.
rard: Microlocal defect measures, Comm. Partial Di. Eq., 16
[G1] Patrick Ge
(1991) 1761{1794.
3
17
H-measures
Rd open.
Theorem
If un * 0 in L2loc (
; Cr ), then there exist a subsequence (un0 ) and
H 2 M(
Sd 1 ; Mr (C)) such that for any '1 ; '2 2 Cc (
) and
2 C(Sd 1 )
lim
n0
Z Rd
\ \
'1 un0 () '2 un0 ()
jj d = hH ; '1 '2 i :
(Unbounded) Radon measure H we call the H-measure corresponding to the
(sub)sequence (un ).
Theorem
un
L2
loc
! 0 () H = 0 :
[T1] Luc Tartar: H-measures, a new approach for studying homogenisation,
oscillations and concentration eects in partial dierential equations,
Proceedings of the Royal Society of Edinburgh, 115A (1990) 193{230.
rard: Microlocal defect measures, Comm. Partial Di. Eq., 16
[G1] Patrick Ge
(1991) 1761{1794.
3
17
Semiclassical measures
Theorem
If un * 0 in L2loc (
; Cr ), !n ! 0+ , then there exist a subsequence (un0 ) and
(sc!n ) 2 M(
Rd ; Mr (C)) such that for any '1 ; '2 2 C1
c (
) and
2 S (Rd )
Z D
E
'1 un0 () '2 un0 () (!n0 ) d = (sc!n ) ; '1 '2 :
lim
0
n Rd
\ \
(!n )
(Unbounded) measure sc
we call the semiclassical measure with
characteristic length (!n ) corresponding to the (sub)sequence (un ).
4
17
Semiclassical measures
Theorem
If un * 0 in L2loc (
; Cr ), !n ! 0+ , then there exist a subsequence (un0 ) and
(sc!n ) 2 M(
Rd ; Mr (C)) such that for any '1 ; '2 2 C1
c (
) and
2 S (Rd )
Z D
E
'1 un0 () '2 un0 () (!n0 ) d = (sc!n ) ; '1 '2 :
lim
0
n Rd
\ \
(!n )
(Unbounded) measure sc
we call the semiclassical measure with
characteristic length (!n ) corresponding to the (sub)sequence (un ).
Theorem
un
L2
loc
! 0 () sc(!n ) = 0 & (un ) is (!n )
oscillatory :
4
17
Semiclassical measures
Theorem
If un * 0 in L2loc (
; Cr ), !n ! 0+ , then there exist a subsequence (un0 ) and
(sc!n ) 2 M(
Rd ; Mr (C)) such that for any '1 ; '2 2 C1
c (
) and
2 S (Rd )
Z D
E
'1 un0 () '2 un0 () (!n0 ) d = (sc!n ) ; '1 '2 :
lim
0
n Rd
\ \
(!n )
(Unbounded) measure sc
we call the semiclassical measure with
characteristic length (!n ) corresponding to the (sub)sequence (un ).
Denition
(un ) is (!n )-oscillatory if
R
un ()j2 d = 0 :
(8 ' 2 C1
c (
)) limR!1 lim supn jj> !R j'd
n
Theorem
un
L2
loc
! 0 () sc(!n ) = 0 & (un ) is (!n )
oscillatory :
4
17
The Wigner transform
(un ) from L2 (Rd ; Cr ), !n ! 0+ ,
Z
Wn (x; ) :=
e 2iy un x + !n y un x !n y dy
2
2
Rd
Theorem
If un * u in L2 (
; Cr ), then there exists (un0 ) such that
0
Wn0 S*(sc!n0 ) :
rard: Mesures semi-classiques et ondes de Bloch, Sem. EDP
[G2] Patrick Ge
1990{91 (exp. 16), (1991)
[LP] Pierre Louis Lions, Thierry Paul: Sur les measures de Wigner, Revista
Mat. Iberoamericana 9, (1993) 553-618
5
17
Example 1: Oscillations - one characteristic length
> 0, k 2 Zd n f0g,
L2
un (x) := e2in x loc* 0 ; n ! 1
k
6
17
Example 1: Oscillations - one characteristic length
> 0, k 2 Zd n f0g,
L2
un (x) := e2in x loc* 0 ; n ! 1
k
=
6
17
Example 1: Oscillations - one characteristic length
> 0, k 2 Zd n f0g,
L2
un (x) := e2in x loc* 0 ; n ! 1
k
=
H = jkkj
6
17
Example 1: Oscillations - one characteristic length
> 0, k 2 Zd n f0g,
L2
un (x) := e2in x loc* 0 ; n ! 1
k
=
H = jkkj
8
<
(sc!n ) = : c
0
0
k
;
limn n !n = 0
; limn n !n = c 2 h0; 1i
;
limn n !n = 1
6
17
Example 1: Oscillations - one characteristic length
> 0, k 2 Zd n f0g,
L2
un (x) := e2in x loc* 0 ; n ! 1
k
=
H = jkkj
8
<
(sc!n ) = : c
0
0
k
;
limn n !n = 0
; limn n !n = c 2 h0; 1i
;
limn n !n = 1
6
17
Example 1: Oscillations - one characteristic length
> 0, k 2 Zd n f0g,
L2
un (x) := e2in x loc* 0 ; n ! 1
k
=
H = jkkj
sc!n )
(
8
<
= : c
0
0
k
;
limn n !n = 0
; limn n !n = c 2 h0; 1i
;
limn n !n = 1
D
hH ; ' i = (sc!n ) ; ' E
jj
6
17
Example 1: Oscillations - one characteristic length
> 0, k 2 Zd n f0g,
L2
un (x) := e2in x loc* 0 ; n ! 1
k
=
H = jkkj
8
<
(sc!n ) = : c
0
0
k
;
limn n !n = 0
; limn n !n = c 2 h0; 1i
;
limn n !n = 1
Theorem
(!n )
If un * u in L2loc (
; Cr ) is (!n )-oscillatory and trsc
(
f0g) = 0, then
D
E
hH ; ' i = (sc!n ) ; ' jj :
6
17
!n )-concentrating property
(
Denition
(un ) is (!n )-oscillatory if
R
(8 ' 2 C1
un ()j2 d = 0 :
c (
)) limR!1 lim supn jj> !R j'd
n
7
17
!n )-concentrating property
(
Denition
(un ) is (!n )-oscillatory if
R
(8 ' 2 C1
un ()j2 d = 0 :
c (
)) limR!1 lim supn jj> !R j'd
n
(un ) is (!n )-concentrating if
R
(8 ' 2 C1
un ()j2 d = 0 :
c (
)) limR!1 lim supn jj6 R!1 j'd
n
7
17
!n )-concentrating property
(
Denition
(un ) is (!n )-oscillatory if
R
(8 ' 2 C1
un ()j2 d = 0 :
c (
)) limR!1 lim supn jj> !R j'd
n
(un ) is (!n )-concentrating if
R
(8 ' 2 C1
un ()j2 d = 0 :
c (
)) limR!1 lim supn jj6 R!1 j'd
n
Lemma
(un ) !n -concentrating () tr(sc!n ) (
f0g) = 0 :
7
17
!n )-concentrating property
(
Denition
(un ) is (!n )-oscillatory if
R
(8 ' 2 C1
un ()j2 d = 0 :
c (
)) limR!1 lim supn jj> !R j'd
n
(un ) is (!n )-concentrating if
R
(8 ' 2 C1
un ()j2 d = 0 :
c (
)) limR!1 lim supn jj6 R!1 j'd
n
Lemma
(un ) !n -concentrating () tr(sc!n ) (
f0g) = 0 :
Theorem
If un * u in L2loc (
; Cr ) is (!n )-oscillatory and (!n )-concentrating, then
D
E
hH ; ' i = sc(!n ) ; ' jj :
7
17
For an arbitrary bounded sequence (un ) in L2loc (
; Cr ) is there a characteristic
length !n ! 0+ such that (un ) is
1) (!n )-oscillatory?
2) (!n )-concentrating?
3) both (!n )-oscillatory and (!n )-concentrating?
8
17
For an arbitrary bounded sequence (un ) in L2loc (
; Cr ) is there a characteristic
length !n ! 0+ such that (un ) is
1) (!n )-oscillatory?
2) (!n )-concentrating?
3) both (!n )-oscillatory and (!n )-concentrating?
Theorem
(1) is valid and (2) is valid under the additional assumption that un * 0 in
L2loc (
; Cr ).
8
17
For an arbitrary bounded sequence (un ) in L2loc (
; Cr ) is there a characteristic
length !n ! 0+ such that (un ) is
1) (!n )-oscillatory?
2) (!n )-concentrating?
3) both (!n )-oscillatory and (!n )-concentrating?
Theorem
(1) is valid and (2) is valid under the additional assumption that un * 0 in
L2loc (
; Cr ).
Theorem
For un * u in L2loc (
; Cr ) we have
un
! u in L2loc (
; Cr ) () (8 !n ! 0+ ) (un ) is (!n )
oscillatory :
8
17
Example 2: Oscillations - two characteristic length
0 < < , k; s 2 Zd n f0g,
L2
un (x) := e2in x loc* 0 ; n ! 1
L2
vn (x) := e2in x loc* 0 ; n ! 1
k
s
9
17
Example 2: Oscillations - two characteristic length
0 < < , k; s 2 Zd n f0g,
L2
un (x) := e2in x loc* 0 ; n ! 1
L2
vn (x) := e2in x loc* 0 ; n ! 1
k
s
(!n )
H (sc
) is H-measure (semiclassical measure with characteristic length
(!n ), !n ! 0+ ) corresponding to (un + vn ).
H = jkkj + jssj
9
17
Example 2: Oscillations - two characteristic length
0 < < , k; s 2 Zd n f0g,
L2
un (x) := e2in x loc* 0 ; n ! 1
L2
vn (x) := e2in x loc* 0 ; n ! 1
k
s
(!n )
H (sc
) is H-measure (semiclassical measure with characteristic length
(!n ), !n ! 0+ ) corresponding to (un + vn ).
H = jkkj + jssj
8 2
>
>
>
< (c + )
(!n )
sc = > >
>
: c
0
s
0
0
k
0
;
limn n !n = 0
;
limn n !n = c 2 h0; 1i
; limn n !n = 1 & limn n !n = 0
;
limn n !n = c 2 h0; 1i
;
limn n !n = 1
9
17
One-scale H-measures
Theorem
If un * 0 in L2loc (
; Cr ), !n ! 0+ , then there exist a subsequence (un0 ) and
(sc!n ) 2 M(
Rd ; Mr (C)) such that for any '1 ; '2 2 Cc (
) and 2 S (Rd )
Z D
E
(!n0 )
0
lim
(
'
(
!
)
d
=
;
'
:
1 un0 )( ) ('2 un0 )( )
1'
2 n
sc
n0 Rd
\ \
(Unbounded) Radon measure (sc!n ) we call the semiclassical measure with
characteristic length (!n ) corresponding to the (sub)sequence (un ).
10
17
One-scale H-measures
1
1
0e
e
0 := f0 : e 2 Sd 1 g
1 := f1 : e 2 Sd 1 g
K0;1 (Rd ) := Rd n f0g [ 0 [ 1
e
e
0
e
Rd
Corollary
a) C0 (Rd ) C(K0;1 (Rd )).
b) 2 C(Sd 1 ), 2 C(K0;1 (Rd )), where () = =jj.
10
17
One-scale H-measures
Theorem
If un * 0 in L2loc (
; Cr ), !n ! 0+ , then there exist a subsequence (un0 ) and
(K!0n;1) 2 M(
K0;1 (Rd ); Mr (C)) such that for any '1 ; '2 2 Cc (
) and
2 C(K0;1 (Rd ))
Z E
D (! 0 )
n ; '1 '2 0
:
lim
(
'
(
!
)
d
=
1 un0 )( ) ('2 un0 )( )
n
K
0;1
n0 R d
\ \
)
we call the one-scale H-measure with
(Unbounded) Radon measure (K!0n;1
characteristic length (!n ) corresponding to the (sub)sequence (un ).
10
17
One-scale H-measures
Theorem
If un * 0 in L2loc (
; Cr ), !n ! 0+ , then there exist a subsequence (un0 ) and
(K!0n;1) 2 M(
K0;1 (Rd ); Mr (C)) such that for any '1 ; '2 2 Cc (
) and
2 C(K0;1 (Rd ))
Z E
D (! 0 )
n ; '1 '2 0
:
lim
(
'
(
!
)
d
=
1 un0 )( ) ('2 un0 )( )
n
K
0;1
n0 R d
\ \
)
we call the one-scale H-measure with
(Unbounded) Radon measure (K!0n;1
characteristic length (!n ) corresponding to the (sub)sequence (un ).
[T2] Luc Tartar: The general theory of homogenization: A personalized
introduction, Springer (2009)
[T3] Luc Tartar: Multi-scale H-measures, Discrete and Continuous Dynamical
Systems, S 8 (2015) 77{90.
, M.E., Martin Lazar: Localisation principle for one-scale
[AEL] Nenad Antonic
H-measures, submitted (arXiv:1504.03956).
10
17
Some properties of
K ;1
0
Theorem
a)
c)
d)
K0;1 = K0;1 ;
L2
loc
un ! 0
trK0;1 (
1 ) = 0
K0;1 > 0
()
()
K0;1 = 0
(un ) is (!n ) oscillatory
Theorem
2 C0 (Rd ), ~ 2 C(Sd 1 ), !n ! 0+ ,
a)
hK(!0n;1) ; '1 '2 i = h(sc!n ) ; '1 '2 i ;
b)
h(K!0n;1) ; '1 '2 ~ i = hH ; '1 '2 ~i ;
where () = =jj.
'1 ; '2 2 Cc (
),
11
17
Example 1 revisited
un (x) = e2in x ,
k
H = jkkj
8
< ;
(sc!n ) = : c ;
0 ;
8
>
< jkkj
(!n )
K0;1 = > c
: k
1 jkj
0
k
0
k
limn n !n = 0
limn n !n = c 2 h0; 1i
limn n !n = 1
;
limn n !n = 0
; limn n !n = c 2 h0; 1i
;
limn n !n = 1
12
17
Localisation principle - assumptions
Let
Rd open, m 2 N, un * 0 in L2loc (
; Cr ) and
X
l6jj6m
"jnj l @ (An un ) = fn in ;
where
l 2 0::m
"n > 0 bounded
An ! Am in C(
;
Mr (C))
fn 2 Hloc (
; Cr ) such that
( 8 ' 2 C1
c (
))
'd
f
Pm n s l s
1 + s=l "n jj
!0
in
L2 (Rd ; Cr )
13
17
Localisation principle - assumptions
Let
Rd open, m 2 N, un * 0 in L2loc (
; Cr ) and
X
l6jj6m
"jnj l @ (An un ) = fn in ;
where
l 2 0::m
"n > 0 bounded
An ! Am in C(
;
Mr (C))
fn 2 Hloc (
; Cr ) such that
( 8 ' 2 C1
c (
))
'd
f
Pm n s l s
1 + s=l "n jj
!0
in
L2 (Rd ; Cr )
For l = 0 the condition on (fn ) is equivalent to
where kuk2Hsh =
u 2 Hs (
; Rd ).
R
( 8 ' 2 C1
c (
))
k'fn kH"nm ! 0 ;
2 s
2
Rd (1 + 2jhj ) ju^()j d is the semiclassical norm of
13
17
Localisation principle - theorem
Theorem
For !n ! 0+ such that c := limn !"nn 2 [0; 1], corresponding one-scale
H-measure K0;1 with characteristic length (!n ) satises
p>
K0;1
= 0;
where
P
8
;
c=0
>
jj=l jjl +jjm A (x)
<P
j
j
(2ic) jjl +jjm A (x) ; c 2 h0; 1i
pc (x; ) :=
> l6jPj6m
:
; c=1
jj=m jjl +jjm A (x)
Moreover, if there exists "0 > 0 such that "n > "0 , n 2 N, we can take
X p1 (x; ) :=
m A (x) :
jj=m jj
14
17
Example: equations with characteristic length (1/2)
R2 be open, and let un := (u1n ; u2n ) * 0 in L2loc (
; C2 ) satises
( 1
un + "n @x1 (a1 u1n ) = fn1
;
u2n + "n @x2 (a2 u2n ) = fn2
where "n ! 0+ , fn := (fn1 ; fn2 ) 2 Hloc1 (
; C2 ) satises
( 8 ' 2 C1
k'fn kH"n1 ! 0 ;
c (
))
while a1 ; a2 2 C(
; R), a1 ; a2 6= 0 everywhere.
Let
15
17
Example: equations with characteristic length (1/2)
R2 be open, and let un := (u1n ; u2n ) * 0 in L2loc (
; C2 ) satises
( 1
un + "n @x1 (a1 u1n ) = fn1
;
u2n + "n @x2 (a2 u2n ) = fn2
where "n ! 0+ , fn := (fn1 ; fn2 ) 2 Hloc1 (
; C2 ) satises
( 8 ' 2 C1
k'fn kH"n1 ! 0 ;
c (
))
while a1 ; a2 2 C(
; R), a1 ; a2 6= 0 everywhere.
By the localisation principle for one-scale H-measure K0;1 with characteristic
length ("n ) (i.e. c = 1) associated to (un ) we get the relation
1
1 0 + 2i1 a1 (x) 0 + 2i2 0 0
>
1 + jj 0 1 1 + jj 0 0 1 + jj 0 a2 (x) K0;1 = 0 ;
Let
15
17
Example: equations with characteristic length (1/2)
R2 be open, and let un := (u1n ; u2n ) * 0 in L2loc (
; C2 ) satises
( 1
un + "n @x1 (a1 u1n ) = fn1
;
u2n + "n @x2 (a2 u2n ) = fn2
where "n ! 0+ , fn := (fn1 ; fn2 ) 2 Hloc1 (
; C2 ) satises
( 8 ' 2 C1
k'fn kH"n1 ! 0 ;
c (
))
while a1 ; a2 2 C(
; R), a1 ; a2 6= 0 everywhere.
By the localisation principle for one-scale H-measure K0;1 with characteristic
length ("n ) (i.e. c = 1) associated to (un ) we get the relation
1
1 0 + 2i1 a1 (x) 0 + 2i2 0 0
>
1 + jj 0 1 1 + jj 0 0 1 + jj 0 a2 (x) K0;1 = 0 ;
whose (1; 1) component reads
1 + i 21 a (x) 11 = 0 ;
K0;1
1 + jj 1 + jj 1
Let
hence
1 11
1 11
1 + jj K0;1 = 0 ; 1 + jj K0;1 = 0
15
17
Example: equations with characteristic length (1/2)
R2 be open, and let un := (u1n ; u2n ) * 0 in L2loc (
; C2 ) satises
( 1
un + "n @x1 (a1 u1n ) = fn1
;
u2n + "n @x2 (a2 u2n ) = fn2
where "n ! 0+ , fn := (fn1 ; fn2 ) 2 Hloc1 (
; C2 ) satises
( 8 ' 2 C1
k'fn kH"n1 ! 0 ;
c (
))
while a1 ; a2 2 C(
; R), a1 ; a2 6= 0 everywhere.
By the localisation principle for one-scale H-measure K0;1 with characteristic
length ("n ) (i.e. c = 1) associated to (un ) we get the relation
1
1 0 + 2i1 a1 (x) 0 + 2i2 0 0
>
1 + jj 0 1 1 + jj 0 0 1 + jj 0 a2 (x) K0;1 = 0 ;
whose (1; 1) component reads
1 + i 21 a (x) 11 = 0 ;
K0;1
1 + jj 1 + jj 1
Let
hence
1 11
supp 11
K0;1 1 ;
1 + jj K0;1 = 0
15
17
Example: equations with characteristic length (1/2)
R2 be open, and let un := (u1n ; u2n ) * 0 in L2loc (
; C2 ) satises
( 1
un + "n @x1 (a1 u1n ) = fn1
;
u2n + "n @x2 (a2 u2n ) = fn2
where "n ! 0+ , fn := (fn1 ; fn2 ) 2 Hloc1 (
; C2 ) satises
( 8 ' 2 C1
k'fn kH"n1 ! 0 ;
c (
))
while a1 ; a2 2 C(
; R), a1 ; a2 6= 0 everywhere.
By the localisation principle for one-scale H-measure K0;1 with characteristic
length ("n ) (i.e. c = 1) associated to (un ) we get the relation
1
1 0 + 2i1 a1 (x) 0 + 2i2 0 0
>
1 + jj 0 1 1 + jj 0 0 1 + jj 0 a2 (x) K0;1 = 0 ;
whose (1; 1) component reads
1 + i 21 a (x) 11 = 0 ;
K0;1
1 + jj 1 + jj 1
Let
hence
supp 11
supp 11
K0;1 1 ;
K0;1 (0 [ f1 = 0g)
15
17
Example: equations with characteristic length (1/2)
R2 be open, and let un := (u1n ; u2n ) * 0 in L2loc (
; C2 ) satises
( 1
un + "n @x1 (a1 u1n ) = fn1
;
u2n + "n @x2 (a2 u2n ) = fn2
where "n ! 0+ , fn := (fn1 ; fn2 ) 2 Hloc1 (
; C2 ) satises
( 8 ' 2 C1
k'fn kH"n1 ! 0 ;
c (
))
while a1 ; a2 2 C(
; R), a1 ; a2 6= 0 everywhere.
By the localisation principle for one-scale H-measure K0;1 with characteristic
length ("n ) (i.e. c = 1) associated to (un ) we get the relation
1
1 0 + 2i1 a1 (x) 0 + 2i2 0 0
>
1 + jj 0 1 1 + jj 0 0 1 + jj 0 a2 (x) K0;1 = 0 ;
whose (1; 1) component reads
1 + i 21 a (x) 11 = 0 ;
K0;1
1 + jj 1 + jj 1
Let
hence
supp 11
supp 11
K0;1 1 ;
K0;1 (0 [ f1 = 0g)
15
17
Example: equations with characteristic length (1/2)
R2 be open, and let un := (u1n ; u2n ) * 0 in L2loc (
; C2 ) satises
( 1
un + "n @x1 (a1 u1n ) = fn1
;
u2n + "n @x2 (a2 u2n ) = fn2
where "n ! 0+ , fn := (fn1 ; fn2 ) 2 Hloc1 (
; C2 ) satises
( 8 ' 2 C1
k'fn kH"n1 ! 0 ;
c (
))
while a1 ; a2 2 C(
; R), a1 ; a2 6= 0 everywhere.
By the localisation principle for one-scale H-measure K0;1 with characteristic
length ("n ) (i.e. c = 1) associated to (un ) we get the relation
1
1 0 + 2i1 a1 (x) 0 + 2i2 0 0
>
1 + jj 0 1 1 + jj 0 0 1 + jj 0 a2 (x) K0;1 = 0 ;
whose (1; 1) component reads
1 + i 21 a (x) 11 = 0 ;
1
K0;1
1 + jj 1 + jj
Let
hence
(0; 1)
supp 11
; 1(0;1) g
K0;1 f1
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Example: equations with characteristic length (1/2)
R2 be open, and let un := (u1n ; u2n ) * 0 in L2loc (
; C2 ) satises
( 1
un + "n @x1 (a1 u1n ) = fn1
;
u2n + "n @x2 (a2 u2n ) = fn2
where "n ! 0+ , fn := (fn1 ; fn2 ) 2 Hloc1 (
; C2 ) satises
( 8 ' 2 C1
k'fn kH"n1 ! 0 ;
c (
))
while a1 ; a2 2 C(
; R), a1 ; a2 6= 0 everywhere.
By the localisation principle for one-scale H-measure K0;1 with characteristic
length ("n ) (i.e. c = 1) associated to (un ) we get the relation
1
1 0 + 2i1 a1 (x) 0 + 2i2 0 0
>
1 + jj 0 1 1 + jj 0 0 1 + jj 0 a2 (x) K0;1 = 0 ;
whose (1; 1) component reads
1 + i 21 a (x) 11 = 0 ;
1
K0;1
1 + jj 1 + jj
Let
hence
(0; 1)
supp 11
; 1(0;1) g
K0;1 f1
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Example: equations with characteristic length (2/2)
Analogously, from the
(2; 2) component we get
(
supp 22
K0;1 f1
hence
;
1 0)
; 1(1;0) g ;
22
12
21
supp 11
K0;1 \ supp K0;1 = ; which implies K0;1 = K0;1 = 0.
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17
Example: equations with characteristic length (2/2)
Analogously, from the
(2; 2) component we get
(
supp 22
K0;1 f1
;
1 0)
; 1(1;0) g ;
22
12
21
supp 11
K0;1 \ supp K0;1 = ; which implies K0;1 = K0;1 = 0.
The very denition of one-scale H-measures gives u1n u2n * 0.
hence
This approach can be systematically generalised by introducing a variant of
compensated compactness suitable for problems with characteristic length.
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Compactness by compensation with a characteristic length
Let un * u in
L2loc (
; Cr ) satisfy
X
l6jj6m
"jnj l @ (An un ) = fn ;
m
+
q
where A
n ! A in C(
; Mqr (C)), let "n ! 0 , and fn 2 Hloc (
; C ) be
1
such that for any ' 2 Cc (
)
'd
fn
1 + kn
is precompact in L2 (Rd ; Cq ). Furthermore, let Q(x; ) := Q(x) , where
Q 2 C(
; Mr (C)), Q = Q, is such that Q(; un ) * in M(
).
Then we have
a) (9 c 2 [0; 1])(8 (x; ) 2 K0;1 (Rd ))(8 2 c;x; ) Q(x; ) > 0 =)
> Q(; u),
b) (9 c 2 [0; 1])(8 (x; ) 2 K0;1 (Rd ))(8 2 c;x; ) Q(x; ) = 0 =)
= Q(; u),
where
c;x; := f 2 Cr : pc (x; ) = 0g ;
and pc is given as before.
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