Taylor M.E. Partial Differential Equations III: Nonlinear Equations

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8 13. Function Space and Operator Theory for Nonlinear Analysis

(Hint: Suppose x D e

.If

is the line segment from 0 to z, followed by the line

segment from z to e

, write

u.e

/ u.0/ D





d u



dS.z/; ˙ D



x 2 B W x



Show that this gives u.e

/  u.0/ D

ru.z/  '.z/dz, with ' 2 L

.B/; 8 q<

n=.n  1/:)

6. Show that H

n;1

/  C.R

/ \ L

(Hint: u.x/ D

1



1

D

u.x C y/ dy

dy

3. Gagliardo–Nirenberg–Moser estimates

In this section we establish further estimates on various L

-norms of derivatives

of functions, which are very useful in nonlinear PDE. Estimates of this sort arose

in work of Gagliardo [Gag], Nirenberg [Ni], and Moser [Mos]. Our ﬁrst such

estimate is the following. We keep the convention (2.8).

Proposition 3.1. Fo r real k  1; 1  p  k, we have

(3.1) kD

2k=p

 C kuk

2k=.p1/

kD

2k=.pC1/

;

for all u 2 C

/, hence for all u 2 L

/ \ H

2;q

,where

(3.2) q

p C 1

p  1

Proof. Given v 2 C

/; q  2,wehavevjvj

q2

2 C

/ and

.vjvj

q2

/ D .q  1/.D

v/jvj

q2

Letting v D D

u,wehave

D D

.u D

ujD

q2

/  .q  1/u D

ujD

q2

Integrating this, we have, by the generalized H¨older inequality (2.1),

(3.3) kD

jq  1jkuk

q2

;

where q D 2k=p and q

and q

are given by (3.2). Dividing by kD

q2

gives

the estimate (3.1)foru 2 C

/, and the proposition follows.

If we apply (3.1)toD

`1

u,weget

(3.4) kD

2k=p

 C kD

`1

2k=.p1/

`C1

2k=.pC1/

;

3. Gagliardo–Nirenberg–Moser estimates 9

for real k  1; p 2 Œ1; k; `  1. Consequently, for any ">0,

(3.5) kD

2k=p

 C"kD

`1

2k=.p1/

C C."/kD

`C1

2k=.pC1/

If p 2 Œ2; k and `  2, we can apply (3.5) with p replaced by p  1 and D

`1

replaced by D

`2

u,toget,forany"

>0,

(3.6) kD

`1

2k=.p1/

 C"

`2

2k=.p2/

C C."

/kD

2k=p

Now we can plug (3.6)into(3.5); ﬁx "

(e.g., "

D 1), and pick " so small that

C"C."

/  1=2,sothetermC"C."

/kD

2k=p

can be absorbed on the left,

to yield

(3.7) kD

2k=p

 C"kD

`2

2k=.p2/

C C."/kD

`C1

2k=.pC1/

;

for real k  2; p 2 Œ2; k; `  2. Continuing in this fashion, we get

(3.8) kD

2k=p

 C"kD

`j

2k=.pj/

C C."/kD

`C1

2k=.pC1/

;

j  p  k; `  j . Similarly working on the last term in (3.8), we have the

following:

Proposition 3.2. If j  p  k C1m; `  j , then (for sufﬁciently small ">0)

(3.9) kD

2k=p

 C"kD

`j

2k=.pj/

C C."/kD

`Cm

2k=.pCm/

Here, j; `,andm must be positive integers, but p and k are real. Of course, the

full content of (3.9) is represented by the case ` D j , which reads

(3.10) kD

2k=p

 C"kuk

2k=.p`/

C C."/kD

`Cm

2k=.pCm/

;

for `  p  k C 1  m.Takingp C m D k, we note the following important

special case.

Corollary 3.3. If `;p, and k are positive integers satisfying `  p  k 1,then

(3.11) kD

2k=p

 C"kuk

2k=.p`/

C C."/kD

kC`p

In particular, taking p D `,if`<k,then

(3.12) kD

2k=`

 C"kuk

C C."/kD

;

for all u 2 C

We want estimates for the left sides of (3.11)and(3.12) which involve prod-

ucts, as in (3.1), rather than sums. The following simple general result produces

such estimates.

10 13. Function Space and Operator Theory for Nonlinear Analysis

Proposition 3.4. Let `;, and m be nonnegative integers satisfying `  max

.; m/, and let q; r, and  belong to Œ1; 1. Suppose the estimate

(3.13) kD

 C



C C



is valid for all u 2 C

/.Then

(3.14) kD

 .C

C C

/kD



ˇ=.˛Cˇ/

kD

˛=.˛Cˇ/



;

with

(3.15) ˛ D



C   `; ˇ D



 m C `;

provided these quantities are not both zero. If (3.13) is valid and the quantities

(3.15) are both nonzero, then they have the same sign.

Proof. Replacing u.x/ in (3.13)byu.sx/ produces from (3.13), which we write

schematically as Q  C

R C C

P , the estimate

`n=q

Q  C

n=r

R C C

mn=

P; for all s>0;

or equivalently,

Q  C

R C C

ˇ

P; for all s>0;

with ˛ and ˇ given by (3.15). If ˛ and ˇ have opposite signs, one can take s ! 0

or s !1to produce the absurd conclusion Q D 0. If they have the same

sign, one can take s so that s

R D s

ˇ

P D P

, which can be done with

a D ˛=.˛ C ˇ/; b D ˇ=.˛ C ˇ/, and the estimate (3.14) results.

Applying Proposition 3.4 to the estimate (3.11), we ﬁnd ˛ D .n 

2k/`=2k; ˇ D .n  2k/.k  p/=2k, which gives the following:

Proposition 3.5. If `; p, and k are positive integers satisfying `  p  k  1,

then

(3.16) kD

2k=p

 C kuk

.kp/=.kC`p/

2k=.p`/

kD

kC`p

`=.kC`p/

In particular, taking p D `,if`<k,then

(3.17) kD

2k=`

 C kuk

1`=k

kD

`=k

One of the principal applications of such an inequality as (3.17) is to bilinear

estimates, such as the following.

3. Gagliardo–Nirenberg–Moser estimates 11

Proposition 3.6. If jˇjCj jDk,then

(3.18) k.D

f /.D



g/k

 C kf k

kgk

C C kf k

kgk

;

for all f; g 2 C

/ \ H

Proof. With jˇjD`; j jDm,and` C m D k,wehave

k.D

f /.D



g/k

kD

f k

2k=`

kD



2k=m

 C kf k

1`=k

kf k

`=k

kgk

1m=k

kgk

m=k

;

(3.19)

using H¨older’s inequality and (3.17). We can write the right side of (3.19)as

(3.20) C



kf k

kgk



m=k



kf k

kgk



`=k

;

and this is readily dominated by the right side of (3.18).

The two estimates of the next proposition are major implications of (3.18).

Proposition 3.7. We have the estimates

(3.21) kf  gk

 C kf k

kgk

C C kf k

kgk

and, for j˛jk,

(3.22) kD

.f  g/  fD

 C kf k

kgk

C C krf k

kgk

k1

Proof. Theestimate(3.21) is an immediate consequence of (3.18). To prove

(3.22), write

(3.23) D

.f  g/ D

ˇC D˛













;

so, if j˛jDk,

.f  g/  fD

g D

ˇC D˛;ˇ>0













jˇjCj jDk1

jˇ

f /.D



g/:

(3.24)

Hence, with u

D D

f ,

(3.25) kD

.fg/  fD

 C

jˇjCj jDk1

k.D

/.D



g/k

12 13. Function Space and Operator Theory for Nonlinear Analysis

From here, the estimate (3.22) follows immediately from (3.18), and Proposition

3.7 is proved. Note that on the right side of (3.22), we can replace kf k

krf k

k1

From Proposition 3.4 there follow further estimates involving products of

norms, which can be quite useful. We record a few here.

Proposition 3.8. We have the estimates

(3.26) kuk

 C kD

mC1

1=2

kD

m1

1=2

; for u 2 C

and

(3.27) kuk

 C kD

mC1

1=2

kD

1=2

; for u 2 C

2mC1

Proof. It is easy to see that

(3.28) kuk

 C kD

mC1

C C kD

m1

; for u 2 C

and

(3.29) kuk

 C kD

mC1

C C kD

; for u 2 C

2mC1

Proposition 3.4 then yields ˛ D ˇ D 1 in case (3.28)and˛ D ˇ D 1=2 in case

(3.29), proving (3.26)and(3.27).

A more delicate L

-estimate will be proved in  8.

It is also useful to have the following estimates on compositions.

Proposition 3.9. Let F be smooth, and assume F.0/D0. Then, for u 2H

(3.30) kF.u/k

 C

.kuk

/.1Ckuk

Proof. The chain rule gives

F.u/ D

Cˇ



D˛

.ˇ

u

.ˇ



./

.u/;

hence

(3.31) kD

F.u/k

 C

.kuk



.ˇ

u

.ˇ





From here, (3.30) is obtained via the following simple generalization of

Proposition 3.6:

Exercises 13

Lemma 3.10. If jˇ

jCCjˇ



jDk,then

(3.32)

.ˇ

f

.ˇ



 C







kf



kf k

Proof. The generalized H¨older inequality dominates the left side of (3.32)by

(3.33) kf

.ˇ

2k=jˇ

kf

.ˇ



2k=jˇ



Then applying (3.17) dominates this by

(3.34) C kf

1jˇ

j=k

kf

jˇ

j=k

kf



1jˇ



j=k

kf



jˇ



j=k

;

which in turn is easily bounded by the right side of (3.32) (with f D

;:::;f



/).

We remark that Proposition 3.9 also works if u takesvaluesinR

. The esti-

mates in Propositions 3.7 and 3.8 are called Moser estimates, and are very useful

in nonlinear PDE. Some extensions will be given in (10.20) and (10.52).

Exercises

1. Show that the proof of Proposition 3.1 yields

(3.35) kD

 C kuk

kD

whenever 2  q<1;1 q

1,and1=q

C 1=q

D 2=q. Show that if q

q<q

,then(3.35)and(3.1) are equivalent. Is (3.35) valid if the hypothesis q  2 is

relaxed to q  1?

2. Show directly that (3.35) holds with q

D q

D q 2 Œ1; 1.(Hint: Do the next

exercise.)

3. Let A generate a contraction semigroup on a Banach space B. Show that

(3.36) kAuk

 8kukkA

uk; for u 2 D.A

(Hint: Use the identity tAu D t.tA/

1

uCt

ut

t.tA/

1

u together with the

estimate kt.tA/

1

k1,fort>0, to obtain the estimate tkAukkA

ukC2t

kuk,

for t>0:) Try to improve the 8 to a 4 in (3.36), in case B is a Hilbert space.

4. Show that (3.10) implies

(3.37) kD

 C

kuk

C C

`Cm



when <q<rare related by

(3.38)

m C `



;

14 13. Function Space and Operator Theory for Nonlinear Analysis

as long as we require furthermore that q>2, in order to satisfy the hypothesis p=k 

1.m1/=k used for (3.10). In how much greater generality can you establish (3.37)?

Note that if Proposition 3.4 is applied to (3.37), one gets

(3.39) kD

 C kuk

m=.mC`/

kD

`Cm

`=.mC`/



;

provided (3.38) holds.

5. Generalize Propositions 3.6 and 3.7, replacing L

and H

by L

and H

k;p

.Use(3.10)

to do this for p  2. Can you also treat the case 1  p<2?

6. Show that in (3.30) you can use C

.kuk

/ with

(3.40) C

./ D sup

jxj;k

./

.x/j:

7. Extend the Moser estimates in Propositions 3.7 and 3.9 to estimates in H

k;p

-norms.

4. Trudinger’s inequalities

The space H

n=2

/ does not quite belong to L

/, although H

n=2

/ 

/ for all p 2 Œ2; 1/. In fact, quite a bit more is true; exponential functions

of u 2 H

n=2

/ are locally integrable. The proof of this starts with the following

estimate of kuk

as p !1.

Proposition 4.1. If u 2 H

n=2

/, then, for p 2 Œ2; 1/,

(4.1) kuk

 C

1=2

kuk

n=2

Proof. We have u D ƒ

n=2

v for v 2 L

/, where, recall,

(4.2)



s



O./ Dhi

s

Ov./:

Hence, with v 2 L

(4.3) u D J

n=2

 v;

where

(4.4)

n=2

./ Dhi

n=2

The behavior of J

n=2

.x/ follows results of Chap. 3. By Proposition 8.2 of Chap. 3,

n=2

.x/ is C

on R

n 0 and vanishes rapidly as jxj!1. By Proposition 9.2

of Chap.3, we have

(4.5) J

n=2

.x/  C jxj

n=2

; for jxj1:

4. Trudinger’s inequalities 15

Consequently, J

n=2

just misses being in L

/;wehave,forı 2 .0; 1,

(4.6) kJ

n=2

2ı

2ı

 C C C

nı=21

dr 

Now the map K

deﬁned by K

f D v  f , with v given in L

/, satisﬁes

(4.7) K

W L

! L

W L

! L

;

both maps having operator norm kvk

. By interpolation,

(4.8) kK

f k

kf k

kvk

; for q 2 Œ1; 2;

where p is deﬁned by 1=q 1=p D 1=2.Takingf D J

n=2

;qD 2 ı,wehave,

for v 2 L

(4.9) kJ

n=2

 vk







1=.2ı/

kvk

;pD

2.2  ı/

;

which gives (4.1).

The following result, known as Trudinger’s inequality, is a direct consequence

of (4.1):

Proposition 4.2. If u 2 H

n=2

/, there is a constant  D .u/>0,oftheform

(4.10) .u/ D



kuk

n=2

;

such that

(4.11)



ju.x/j

 1



dx < 1:

If M is a compact manifold, possibly with boundary, of dimension n, and if u 2

n=2

.M /, then there exists  D .M/=kuk

n=2

.M /

such that

(4.12)

ju.x/j

dV.x/ < 1:

Proof. We have

ju.x/j

 1 D ju.x/j



ju.x/j

CC



mŠ

ju.x/j

C:

16 13. Function Space and Operator Theory for Nonlinear Analysis

By (4.1),

(4.13)



mŠ

ju.x/j

dV.x/  C



mŠ

.2m/

kuk

n=2

;

which is bounded by C



,forsome<1,if has the form (4.10), with



< 1=.2eC

/, as can be seen via Stirling’s formula for mŠ. This proves the

proposition.

We note that the same argument involving (4.2)–(4.8) also shows that, for any

p 2 Œ2; 1/,thereisan">0such that

(4.14) H

n=2"

/  L

Similarly, we have H

n=2"

.M /  L

.M /,whenM is a compact manifold, per-

haps with boundary, of dimension n. By virtue of Rellich’s theorem, we have for

such M that the natural inclusion

(4.15)  W H

n=2

.M / ,! L

.M / is compact, for all p<1:

Using this, we obtain the following result:

Proposition 4.3. If M is a compact manifold (with boundary) of dimension

n; ˛ 2 R,then

(4.16) u

! u weakly in H

n=2

.M / H) e

˛u

! e

˛u

in L

.M /-norm:

Proof. We have

˛u

 e

˛u



m1

j˛j

mŠ

.x/j

ju.x/j

If ku

n=2

.M /

 A, we obtain

˛u

 e

˛u



mk

j˛j

mŠ

 uk

 m

m1

Ckuk

m1

C C

m>k

m=2

mŠ

jAC

˛j

;

(4.17)

where we use

juj

 mju

 uj



m1

Cjuj

m1



to estimate the sum over m  k, and we use (4.1) to estimate the sum over m>k.

By (4.15), for any k, the ﬁrst sum on the right side of (4.17) goes to 0 as j !1.

Meanwhile the second sum vanishes as k !1,so(4.16) follows.

5. Singular integral operators on L

Exercises

1. Partially generalizing (4.10), let p 2 .1; 1/,andletu 2 H

k;p

/, with kp D n; k 2

. Show that there exists  D 

.u/ such that

(4.18)

jxjR

ju.x/j

p=.p1/

dx  C

For a more complete generalization, see Exercise 5 of  6.

Note: Finding the best constant  in (4.18) is subtle and has some important uses; see

[Mos2], [Au], particularly for the case k D 1; p D n.

5. Singular integral operators on L

One way the Fourier transform makes analysis on L

/ easier than analysis

on other L

-spaces is by the deﬁnitive result the Plancherel theorem gives as a

condition that a convolution operator k u D P.D/u be L

-bounded, namely that

k./ D P./ be a bounded function of . A replacement for this that advances

our ability to pursue analysis on L

is the next result, established by S. Mikhlin,

following related work for L

/ by J. Marcinkiewicz.

Theorem 5.1. Suppose P./ satisﬁes

(5.1) jD

P./jC

hi

j˛j

;

for j˛jn C 1.Then

(5.2) P.D/ W L

/ ! L

/; for 1<p<1:

Stronger results have been proved; one needs (5.1) only for j˛jŒn=2C1,and

one can use certain L

-estimates on the derivatives of P./. These sharper results

can be found in [H1]and[S1]. Note that the characterization of P./ 2 S

is that (5.1) hold for all ˛.

The theorem stated above is a special case of a result that applies to pseu-

dodifferential operators with symbols in S

1;ı

/.Asshownin 2 of Chap. 7, if

p.x; / satisﬁes the estimates

(5.3) jD



p.x; /jC

˛ˇ

hi

j˛jCjˇj

;

for

(5.4) jˇj1; j˛jn C 1 Cjˇj;

then the Schwartz kernel K.x; y/ of P D p.x;D/ satisﬁes the estimates

(5.5) jK.x; y/jC jx  yj

n