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

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138 14. Nonlinear Elliptic Equations

Theorem 4.4. If u solves the quasi-linear elliptic PDE (4.21), then

(4.22) u 2 C

m1Cs

\ C

msC"

;f2 C



H) u 2 C

mCr



;

provided s>0;">0, and s<r. Thus

(4.23) u 2 C

m1Cs

;f 2 C



H) u 2 C

mCr



;

provided

(4.24) s>

;r>s 1:

We can sharpen Theorem 4.4 a bit as follows. Replace the hypothesis in

(4.22)by

(4.25) u 2 C

m1Cs

\ H

m1C;p

;

with p 2 .1; 1/. Recall that Proposition 9.10 of Chap.13 gives both (4.6) and,

for p 2 .1; 1/,

(4.26)

p.x; / 2 C



1;ı

H) p.x; D/ W H

rCm;p

! H

r;p

;

.1  ı/s < r < s:

Parallel to (4.14), we have

(4.27) a 2 C

; u 2 H

;p

H) au 2 H

;p

; for  s<<s;

as a consequence of (4.26), so we see that the left side of (4.21) is well deﬁned

provided s C >1.Wehave(4.8) and, by (4.26),

(4.28) A

.x; D/ W H

mCrıs;p

! H

r;p

; for  .1  ı/s < r < s;

parallel to (4.7). Thus, if (4.25) holds, we obtain

(4.29) p.x; D/A

.x; D/u 2 H

m1CCıs;p

;

provided .1  ı/s < ıs  1 C <s, i.e., provided

(4.30) s C >1 and 1 C  C ıs < s:

Thus, if f 2 H

;p

with > 1, we manage to improve the regularity of u

over the hypothesized (4.25). One way to record this gain is to use the Sobolev

imbedding theorem:

(4.31) H

m1CCıs;p

 H

m1C;p

n  ıs



1 C

ısp



4. Elliptic regularity I (interior estimates) 139

If we assume f 2 C



with r> 1, we can iterate this argument sufﬁciently

often to obtain u 2 C

m1C"

, for arbitrary ">0. Now we can arrange s C >

1C", so we are now in a position to apply Theorem 4.4. This proves the following:

Theorem 4.5. If u solves the quasi-linear elliptic PDE (4.21), then

(4.32) u 2 C

m1Cs

\ H

m1C;p

;f2 C



H) u 2 C

mCr



;

provided 1<p<1 and

(4.33) s>0; sC >1; r> 1:

Note that if u 2 H

m;p

for some p>n,thenu 2 C

m1Cs

for s D 1n=p > 0,

and then (4.32) applies, with  D 1, or even with  D n=p C ".

We next obtain a result regarding the regularity of solutions to a completely

nonlinear elliptic system

(4.34) F.x;D

u/ D f:

We could apply Theorems 4.2 and 4.3 to the equation for u

D @u=@x

(4.35)

j˛jm

@

.x; D

u/D

DF

.x; D

u/ C

D f

Suppose u 2 C

mCs

;s>0, so the coefﬁcients a

.x/ D .@F=@

/.x; D

u/ 2

.Iff 2 C



,thenf

2 C

r1



. We can apply Theorems 4.2 and 4.3 to u

provided u 2 C

mC1sC"

, to conclude that u 2 C

mCsC1



mCr



. This implication

can be iterated as long as s C 1<r, until we obtain u 2 C

mCr



This argument has the drawback of requiring too much regularity of u, namely

that u 2 C

mC1sC"

as well as u 2 C

mCs

. We can ﬁx this up by considering

difference quotients rather than derivatives @

u. Thus, for y 2 R

; jyj small, set

.x/ Djyj

1



u.x C y/  u.x/



satisﬁes the PDE

(4.36)

j˛jm

˛y

.x/D

.x/ D G

.x; D

u/;

where

(4.37) ˆ

˛y

.x/ D

.@F=@



x; tD

u.x/ C .1  t/D

u.x C y/



140 14. Nonlinear Elliptic Equations

and G

is an appropriate analogue of the right side of (4.35). Thus ˆ

˛y

is in C

uniformly as jyj!0,ifu 2 C

mCs

, while this hypothesis also gives a uniform

bound on the C

m1Cs

-norm of v

. Now, for each y, Theorems 4.2 and 4.3 apply

to v

, and one can get an estimate on kv

mC

; D min.s; r  1/, uniform as

jyj!0. Therefore, we have the following.

Theorem 4.6. If u solves the elliptic PDE (4.34), then

(4.38) u 2 C

mCs

;f2 C



H) u 2 C

mCr



;

provided

(4.39) 0<s<r:

We shall now give a second approach to regularity results for nonlinear elliptic

PDE, making use of the paradifferential operator calculus developed in  10 of

Chap. 13. In addition to giving another perspective on interior estimates, this will

also serve as a warm-up for the work on boundary estimates in  8.

If F is smooth in its arguments, then, as shown in (10.53)–(10.55) of Chap. 13,

(4.40) F.x;D

u/ D

j˛jm

.x; D/D

u C F.x;D

‰

.D/u/;

where F



x; D

‰

.D/u



2 C

and

(4.41) M

.x; / D

.x/

kC1

./;

with

(4.42) m

.x/ D

@



‰

.D/D

u C t

kC1

.D/D



dt:

As shown in Proposition 10.7 of Chap. 13, we have, for r  0,

(4.43) u 2 C

mCr

H) M

.x; / 2 A

1;1

 S

1;1

\ C

1;0

We recall from (10.31) of Chap. 13 that

(4.44) p.x; / 2 A

1;ı

”kD



p.;/k

rCs

 C

˛s

hi

mj˛jCıs

;s 0:

Consequently, if we set

(4.45) M.uIx; D/ D

j˛jm

.x; D/D

;

we obtain

4. Elliptic regularity I (interior estimates) 141

Proposition 4.7. If u 2 C

mCr

;r 0,then

(4.46) F.x;D

u/ D M.uIx; D/u C R;

with R 2 C

and

(4.47) M.uIx; / 2 A

1;1

 S

1;1

\ C

1;0

Decomposing each M

.x; /,wehave,by(10.60)–(10.61) of Chap. 13,

(4.48) M.uIx; / D M

.x; / C M

.x; /;

with

(4.49) M

.x; / 2 A

1;ı

 S

1;ı

and

(4.50) M

.x; / 2 C

mır

1;ı

\ A

1;1

 S

mrı

1;1

Let us explicitly recall that (4.49) implies

(4.51)

.x; / 2 S

1;ı

; jˇjr;

mCı.jˇjr/

1;ı

; jˇjr:

Note that the linearization of F.x;D

u/ at u is given by

(4.52) Lv D

j˛jm

.x/D

where

(4.53)

.x/ D

@

.x; D

u/:

Comparison with (4.40)–(4.42) gives (for u 2 C

mCr

)

(4.54) M.uIx; /  L.x; / 2 C

mr

1;1

;

by the same analysis as in the proof of the ı D 1 case of (9.35) of Chap. 13. More

generally, the difference in (4.54) belongs to C

mrı

1;ı

;0 ı  1. Thus L.x; /

and M.uIx; / have many qualitative properties in common.

Consequently, given u 2 C

mCr

, the operator M

.x; D/ 2 OP S

1;ı

microlocally elliptic in any direction .x

;

/ 2 T



n0 that is noncharacteristic

142 14. Nonlinear Elliptic Equations

for F.x;D

u/, which by deﬁnition means noncharacteristic for L. In particular,

.x; D/ is elliptic if F.x;D

u/ is. Now if

(4.55) F.x;D

u/ D f

is elliptic and Q 2 OP S

m

1;ı

is a parametrix for M

.x; D/,wehave

(4.56) u D Q.f  M

.x; D/u/; mod C

By (4.50)wehave

(4.57) QM

.x; D/ W H

mrıCs;p

! H

mCs;p

;s>0:

(In fact s>.1  ı/r sufﬁces.) We deduce that

(4.58) u 2 H

mırCs;p

;f2 H

s;p

H) u 2 H

mCs;p

;

granted r>0;s>0,andp 2 .1; 1/. There is a similar implication, with

Sobolev spaces replaced by H¨older (or Zygmund) spaces. This sort of implication

can be iterated, leading to a second proof of Theorem 4.6. We restate the result,

including Sobolev estimates, which could also have been obtained by the ﬁrst

method used to prove Theorem 4.6.

Theorem 4.8. Suppose, given r>0, that u 2 C

mCr

satisﬁes (4.55) and this

PDE is elliptic. Then, for each s>0;p2 .1; 1/,

(4.59) f 2 H

s;p

H) u 2 H

mCs;p

and f 2 C



H) u 2 C

mCs



By way of further comparison with the methods used earlier in this section,

we now rederive Theorem 4.5, on regularity for solutions to a quasi-linear elliptic

PDE. Note that, in the quasi-linear case,

(4.60) F.x;D

u/ D

j˛jm

.x; D

m1

u/D

u D f;

the construction above gives F.x;D

u/ D M.uIx; D/u C R

.u/ with the

property that, for r  0,

(4.61) u 2 C

mCr

H) M.uIx; / 2 C

rC1

1;0

\ S

1;1

C C

m1

1;0

\ S

m1

1;1

Of more interest to us now is that, for 0<r<1,

(4.62) u 2 C

m1Cr

H) M.uIx; / 2 C

1;0

\ S

1;1

C S

mr

1;1

;

4. Elliptic regularity I (interior estimates) 143

which follows from (10.23) of Chap. 13. Thus we can decompose the term in

1;0

1;1

via symbol smoothing, as in (10.60)–(10.61)of Chap. 13, and throw

the term in S

mr

1;1

into the remainder, to get

(4.63) M.uIx; / D M

.x; / C M

.x; /;

with

(4.64) M

.x; / 2 S

1;ı

.x; / 2 S

mrı

1;1

If P.x;D/ 2 OP S

m

1;ı

is a parametrix for the elliptic operator M

.x; D/,then

whenever u 2 C

m1Cr

\ H

m1C;p

is a solution to (4.60), we have, mod C

(4.65) u D P.x;D/f  P.x;D/M

.x; D/u:

Now

(4.66) P.x;D/M

.x; D/ W H

m1C;p

! H

m1CCrı;p

if r C >1;

by the last part of (4.64). As long as this holds, we can iterate this argument and

obtain Theorem 4.5, with a shorter proof than the one given before.

Next we look at one example of a quasi-linear elliptic system in divergence

form, with a couple of special features. One is that we will be able to assume less

regularity a priori on u than in results above. The other is that the lower-order

terms have a more signiﬁcant impact on the analysis than above. After analyzing

the following system, we will show how it arises in the study of the Ricci tensor.

We consider second-order elliptic systems of the form

(4.67)

.x; u/@

u C B.x; u; ru/ D f:

We assume that a

.x; u/ and B.x; u;p/ are smooth in their arguments and that

(4.68) jB.x; u;p/jC hpi

Proposition 4.9. Assumethatasolutionuto(4.67) satisﬁes

(4.69) ru 2 L

; for some q>n; hence u 2 C

;

for some r 2 .0; 1/. Then, if p 2 .q; 1/ and s 1, we have

(4.70) f 2 H

s;p

H) u 2 H

sC2;p

To begin the proof of Proposition 4.9, we write

(4.71)

.x; u/@

u D A

.uIx; D/u

144 14. Nonlinear Elliptic Equations

mod C

, with

(4.72) u 2 C

H) A

.uIx; / 2 C

1;0

\ S

1;1

C S

1r

1;1

;

as established in Chap. 13. Hence, given ı 2 .0; 1/,

(4.73)

.uIx; / D A

.x; / C A

.x; /;

.x; / 2 S

1;ı

.x; / 2 S

1rı

1;1

It follows that we can write

(4.74)

.x; u/@

u D P

u C P

with

(4.75) P

.x; D/ 2 OP S

1;ı

; elliptic;

and

(4.76) P

.x; D/:

By Theorem 9.1 of Chap. 13, we have

(4.77) A

.x; D/ W H

1rıC;p

! H

;p

; for >0; 1<p

< 1:

In particular (taking  D rı; p

D q),

(4.78) ru 2 L

H) P

u 2 H

1Crı;q

Now, if

(4.79) E

2 OPS

2

1;ı

denotes a parametrix of P

,wehave,modC

(4.80) u D E

f  E

B.x; u; ru/  E

and we see that under the hypothesis (4.69), we have some control over the last

term:

(4.81) E

u 2 H

1Crı;q

 H

1; Qq

;



rı

4. Elliptic regularity I (interior estimates) 145

Note also that under our hypothesis on B.x; u;p/,

(4.82) ru 2 L

H) B.x; u; ru/ 2 L

q=2

Now, by Sobolev’s imbedding theorem,

(4.83) E

B.x; u; ru/ 2 H

1; Qp

;

with Qp D q=.2  q=n/ if q<2nand for all Qp<1 if q  2n. Note that

Qp>q.1C a=n/ if q D n C a. This treats the middle term on the right side of

(4.80). Of course, the hypothesis on f yields

(4.84) E

f 2 H

sC2;p

;sC 2  1;

which is just where we want to place u.

Having thus analyzed the three terms on the right side of (4.80), we have

(4.85) u 2 H

1;q

D min. Qp; p; Qq/:

Iterating this argument a ﬁnite number of times, we get

(4.86) u 2 H

1;p

If s D1 in (4.70), our work is done.

If s>1 in (4.70), we proceed as follows. We already have u 2 H

1;p

,so

ru 2 L

. Thus, on the next pass through estimates of the form (4.78)–(4.83), we

obtain

(4.87)

u 2 H

1Crı;p

;

B.x; u; ru/ 2 H

2;p=2

 H

2n=p;p

;

and hence

(4.88) u 2 H

1C;p

;D min



rı;1 

;1C s



We can iterate this sort of argument a ﬁnite number of times until the conclusion

in (4.70) is reached.

Further results on elliptic systems of the form (4.67) will be given in  12B.

We now apply Proposition 4.9 to estimates involving the Ricci tensor. Consider a

Riemannian metric g

deﬁned on the unit ball B

 R

. We will work under the

following hypotheses:

(i) For some constants a

2 .0; 1/, there are estimates

(4.89) 0<a

I 



.x/



 a

146 14. Nonlinear Elliptic Equations

(ii) The coordinates x

;:::;x

are harmonic, namely

(4.90) x

D 0:

Here,  is the Laplace operator determined by the metric g

. In general,

(4.91) v D g

v  

v; 

D g



Note that x

D

, so the coordinates are harmonic if and only if 

D 0.

Thus, in harmonic coordinates,

(4.92) v D g

We will also assume some bounds on the Ricci tensor, and we desire to see

how this inﬂuences the regularity of g

in these coordinates. Generally, as can

be derived from formulas in  3 of Appendix C, the Ricci tensor is given by

(4.93)

Ric



@

 @

C @



C M

.g; rg/

D



C H

.g; rg/;

with 

as in (4.91). In harmonic coordinates, we obtain

(4.94) 

.x/ @

C Q

.g; rg/ D Ric

;

and Q

.g; rg/ is a quadratic form in rg, with coefﬁcients that are smooth

functions of g, as long as (4.89) holds. Also, when (4.89) holds, the equation

(4.94) is elliptic, of the form (4.67). Thus Proposition 4.9 implies the following.

Proposition 4.10. Assume the metric tensor satisﬁes hypotheses (i) and (ii). Also

assume that, on B

(4.95) rg

2 L

; for some q>n;

and

(4.96) Ric

2 H

s;p

;

for some p 2 .q; 1/; s 1. Then, on the ball B

9=10

(4.97) g

2 H

sC2;p

In [DK] it was shown that if g

2 C

, in harmonic coordinates, then, for

k 2 Z

;˛2 .0; 1/,Ric

2 C

kC˛

) g

2 C

kC2C˛

. Such results also follow

5. Isometric imbedding of Riemannian manifolds 147

by the methods used to prove Proposition 4.10. A result stronger than Proposition

4.10, using Morrey spaces, is proved in [T2].

Exercises

1. Consider the system F.x;D

u/ D f when

F.x;D

u/ D

j˛jm

.x; D

u/D

for some j such that 0  j<m. Assume this quasi-linear system is elliptic. Given

p; q 2 .1; 1/; r > 0, assume

u 2 C

j Cr

\ H

m1C;p

;rC>1:

Show that

f 2 H

s;q

H) u 2 H

sCm;q

5. Isometric imbedding of Riemannian manifolds

In this section we will establish the following result.

Theorem 5.1. If M is a compact Riemannian manifold, there exists a C

-map

(5.1) ˆ W M ! R

;

which is an isometric imbedding.

This was ﬁrst proved by J. Nash [Na1], but the proof was vastly simpliﬁed by

M. G¨unther [Gu1]–[Gu3]. These works also deal with noncompact Riemannian

manifolds and derive good bounds for N , but to keep the exposition simple we

will not cover these results.

To prove Theorem 5.1, we can suppose without loss of generality that M is a

torus T

. In fact, imbed M smoothly in some Euclidean space R

I M will sit

inside some box; identify opposite faces to have M  T

. Then smoothly extend

the Riemannian metric on M to one on T

If R denotes the set of smooth Riemannian metrics on T

and E is the set of

such metrics arising from smooth imbeddings of T

into some Euclidean space,

our goal is to prove

(5.2) E D R:

Now R is clearly an open convex cone in the Fr´echet space

V D C

