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

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38 13. Function Space and Operator Theory for Nonlinear Analysis

We mention that an interpolation argument yields that e

z

is a holomorphic

semigroup on L

./ on a cone K that is symmetric about R

and has angle





1 j2=p  1j



.(See[RS], Vol. 2, p. 255.) This result is valid even if  has

nasty boundary, as well as in other settings. On the other hand, ingredients of the

argument used above will also be useful for other results, presented below.

Note that once we have the holomorphy of e

t

on L

./,forallp 2 .1; 1/,

we can apply Proposition 7.2. In particular, suppose we carry out the construction

of the u

above, not stopping as soon as p

 2, but letting p

become arbitrarily

large. Then (7.44) is replaced by g

2 C



K;L

./



, and we can now apply

Proposition 7.2 to improve (7.47)to

(7.49) w 2 C



K;H

2";p

./



;

making use of (7.2), (7.11), and interpolation to estimate the norm of e

t

./ ! H

2";p

./.

We now consider the construction (7.24)–(7.44)whenu.0/ D f 2 L

./.

We will restrict attention to t 2 R

. A direct inspection of the parametrix

for the heat kernel, constructed in Chap. 7,  13, shows that e

t

W L

.M /

! C

.M /, with norm  Ct

1=2

,fort 2 .0; 1,sov in (7.26) satisﬁes the

estimate kv.t/k

.M /

 Ct

1=2

kf k

./

,andku

.t/k

./

satisﬁes a similar

estimate. Thus g in (7.29) satisﬁes the estimate (7.30), with p D1, and conse-

quently v

in (7.32) satisﬁes kv

.t/k

.M /

 C . Hence ku

.t/k

./

 C ,and

in (7.35) satisﬁes (7.36) with p D1. Thus u D u

Cw,wherew satisﬁes

(7.50)

D w  g

on R

 ; w.0/ D 0; w

@

D 0:

By the holomorphy of e

t

on L

./ for p 2 .1; 1/,wehave

(7.51) w 2 C



Œ0; 1/; H

2";p

./



;

for any ">0and arbitrarily large p<1, hence w 2 C



2ı

./



,forany

ı>0. We deduce that

(7.52) ke

t

f k

./

 Ct

1=2

kf k

./

;0<t 1:

Theestimate(7.52), together with the following result, will be useful for the

study of semilinear parabolic equations on domains with boundary, in  3 of

Chap. 15.

Proposition 7.4. If

 is a compact Riemannian manifold with boundary, on

which the Dirichlet condition is placed, then e

t

deﬁnes a strongly continuous

semigroup on the Banach space

(7.53) C

./ D

f 2 C

./ W f j

@

D 0



7. L

-spectral theory of the Laplace operator 39

Proof. It is easy to verify that, for N>1C .dim M/=4,

D.

/  C

./; densely:

Since e

t

is a strongly continuous semigroup on D.

/, it sufﬁces to show that

for each f 2 C

./, fe

t

f W 0  t  1g is uniformly bounded in Lip./.To

see this, we analyze solutions to

D u; for x 2 ; u.0; x/ D f.x/; u.t; x/ D 0; for x 2 @;

when

(7.54) f 2 C

./; f

@

D 0:

We will to some extent follow the proof of Proposition 7.3, and also use that result.

In this case, for

f equal to f on

 and to zero on M n,wehave

f 2 Lip.M /.

Thus, for v deﬁned by

D v on R

 M; v.0/ D

we have

(7.55) v 2 C



; Lip.M /



;

where the “C” stands for “weak” continuity in t,(i.e.,v.t/ is bounded in Lip.M /

and continuous in t, with values in H

1;p

.M /, for each p<1). Hence

.t; x/ D v.t; x/  v



t;R.x/





satisﬁes

(7.56) u

2 C



; Lip./



We have

D u

C g; u

.0/ D f; u

@

D 0;

where

g D L



Here, as in (7.15), L

is a second-order differential operator whose principal sym-

bol vanishes on @,andv

.x/ D v



R.x/



. Consequently, again an analogue of

(5.49)gives

(7.57) g 2 C



./



40 13. Function Space and Operator Theory for Nonlinear Analysis

Now, we have u D u

C w,wherew satisﬁes

(7.58)

D w  g; w.0/ D 0; w

@

D 0;

and, by (7.57), g 2 C



./



,forallp<1. This implies

(7.59) w 2 C



2";p

./



; 8p<1;">0;

since e

t

is a holomorphic semigroup on L

./. This proves Proposition 7.4.

Exercises

1. Extend results of this section to the Neumann boundary condition.

In Exercises 2 and 3, let  be an open subset, with smooth boundary, of a com-

pact Riemannian manifold M . Assume there is an isometry  W M ! M that is an

involution, ﬁxing @,soM is the isometric double of

.

2. Suppose X

are smooth vector ﬁelds on , f

2 L

./ for some p 2 Œ2; 1/,andu

is the unique solution in H

1;2

./ to

u D

Show that u 2 H

1;p

./.(Hint: Reduce to the case where each X

is a smooth vector

ﬁeld on M , such that 

D˙X

.Extendf

to f

2 L

.M /,sothat



Df

Thus

2 H

1;p

.M / is odd under :)

3. Extend the result of Exercise 2 to the case f

2 L

./ when 1<p<2, appropriately

weakening the a priori hypothesis on u.

4. Try to extend the results of Exercises 2 and 3 to general, compact, smooth

, not nec-

essarily having an isometric double.

5. Show that (7.5) can be improved to



W L

./ ! C./;

for   0.(Hint:Use(7.11). Show that, in fact, for   0,



W L

./ ! C

./; 8 r<2:/

A sharper result will be contained in (8.54)–(8.55).

8. H

older spaces and Zygmund spaces

If 0<s<1, we deﬁne the space C

/ of H¨older-continuous functions on R

to consist of bounded functions u such that

(8.1) ju.x C y/  u.x/jC jyj

8. H¨older spaces and Zygmund spaces 41

For k D 0; 1; 2; : : : ,wetakeC

/ to consist of bounded, continuous functions

u such that D

u is bounded and continuous, for jˇjk.Ifs D k C r; 0 <

r<1,wedeﬁneC

/ to consist of functions u 2 C

/ such that, for

jˇjDk; D

u belongs to C

For nonintegral s,theH¨older spaces C

/ have a characterization similar

to that for L

and more generally H

s;p

,in(5.46)and(6.23), via the Littlewood–

Paley partition of unity used in (5.37),

1 D

j D0

./

;

with '

supported on hi2

,and'

./ D '

1j

/ for j  1.Let

./ D

./

Proposition 8.1. If u 2 C

/,then

(8.2) sup

.D/uk

< 1:

Proof. To see this, ﬁrst note that it is obvious for s D 0.Fors D ` 2 Z

,itthen

follows from the elementary estimate

(8.3)

.D/u.x/k



j˛j`

.D/D

u.x/k

 C

.D/u.x/k

Thus it sufﬁces to establish that u 2 C

implies (8.2)for0<s<1.Since

.x/

has zero integral, we have, for k  1,

.D/u.x/jD

.y/



u.x  y/  u.x/



 C

.y/jjyj

dy;

(8.4)

which is readily bounded by C  2

ks

This result has a partial converse.

Proposition 8.2. If s is not an integer, ﬁniteness in (8.2) implies u 2 C

Proof. It sufﬁces to demonstrate this for 0<s<1. With ‰

./ D

j k

./,

if jyj2

k

, write

u.x Cy/  u.x/ D

y r‰

.D/u.x C ty/ dt



I  ‰

.D/



u.x C y/  u.x/



(8.5)

42 13. Function Space and Operator Theory for Nonlinear Analysis

and use (8.2)and(8.3) to dominate the L

-norm of both terms on the right by

C  2

sk

,sincekr‰

.D/uk

 C  2

.1s/k

This converse breaks down if s 2 Z

.WedeﬁnetheZygmund space C



to consist of u such that (8.2) is ﬁnite, using that to deﬁne the C



-norm, namely,

(8.6) kuk



D sup

.D/uk

Thus

(8.7) C

D C



if s 2 R

n Z

 C



;k2 Z

The class C



/ can be deﬁned for any s 2 R, as the set of elements u 2 S

such that (8.6) is ﬁnite.

The following complements previous boundedness results for Fourier multi-

pliers P.D/ on L

/ and on H

s;p

Proposition 8.3. If P./ 2 S

/, then, for all s 2 R,

(8.8) P.D/ W C



! C

sm



Proof. Consider ﬁrst the case m D 0.Pick

./ 2 C

/ such that

./ D 1 on supp

and

./ D

1j

/,forj  2. It follows readily

from the analysis of the Schwartz kernel of P.D/ made in  2 of Chap. 7, partic-

ularly in the proof of Proposition 2.2 there, that

(8.9) P./ 2 S

/ H) sup

./P ./k

F L

< 1;

where kQk

F L

. Also, it is clear that

(8.10) k

.D/P .D/uk

 C k

P k

F L

k

.D/uk

;

which implies (8.8)form D 0. The extension to general m 2 R is straightforward.

In particular, with ƒ D .1 /

1=2

(8.11) ƒ

W C



! C

sm



is an isomorphism.

Note that in light of (8.9)and(8.10), we have

(8.12) kP.D/uk



 C sup

2R

;j˛jŒn=2C1

.˛/

./hi

j˛j

kuk



8. H¨older spaces and Zygmund spaces 43

In particular, for y 2 R,

(8.13) kƒ



 C hyi

n=2C1

kuk



Compare with (5.47).

The Sobolev imbedding theorem, Proposition 6.3, can be sharpened and ex-

tended to the following:

Proposition 8.4. Fo r al l s 2 R;p2 .1; 1/,

(8.14) H

s;p

/  C



/; r D s 

Proof. In light of (8.11), it sufﬁces to consider the case s D n=p.LetL

./ 2

/ be nowhere vanishing and satisfy L

./ Djj

,forjj1=100.It

sufﬁces to show that, for p 2 .1; 1/,

(8.15) k

.D/L

n=p

.D/uk

 C kuk

;

with C independent of k. We can restrict attention to k  2.ThenA

./ D

./L

n=p

./ satisﬁes

kC1

./ D 2

nk=p

k

/:

Hence

kC1

.x/ 2 S.R

/ and

(8.16) k

kC1

D C; independent of k  2:

Thus the left side of (8.15) is dominated by k

kuk

, which in turn is

dominated by the right side of (8.15). This completes the proof.

It is useful to extend Proposition 8.3 to the following.

Proposition 8.5. If p.x; / 2 S

1;0

/, then, for s 2 R,

(8.17) p.x; D/ W C



/ ! C

sm



Proof. In light of (8.11), it sufﬁces to consider the case m D 0. Also, it sufﬁces

to consider one ﬁxed s, which we can take to be positive. First we prove (8.17)in

the special case where p.x; / has compact support in x. Then we can write

(8.18) p.x; D/u D

ix



.D/u d;

with

(8.19) q



./ D .2/

n

ix

p.x; / dx:

44 13. Function Space and Operator Theory for Nonlinear Analysis

Via the estimates used to prove Proposition 8.3, it follows that, for any given

s 2 R, q



.D/ 2 L







has an operator norm that is a rapidly decreasing

function of . It is easy to establish the estimate

(8.20) ke

ix



 C.s/ hi

kuk



.s > 0/;

ﬁrst for s … Z

, by using the characterization (8.1)ofC

D C



, then for general

s>0by interpolation. The desired operator bound on (8.18) follows easily.

To do the general case, one can use a partition of unity in the x-variables, of

the form

1 D

j 2Z

.x/; '

.x/ D '

.x C j/; '

2 C

and exploit the estimates on p

.x; D/u D '

.x/p.x; D/u obtained by the ar-

gument above, in concert with the rapid decrease of the Schwartz kernel of the

operator p.x; D/ away from the diagonal. Details are left to the reader.

In  9 we will establish a result that is somewhat stronger than Proposition 8.5,

but this relatively simple result is already useful for H¨older estimates on solutions

to linear, elliptic PDE.

It is useful to note that we can deﬁne Zygmund spaces C



/ on the torus

just as in (8.6), but using Fourier series. We again have (8.7) and Propositions

8.3–8.5.

The issue of how Zygmund spaces form a complex interpolation scale is more

subtle than the analogous situation for L

-Sobolev spaces, treated in  6.Adiffer-

ent type of complex interpolation functor, ŒX; Y 



, deﬁned in Appendix A at the

end of this chapter, does a better job than ŒX; Y 



. We have the following result

established in Appendix A.

Proposition 8.6. Fo r r; s 2 R;2 .0; 1/,

(8.21) ŒC



/; C



/



D C

sC.1/r



It is straightforward to extend the notions of H¨older and Zygmund spaces to

spaces C

.M / and C



.M / when M is a compact manifold without boundary.

Furthermore, the analogue of (8.14) is readily established, and we have

(8.22) P W C



.M / ! C

sm



.M / if P 2 OPS

1;0

.M /:

 is a compact manifold with boundary, there is an obviousnotion of C

./,

for s  0. We will deﬁne C



./ below, for s  0. For now we look further at

./. The following simple observation is useful. Give  a Riemannian metric

and let ı.x/ D dist.x; @/.

8. H¨older spaces and Zygmund spaces 45

Proposition 8.7. Let r 2 .0; 1/. Assume f 2 C

./ satisﬁes

(8.23) jrf.x/jCı.x/

r1

;x2 :

Then f extends continuously to

, as an element of C

./.

Proof. There is no loss of generality in assuming that  is the unit ball in R

When estimating f.x

/  f.x

/, we may as well assume that x

and x

are a

distance  1=4 from @ and jx

 x

j1=4. Write

f.x

/  f.x

/ D



df .x/;

where  is a path from x

to x

of the following sort. Let y

lie on the ray segment

from 0 to x

, a distance d Djx

 x

j from x

.Then goes from x

to y

on a

line, from y

to y

on a line, and from y

to x

on a line, as illustrated in Fig. 8.1.

Then

(8.24) jf.x

/  f.y

/jC

1d

.1  /

r1

d D C



r1

d  C

;

while

(8.25) jf.y

/  f.y

/jC jy

 y

r1

 C

;

(8.26) jf.x

/  f.x

/jC jx

 x

;

as asserted.

FIGURE 8.1 Path from x

to x

46 13. Function Space and Operator Theory for Nonlinear Analysis

Now consider  of the form  D Œ0; 1  M ,whereM is a compact

Riemannian manifold without boundary. We want to consider the action on

f 2 C

.M / of a family of operators of Poisson integral type, such as were

studied in Chap. 7,  12, to construct parametrices for regular elliptic boundary

problems. We recall from (12.35) of Chap. 7 the class OP P

j

consisting of fam-

ilies A.y/ of pseudodifferential operators on M , parameterized by y 2 Œ0; 1:

(8.27) A.y/ 2 OP P

j

” y

A.y/ bounded in OPS

j kC`

1;0

.M /:

Furthermore, if L 2 OP S

.M / is a positive, self-adjoint, elliptic operator, then

operators of the form A.y/e

yL

, with A.y/ 2 OP P

j

, belong to OP P

j

.In

addition (see (12.50)), any A.y/ 2 OP P

j

can be written in the form e

yL

B.y/

for some such elliptic L and some B.y/ 2 OP P

j

. The following result is useful

for H¨older estimates on solutions to elliptic boundary problems.

Proposition 8.8. If A.y/ 2 OP P

j

and f 2 C



.M /,then

(8.28) u.y; x/ D A.y/f .x/ H) u 2 C

j Cr

.I  M/;

provided j C r 2 R

n Z

Note that we allow r<0if j>0.

Proof. First consider the case j D 0; 0 < r < 1, and write

(8.29) A.y/f D e

yƒ

B.y/f; B.y/ 2 OP P

We can assume without loss of generality that ƒ D .1/

1=2

, and we can replace

M by R

. In such a case, we will show that

(8.30) jr

y;x

u.y; x/jCy

r1

kuk

if 0<r<1, which by Proposition 8.7 will yield u 2 C

.I  M/.Nowifweset

D @=@x

for 1  j  n; @

D @=@y, then we can write

(8.31) y@

u.y; x/ D yƒe

yƒ

.y/f; B

.y/ 2 OP P

Now, given f 2 C

.M /; 0 < r < 1,wehaveB

.y/f bounded in C

.M /,for

y 2 Œ0; 1. Then the estimate (8.30) follows from

(8.32) k'.yƒ/gk

 Cy

kgk



;

for 0<r<1,where'./ D e



, which vanishes at  D 0 and is rapidly

decreasing as  !C1. In turn, this follows easily from the characterization

(8.6)oftheC



-norm.

8. H¨older spaces and Zygmund spaces 47

If f 2 C

kCr

.M /; k 2 Z

; 0<r<1,andj D 0 then given j˛jk,

(8.33) D

y;x

u D e

yƒ

.y/ƒ

f; B

.y/ 2 OP P

;

so the analysis of (8.29), with f replaced by ƒ

f , applies to yield D

y;x

u 2

.I M/,forj˛jk.

Similarly, the extension from j D 0 to general j 2 Z

is straightforward, so

Proposition 8.8 is proved.

As we have said above, Proposition 8.8 is important because it yields H¨older

estimates on solutions to elliptic boundary problems, as deﬁned in Chap. 5,  11.

The principal consequence is the following:

Theorem 8.9. Let .P; B

;1  j  `/ be a regular elliptic boundary problem.

Suppose P has order m and each B

has order m

. If u solves

(8.34) P u D 0 on ; B

u D g

on @;

then, for r 2 R

n Z

(8.35) g

2 C

rm



.@/ H) u 2 C

./:

Proof. Of course, u 2 C

./. On a collar neighborhood of @, diffeomorphic

to Œ0; 1  @, we can write, modulo C



Œ0; 1  @



(8.36) u D

.y/g

.y/ 2 OP P

m

;

by Theorem 12.6 of Chap. 7, so the implication (8.35) follows directly from

(8.28).

We next want to deﬁne Zygmund spaces on domains with boundary. Let  be

an open set with smooth boundary (and closure

) in a compact manifold M .We

want to consider Zygmund spaces C



./; r > 0. The approach we will take is

to deﬁne C



./ by interpolation:

(8.37) C



./ D



./; C

./





;

where 0<s

<r<s

;0<<1;rD .1  /s

C s

(and s

… Z). As in

(8.21), we are using the complex interpolation functor deﬁned in Appendix A. We

need to show that this is independent of choices of such s

. Using an argument

parallel to one in  6,foranyN 2 Z

, we have an extension operator

(8.38) E W C

./ ! C

.M /; s 2 .0; N / n Z;