Taylor M.E. Partial Differential Equations III: Nonlinear Equations
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78 13. Function Space and Operator Theory for Nonlinear Analysis
and
(11.31) kyk
L
p
.R;d
j
/
C<1:
As before, (11.29) is actually a consequence of (11.30)and(11.31). Now (11.28)
is easily extended to
(11.32) .f
j
;
j
/ ! .f; / in Y
1
./ H) ˆ.f
j
;
j
/ ! ˆ.f; / in Y
1
./;
for continuous ˆ W R ! R. There is a similar extension of (11.25), granted the
bound (11.26)onˆ.y/.
We say that f
j
(or more generally .f
j
;
j
/)convergessharply in Y
p
./,if
it converges, in the sense defined above, to .f; / with D
f
. It is of interest
to specify conditions under which we can guarantee sharp convergence. We will
establish some results in that direction a bit later.
When one has a fuzzy function .f; /, it can be conceptually useful to pass
from the measure on
R to a family of probability measures
x
on R,
defined for a.e. x 2 . We discuss how this can be done. From (11.4)wehave
(11.33)
ˇ
ˇ
ˇ
“
ER
.y/ d.x; y/
ˇ
ˇ
ˇ
sup j j L
n
.E/;
and hence
(11.34)
ˇ
ˇ
ˇ
“
R
'.x/ .y/ d.x; y/
ˇ
ˇ
ˇ
sup j jk'k
L
1
./
:
It follows that there is a linear transformation
(11.35) T W C.
R/ ! L
1
./; kT k
L
1
./
sup j j;
such that
(11.36)
“
R
'.x/ .y/ d.x; y/ D
Z
'.x/T .x/ dx:
Using the separability of C.
R/, we can deduce that there is a set S ,of
Lebesgue measure zero, such that, for all 2 C.
R/; T .x/ is defined pointwise,
for x 2 n S. Note that T is positivity preserving and T.1/ D 1. Thus for each
x 2 n S , there is a probability measure
x
on R such that
(11.37) T .x/D
Z
R
.y/ d
x
.y/:
11. Young measures and fuzzy functions 79
Hence
(11.38)
“
R
'.x/ .y/ d.x; y/ D
Z
Z
R
'.x/ .y/ d
x
.y/
!
dx:
From this it follows that
(11.39)
“
R
.x; y/ d .x; y/ D
Z
Z
R
.x; y/ d
x
.y/
!
dx;
for any Borel-measurable function that is either positive or integrable with re-
spect to d. Thus we can reformulate Proposition 11.1:
Corollary 11.2. If ˆ W R ! R is continuous and f
j
! .f; / in Y
1
./,then
(11.40) ˆ.f
j
/ ! g weak
in L
1
./;
where
(11.41) g.x/ D
Z
R
ˆ.y/ d
x
.y/; a.e. x 2 :
One key feature of the notion of convergence of a sequence of fuzzy func-
tions is that, while it is preserved under nonlinear maps, we also retain the sort of
compactness property that weak
convergence has.
Proposition 11.3. Let .f
j
;
j
/ 2 Y
1
./, and assume kf
j
k
L
1
./
M .Then
there exist .f; / 2 Y
1
./ and a subsequence .f
j
;
j
/ such that
(11.42) .f
j
;
j
/ ! .f; /:
Proof. The well-known weak
compactness (and metrizability) of fg 2 L
1
./ W
kgk
L
1
M g implies that one can pass to a subsequence (which we continue to
denote by .f
j
;
j
/) such that f
j
! f weak
in L
1
./.
Each measure
j
is supported on I; I D ŒM; M. Now we exploit the
weak
compactness and metrizability of f 2 M.K I/ WkkL
n
.K/g,
for each compact K , together with a standard diagonal argument, to obtain
a further subsequence such that
j
! weak
in M. I/. The identities
(11.6)and(11.7) are preserved under passage to such a limit, so the proposition
is proved.
So far we have dealt with real-valued fuzzy functions, but we can as easily
consider fuzzy functions with values in a finite-dimensional, normed vector space
V .WedefineY
p
.; V / to consist of pairs .f; /,wheref 2 L
p
.; V / is a
80 13. Function Space and Operator Theory for Nonlinear Analysis
V -valued L
p
function and is a positive Borel measure on V (V D V plus
the sphere S
1
at infinity), having the properties
(11.43) jyj2L
p
V;d.x;y/
;
so in particular S
1
has measure zero,
(11.44) .E V/D L
n
.E/;
for Borel sets E ,and
(11.45)
“
EV
yd.x;y/D
Z
E
f.x/ dx 2 V;
for each Borel set E .
All of the preceding results of this section extend painlessly to this case. Instead
of considering ˆ W R ! R,wetakeˆ W V
1
! V
2
,whereV
j
are two normed
finite-dimensional vector spaces. This time, a Young measure “disintegrates”
into a family
x
of probability measures on V .
There is a natural map
(11.46) & W Y
1
.; V
1
/ Y
1
.; V
2
/ ! Y
1
.; V
1
˚ V
2
/
defined by
(11.47) .f
1
;
1
/&.f
2
;
2
/ D .f
1
˚ f
2
;/;
where, for a.e. x 2 ,BorelF
j
V
j
,
(11.48)
x
.F
1
F
2
/ D
1x
.F
1
/
2x
.F
2
/:
Using this, we can define an “addition” on elements of Y
1
.; V /:
(11.49) .f
1
;
1
/ C .f
2
;
2
/ D S
.f
1
;
1
/&.f
2
;
2
/
;
where S W V ˚ V ! V is given by S.v;w/ D v C w, and we extend S to a map
S W Y
1
.; V ˚ V/! Y
1
.; V / by the same process as used in (11.27).
Of course, multiplication by a scalar a 2 R;M
a
W V ! V , induces
amapM
a
on Y
1
.; V /, so we have what one might call a “fuzzy linear
structure” on Y
1
.; V /. It is not truly a linear structure since certain basic re-
quirements on vector space operations do not hold here. For example (in the case
11. Young measures and fuzzy functions 81
V D R), .f; / 2 Y
1
./ has a natural “negative,” namely .f;
L
/,where
L
.E/ D .E/.However,.f; / C .f;
L
/ ¤ .0;
0
/ unless .f; / is sharply
defined. Similarly, .f; / C .f; / ¤ 2.f; / unless .f; / is sharply defined, so
the distributive law fails.
We now derive some conditions under which, for a given sequence u
j
! .u;/
in Y
1
./ and a given nonlinear function F ,wealsohaveF.u
j
/ ! F.u/ weak
in L
1
./, which is the same here as F.u/ D F . The following result is of
the nature that weak
convergence of the dot product of the R
2
-valued functions
u
j
;F.u
j
/
with a certain family of R
2
-valued functions V.u
j
/ to .u; F/V will
imply
F D F.u/. The specific choice of V.u
j
/ will perhaps look curious; we will
explain below how this choice arises.
Proposition 11.4. Suppose u
j
! .u;/ in Y
1
./, and let F W R ! R be C
1
.
Suppose you know that
(11.50) u
j
q.u
j
/ F.u
j
/.u
j
/ ! uq F weak
in L
1
./;
for every convex function W R ! R, with q given by
(11.51) q.y/ D
Z
y
c
0
.s/F
0
.s/ ds;
and where
(11.52) q.u;/D .
q;
1
/; F .u;/D .F;
2
/; .u;/ D .;
3
/:
Then
(11.53) F.u
j
/ ! F.u/ weak
in L
1
./:
Proof. It suffices to prove that
F D F.u/ a.e. on . Now, applying Corollary
11.2 to ˆ.y/ D yq.y/ F.y/.y/, we have the left side of (11.50)converging
weak
in L
1
./ to
v.x/ D
Z
yq.y/ F.y/.y/
d
x
.y/;
so the hypothesis (11.50) implies
v D u
q F ; a.e. on :
Rewrite this as
(11.54)
Z
n
F.y/
F.x/
.y/
u.x/ y
q.y/
o
d
x
.y/ D 0; a.e. x 2 :
82 13. Function Space and Operator Theory for Nonlinear Analysis
Now we make the following special choices of functions and q:
(11.55)
a
.y/ Djy aj;q
a
.y/ D sgn.y a/
F.y/ F.a/
:
We use these in (11.54), with a D u.x/, obtaining, after some cancellation,
(11.56)
F
u.x/
F.x/
Z
jy u.x/j d
x
.y/ D 0; a.e. x 2 :
Thus, for a.e. x 2 , either
F.x/ D F
u.x/
or
x
D ı
u.x/
, which also implies
F.x/ D F
u.x/
. The proof is complete.
Why is one motivated to work with such functions .u/ and q.u/?Theyarisein
the study of solutions to some nonlinear PDE on R
2
. Let us use coordinates
.t; x/ on . As long as u is a Lipschitz-continuous, real-valued function on ,it
follows from the chain rule that
(11.57) u
t
C F.u/
x
D 0 H) .u/
t
C q.u/
x
D 0;
provided q
0
.y/ D
0
.y/F
0
.y/,thatis,q is given by (11.52). (For general u 2
L
1
./, the implication (11.57) does not hold.) Our next goal is to establish the
following:
Proposition 11.5. Assume u
j
2 L
1
./, of norm M<1. Assume also that
(11.58) @
t
u
j
C @
x
F.u
j
/ ! 0 in H
1
loc
./
and
(11.59) @
t
.u
j
/ C @
x
q.u
j
/ precompact in H
1
loc
./;
for each convex function W R ! R, with q given by (11.51). If u
j
! uweak
in
L
1
./,then
(11.60) @
t
u C @
x
F.u/ D 0:
Proof. By Proposition 11.3, passing to a subsequence, we have u
j
! .u;/ in
Y
1
./. Then, by Proposition 11.1, F.u
j
/ ! F; q.u
j
/ ! q,and.u
j
/ !
weak
in L
1
./. Consider the vector-valued functions
(11.61) v
j
D
u
j
;F.u
j
/
;w
j
D
q.u
j
/; .u
j
/
:
Thus v
j
! .u; F/; w
j
! .q; / weak
in L
1
./. The hypotheses (11.58)–
(11.59) are equivalent to
(11.62) div v
j
; rot w
j
precompact in H
1
loc
./:
11. Young measures and fuzzy functions 83
Also, the hypothesis on ku
j
k
L
1
implies that v
j
and w
j
are bounded in L
1
./,
and a fortiori in L
2
loc
./. The div-curl lemma hence implies that
(11.63) v
j
w
j
! v w in D
0
./; v D .u; F/; w D .q; /:
In view of the L
1
-bounds, we hence have
(11.64) u
j
q.u
j
/ F.u
j
/.u
j
/ ! uq F weak
in L
1
./:
Since this is the hypothesis (11.50) of Proposition 11.4, we deduce that
(11.65) F.u
j
/ ! F.u/ weak
in L
1
./:
Hence @
t
u
j
C @
x
F.u
j
/ ! @
t
u C @
x
F.u/ in D
0
./,sowehave(11.60).
One of the most important cases leading to the situation dealt with in Proposi-
tion 11.5 is the following; for " 2 .0; 1, consider the PDE
(11.66) @
t
u
"
C @
x
F.u
"
/ D "@
2
x
u
"
on D .0; 1/ R; u
"
.0/ D f:
Say f 2 L
1
.R/. The unique solvability of (11.66), for t 2 Œ0; 1/, for each
">0, will be established in Chap. 15, and results there imply
u
"
2 C
1
./;(11.67)
ku
"
k
L
1
./
kf k
L
1
;(11.68)
and
(11.69) "
Z
1
0
Z
1
1
.@
x
u
"
/
2
dx dt
1
2
kf k
2
L
2
:
The last result implies that
p
"@
x
u
"
is bounded in L
2
./. Hence "@
2
x
u
"
! 0
in H
1
./,as" ! 0.Thus,ifu
"
j
is relabeled u
j
, with "
j
! 0,wehave
hypothesis (11.58) of Proposition 11.5. We next check hypothesis (11.59).
Using the chain rule and (11.66), we have
(11.70) @
t
.u
"
/ C @
x
q.u
"
/ D "@
2
x
.u
"
/ "
00
.u
"
/.@
x
u
"
/
2
;
at least when is C
2
and q satisfies (11.52). Parallel to (11.69), we have
(11.71) "
Z
T
0
Z
00
.u
"
/.@
x
u
"
/
2
dx dt D
Z
f.x/
dx
Z
u
"
.T; x/
dx:
84 13. Function Space and Operator Theory for Nonlinear Analysis
A simple approximation argument, taking smooth
ı
! , shows that whenever
is nonnegative and convex, C
2
or not,
(11.72) @
t
.u
"
/ C @
x
q.u
"
/ D "@
2
x
.u
"
/ R
"
;
with
(11.73) R
"
bounded in M./:
Since @
x
.u
"
/ D
0
.u
"
/@
x
u
"
, and any convex is locally Lipschitz, we deduce
from (11.68)and(11.69)that
p
"@
x
.u
"
/ is bounded in L
2
./. Hence
(11.74) "@
2
x
.u
"
/ ! 0 in H
1
./; as " ! 0:
We thus have certain bounds on the right side of (11.72), by (11.73)and(11.74).
Meanwhile, the left side of (11.72) is certainly bounded in H
1;p
loc
./; 8 p<1.
This situation is treated by the following lemma of F. Murat.
Lemma 11.6. Suppose F is bounded in H
1;p
loc
./,forsomep>2, and F
G C H,whereG is precompact in H
1
loc
./ and H is bounded in M
loc
./.Then
F is precompact in H
1
loc
./.
Proof. Multiplying by a cut-off 2 C
1
0
./, we reduce to the case where all
f 2 F are supported in some compact K, and the decomposition f D g C h;
g 2 G;h2 H also has g; h supported in K. Putting K in a box and identifying
opposite sides, we are reduced to establishing an analogue of the lemma when
is replaced by T
n
.
Now Sobolev imbedding theorems imply
M.T
n
/ H
s;q
.T
n
/; s 2 .0; n/; q 2
1;
n
n s
:
Via Rellich’s compactness result (6.9), it follows that
(11.75) W M.T
n
/,! H
1;q
.T
n
/; compact 8 q 2
1;
n
n 1
:
Hence H is precompact in H
1;q
.T
n
/,foranyq<n=.n 1/,sowehave
(11.76) F precompact in H
1;q
.T
n
/; bounded in H
1;p
.T
n
/; p > 2:
By a simple interpolation argument, (11.76) implies that F is precompact in
H
1
.T
n
/, so the lemma is proved.
We deduce that if the family fu
"
W 0<" 1g of solutions to (11.66) satisfies
(11.67)–(11.69), then
11. Young measures and fuzzy functions 85
(11.77) @
t
.u
"
/ C @
x
q.u
"
/ precompact in H
1
loc
./;
which is hypothesis (11.59) of Proposition 11.5. Therefore, we have the following:
Proposition 11.7. Given solutions u
"
;0<" 1 to (11.66), satisfying (11.67)–
(11.69), a weak
limit u in L
1
./,as" D "
j
! 0, satisfies
(11.78) @
t
u C @
x
F.u/ D 0:
The approach to the solvability of (11.78) used above is given in [Tar].
In Chap.16, 6, we will obtain global existence results containing that of
Proposition 11.7, using different methods, involving uniform estimates of
k@
x
u
"
.t/k
L
1
.R/
. On the other hand, in 9 of Chap. 16 we will make use of
techniques involving fuzzy functions and the div-curl lemma to establish some
global solvability results for certain 2 2 hyperbolic systems of conservation
laws, following work of R. DiPerna [DiP].
The notion of fuzzy function suggests the following notion of a “fuzzy solu-
tion” to a PDE, of the form
(11.79)
X
j
@
@x
j
F
j
.u/ D 0:
Namely, .u;/2 Y
1
./ is a fuzzy solution to (11.79)if
(11.80)
X
j
@
@x
j
F
j
D 0 in D
0
./; F
j
.x/ D
Z
F
j
.y/ d
x
.y/:
This notion was introduced in [DiP], where .u;/ is called a “measure-valued
solution” to (11.79). Given jF
j
.y/jC hyi
p
, we can also consider the concept
of a fuzzy solution .u;/ 2 Y
p
./. Contrast the following simple result with
Proposition 11.5:
Proposition 11.8. Assume .u
j
;
j
/ 2 Y
1
./; ku
j
k
L
1
M , and .u
j
;
j
/ !
.u;/in Y
1
./.If
(11.81)
X
k
@
k
F
k
.u
j
/ ! 0 in D
0
./;
as j !1, then u is a fuzzy solution to (11.79).
Proof. By Proposition 11.1, F
k
.u
j
/ ! F
k
weak
in L
1
./. The result follows
immediately from this.
In [DiP] there are some results on when one can say that, when .u;/ 2
Y
1
./ is a fuzzy solution to (11.79), then u 2 L
1
./ is a weak solution to
(11.79), results that in particular lead to another proof of Proposition 11.7.
86 13. Function Space and Operator Theory for Nonlinear Analysis
Exercises
1. If f
j
! .f; / in Y
1
./, we say the convergence is sharp provided D
f
.Show
that sharp convergence implies
f
j
! f in L
2
.
0
/;
for any
0
.
(Hint: Sharp convergence implies jf
j
j
2
!jf j
2
weak
in L
1
./. Thus f
j
! f
weakly in L
2
and also kf
j
k
L
2
.
0
/
!kf k
L
2
.
0
/
:)
2. Deduce that, given f
j
! .f; / in Y
1
./, the convergence is sharp if and only if, for
some subsequence, f
j
! f a.e. on .
3. Given .f; / 2 Y
1
./ and the associated family of probability measures
x
;x2 ,
as in (11.37)–(11.39), show that D
f
if and only if, for a.e. x 2 ;
x
is a point
mass.
4. Complete the interpolation argument cited in the proof of Lemma 11.6. Show that (with
X D ƒ
1
.F/)ifq<2<p,
X precompact in L
q
.T
n
/; bounded in L
p
.T
n
/ H) X precompact in L
2
.T
n
/:
(Hint:Iff
n
2 X; f
n
! f in L
q
.T
n
/,use
kf
n
f k
L
2
kf
n
f k
˛
L
q
kf
n
f k
1˛
L
p
:/
5. Extend various propositions of this section from Y
1
./ to Y
p
./; 1 < p 1.
12. Hardy spaces
The Hardy space H
1
.R
n
/ is a subspace of L
1
.R
n
/ defined as follows. Set
(12.1) .Gf /.x/ D sup
˚
j'
t
f.x/jW' 2 F;t>0
;
where '
t
.x/ D t
n
'.x=t/ and
(12.2) F D
˚
' 2 C
1
0
.R
n
/ W '.x/ D 0 for jxj1; kr'k
L
1
1
:
This is called the grand maximal function of f . Then we define
(12.3) H
1
.R
n
/ Dff 2 L
1
.R
n
/ W Gf 2 L
1
.R
n
/g:
A related (but slightly larger) space is h
1
.R
n
/,definedasfollows.Set
(12.4) .G
b
f /.x/ D sup
˚
j'
t
f.x/jW' 2 F;0<t 1
;
and define
(12.5) h
1
.R
n
/ Dff 2 L
1
.R
n
/ W G
b
f 2 L
1
.R
n
/g:
12. Hardy spaces 87
An important tool in the study of Hardy spaces is another maximal function,
the Hardy-Littlewood maximal function,definedby
(12.6) M.f /.x/ D sup
r>0
1
vol.B
r
/
Z
B
r
.x/
jf.y/j dy:
The basic estimate on this maximal function is the following weak type-(1,1)
estimate:
Proposition 12.1. There is a constant C D C.n/ such that, for any >0;f 2
L
1
.R
n
/, we have the estimate
(12.7) meas
fx 2 R
n
W M.f /.x/ > g
C
kf k
L
1
:
Note that the estimate
meas
fx 2 R
n
Wjf.x/j >g
1
kf k
L
1
follows by integrating the inequality jf j
S
,whereS
Dfjf j >g.
To begin the proof of Proposition 12.1,let
(12.8) F
Dfx 2 R
n
W Mf.x/ > g:
We remark that, for any f 2 L
1
.R
n
/ and any >0; F
is open. Given x 2 F
,
pick r D r
x
such that A
r
jf j.x/ > ,andletB
x
D B
r
x
.x/. Thus fB
x
W x 2 F
g
is a covering of F
by balls. We will be able to obtain the estimate (12.7) from the
following “covering lemma,” due to N. Wiener.
Lemma 12.2. If C DfB
˛
W ˛ 2 Ag is a collection of open balls in R
n
, with
union U , and if m
0
< meas.U /, then there is a finite collection of disjoint balls
B
j
2 C;1 j K, such that
(12.9)
X
meas.B
j
/>3
n
m
0
:
We show how the lemma allows us to prove (12.7). In this case, let C Df
ı
B
x
W
x 2 F
g.Thus,ifm
0
< meas.F
/, there exist disjoint balls B
j
D
ı
B
r
j
.x
j
/ such
that meas.[B
j
/>3
n
m
0
. This implies
(12.10) m
0
<3
n
X
meas.B
j
/
3
n
X
Z
B
j
jf.x/j dx
3
n
Z
jf.x/j dx;
for all m
0
< meas.F
/, which yields (12.7), with C D 3
n
.