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

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278 14. Nonlinear Elliptic Equations

Hence, for any ` 2f1;:::;Ng,

(14.31)

.y/  m







CeR C ƒ



k¤`



 h

.y/



 C

eR C

k¤`



 h

.y/



;

where C D C.n;ƒ=/. We can use (14.24) to estimate the right side of (14.31),

obtaining

(14.32) ˆ

p;R

 m

/  C

.1 C "/!.2R/  !.R/ C eR C R



Setting ` D k, adding (14.32)to(14.23), and then summing over k, we obtain

(14.33) !.2R/  C

.1 C "/!.2R/  !.R/ C eR C

R



;

and hence

(14.34) !.R/ 



1 

C "



!.2R/ C



eR C

R



Now C is independent of ", though

 is not. Thus ﬁx " D 1=2C , to obtain

(14.35) !.R/ 



1 



!.2R/ C



eR C

R



From this it follows that if B

  and R  R

,wehave

(14.36) osc

u  C





.1 C M/



1 CeR

C R



;

where C and ˛ are positive constants depending only on n and ƒ=.Wehave

proved the following interior estimate:

Proposition 14.2. Let u 2 C

./ satisfy (14.1), and assume that (14.2) and

(14.3) hold. Then, for any O  , there is an estimate

(14.37) k@

.O/

 C



O;;n;;ƒ;kF k

; kuk

./



In fact, examining the derivation of (14.36), we can specify the dependence on

O;.IfO is a ball, and jx  yj for all x 2 O;y2 @,then

(14.38) k@

.O/

 C



n; ; ƒ; kF k

; kuk

./





˛

14. Regularity for a class of completely nonlinear equations 279

We now tackle global estimates on  for solutions to the Dirichlet problem for

(14.1). We ﬁrst obtain estimates for @

@

Lemma 14.3. Under the hypotheses of Proposition 14.2,ifu

@

D ', there is an

estimate

(14.39) k@

.@/

 C



;n;;ƒ; kF k

; kuk

./

; k'k

.@/



Proof. Let Y D b

.x/@

be a smooth vector ﬁeld tangent to @, and consider

v D Y u, which solves the boundary problem

(14.40) F



v D G.x/; v

@

D Y';

where

(14.41)

G.x/ D 2F



/.@

u/ C F



C F

/.@

u/  F

v  F

v  b

The hypotheses give a bound on kGk

./

in terms of the right side of (14.39).

If 2 C

./ denotes an extension of Y' from @ to , then Proposition 13.11,

applied to v  , yields an estimate

(14.42) k@



Y uk

.@/

 C;

where C is of the form (14.39). On the other hand, the ellipticity of (14.1) allows

one to solve for @



@

in terms of quantities estimated in (14.42), plus u

@

and

@

, and second-order tangential derivatives of u,so(14.39) is proved.

We now want to estimate j@



u.x/ @



u.x

/j,givenx

2 @; x 2 ;  2 R

a unit vector. For simplicity, we will strengthen the concavity hypothesis (14.3)to

strong concavity:

(14.43) @



F.x;u;p;/„

„



j„j

;„D „

;

for some 

>0,when D @

u;pDru. Then we can improve (14.8)to

(14.44) F





u/ A



u  B



 



C

;

by Cauchy’s inequality, where

D C



n; ; ƒ; 

; kA



.x; D

u/k

; kB



.x; D

u/k



Now the function

(14.45) W.x/ D C

jx x

.0 < ˛ < 1/

280 14. Nonlinear Elliptic Equations

is concave on R

nfx

g,andifC

is sufﬁciently large, compared to C



diam./

2˛

=,wehave

(14.46) LW C

;LvD F



Hence, by the maximum principle,

(14.47) @



u  @



/ C W on @ H) @



u  @



u.x

/ C W on :

Now the estimate (14.39) implies that the hypothesis of (14.47) is satisﬁed, pro-

vided that also C

k@

.@/

, so we have the one-sided estimate given by

the conclusion of (14.47).

For the reverse estimate, use (14.25), with y D x

, together with (14.29), to

write

(14.48)

kD1





u.x

/  @



u.x/



 D

jx x

Recall that ˇ

/ 2 Œ



;ƒ



; 



>0. This together with (14.47) implies

(14.49) j@



u.x/  @



u.x

/jC

jx x

;

with C

of the form (14.39), and we can express any @

u as a linear combination

of the @



u, to obtain the following:

Lemma 14.4. If we have the hypotheses of Lemma 14.3, and we also assume

(14.43), then there is an estimate

(14.50) j@

u.x/ @

u.x

/jC jx  x

2 @; x 2 ;

with

(14.51) C D C



;n;;ƒ; 

; kF k

; kuk

./

; k'k

.@/



We now put (14.38)and(14.50) together to obtain a H¨older estimate for @

. To estimate j@

u.x/  @

u.y/j,givenx; y 2 , suppose dist.x; @/ C

dist.y; @/ D 2, and consider two cases:

(i) jx  yj <

(ii) jx  yj

In case (i), we can use (14.38) to deduce that

(14.52) j@

u.x/  @

u.y/jC jx  yj



˛

 C jx  yj

˛=2

14. Regularity for a class of completely nonlinear equations 281

In case (ii), let x

2 @ minimize the distance from x to @,andlety

2 @

minimize the distance from y to @. Thus

(14.53)

jx  x

j2  2jx  yj

1=2

; jy  y

j2  2jx  yj

1=2

;

 y

jjx  yjCjx

 xjCjy

 yjjx  yjC4jx  yj

1=2

Thus

(14.54)

u.x/  @

u.y/jj@

u.x/  @

u.x

/jCj@

u.x

/  @

u.y

Cj@

u.y

/  @

u.y/j



C jx  x

C jx

 y

C jy

 yj

 C jx  yj

˛=2

In (14.52)and(14.54), C has the form (14.51). Taking r D ˛=2,wehavethe

following global estimate:

Proposition 14.5. Let u 2 C

./ satisfy (14.1), with u

@

D '. Assume the

ellipticity hypothesis (14.2) and the strong concavity hypothesis (14.43). Then

there is an estimate

(14.55) kuk

2Cr

./

 C



;n;;ƒ; 

; kF k

; kuk

./

; k'k

.@/



;

for some r>0, depending on the same quantities as C .

Now that we have this estimate, the continuity method yields the following

existence result. For  2 Œ0; 1, consider a family of boundary problems

(14.56) F



.x; D

u/ D 0 on ; u

@

D '



Assume F



and '



are smooth in all variables, including  . Also, assume that

the ellipticity condition (14.2) and the strong concavity condition (14.43) hold,

uniformly in , for any smooth solution u



Theorem 14.6. Assume there is a uniform bound in C

./ for any solution u



./ of (14.56). Also assume that @



 0. Then, if (14.56) has a solution in

./ for  D 0, it has a smooth solution for  D 1.

With some more work, one can replace the strong concavity hypothesis (14.43)

by (14.3); see [CKNS].

There is an interesting class of elliptic PDE, known as Bellman equations,

for which F.x;u;p;/ is concave but not strongly concave in , and also it is

Lipschitz but not C

in its arguments; see [Ev2] for an analysis.

Verifying the hypothesis in Theorem 14.6 that u



is bounded in C

./ can be

a nontrivial task. We will tackle this, for Monge–Ampere equations, in the next

section.

282 14. Nonlinear Elliptic Equations

Exercises

1. Discuss the Dirichlet problem for

u C @

u C



1 C .u/



1=2

D e

;

for   0.

15. Monge–Ampere equations

Here we look at equations of Monge–Ampere type:

(15.1) det H.u/  F.x;u; ru/ D 0 on ; u D ' on @;

where  is a smoothly bounded domain in R

, which we will assume to be

strongly convex. As in (3.7a), H.u/ D .@

u/ is the Hessian matrix. We assume

F.x;u; ru/>0,sayF.x;u; ru/ D exp f.x;u; ru/, and look for a convex

solution to (15.1). It is convenient to set

(15.2) G.u/ D log det H.u/  f.x;u; ru/;

so (15.1) is equivalent to G.u/ D 0 on ; u D ' on @. Note that

(15.3) DG.u/v D g

v  .@

f /.x; u; ru/@

v  .@

f /.x; u; ru/v;

where .g

/ is the inverse matrix of .@

u/, which we will also denote as .g

We will assume

(15.4) .@

f /.x; u;p/ 0;

this hypothesis being equivalent to .@

F /.x; u;p/  0.

The hypotheses made above do not sufﬁce to guarantee that (15.1)hasa

solution. Consider the following example:

(15.5) det H.u/  K



1 Cjruj



D 0 on ; u D 0 on @;

where  is a domain in R

. Compare with (3.41). Let K be a positive constant. If

there is a convex solution u, the surface † Df.x; u.x// W x 2 g is a surface in

with Gauss curvature K.If is convex, then the Gauss map N W † ! S

one-to-one and the image N.†/ has area equal to K Area./.ButN.†/ must be

contained in a hemisphere of S

,sowemusthaveK Area./  2. We deduce

that if K  Area./ > 2,then(15.5) has no solution.

To avoid this obstruction to existence, we hypothesize that there exists u

./, which is convex and satisﬁes

(15.6) log det H.u

/  f.x;u

; ru

/  0 on ; u

D ' on @:

15. Monge–Ampere equations 283

We call u

a lower solution to (15.1). Note that the ﬁrst part of (15.6) is equivalent

to det H.u

/  F.x;u

; ru

/. In such a case, we will use the method of conti-

nuity and seek a convex u



2 C

./ solving

(15.7)

log det H.u



/  f.x;u



; ru



D .1  /



log det H.u

/  f.x;u

; ru



D .1  /h.x/;

for  2 Œ0; 1 and u



D ' on @. Note that u

D u

solves (15.7)for D 0.If

such u



exists for all  2 Œ0; 1,thenu D u

is the desired solution to (15.1).

Let J be the largest interval in Œ0; 1, containing 0, such that (15.7)hasaconvex

solution u



2 C

./ for all  2 J . Since the linear operator in (15.3) is elliptic

and invertible (by the maximum principle) under the hypothesis (15.4), the same

sort of argument used in the proof of Lemma 10.1 shows that J is open, and the

real work is to show that J is closed. In this case, we need to obtain bounds on u



in C

2C

./,forsome>0, in order to apply the regularity theory of  8 and

conclude that J is closed.

Lemma 15.1. Given    2 J , we have

(15.8) u

 u



 u



on :

Proof. The operator G.u/ satisﬁes the hypotheses of Proposition 10.8;sinceu



D u



on @,(15.8) follows.

In particular, taking  D ,wehaveuniqueness of the solution u



2 C

./

to (15.7).

Next we record some estimates that are simple consequences of convexity

alone:

Lemma 15.2. Assume  is convex. For any  2 J ,

(15.9) u



 sup

@

' on 

and

(15.10) sup

x2

jru



.x/j sup

y2@

jru



.y/j:

Thus we will have a bound on u



in C

./ if we bound ru



on @.Since



@

D ' 2 C

.@/, it remains to bound the normal derivative @





on @.

Assume @



points out of .Then(15.8) implies

(15.11) @





.y/  @



.y/; 8 y 2 @:

284 14. Nonlinear Elliptic Equations

On the other hand, a lower bound on @





.y/ follows from convexity alone. In

fact, if .y/ is the outward normal to @ at y,sayey D y `.y/.y/ is the other

point in @ through which the normal line passes. Then convexity of u



implies

(15.12) u





sy C .1  s/ey



 s'.y/ C .1  s/'.ey/;

for 0  s  1. Noting that `.y/ Djy eyj,wehave





.y/ 

'.ey/ '.y/

jey  yj

Thus we have the next result:

Lemma 15.3. If  is convex, then, for any  2 J ,

(15.13) sup



jru



jLip

.'/ C sup



jru

Here, Lip

.'/ denotes the Lipschitz constant of ':

(15.14) Lip

.'/ D sup

y;y

2@

j'.y/  '.y

jy  y

We now look for C

-bounds on solutions to (15.7). For notational simplicity,

we write (15.7)as

(15.15) log det H.u/  f.x;u; ru/ D 0; u

@

D ';

where the second term on the left is



.x; u; ru/ D f.x;u; ru/ C .1  /h.x/;

and we drop the .By(15.4)and(15.6), we have f.x;u;p/ > 0 and

f /.x; u;p/ 0.

Since u is convex, it sufﬁces to estimate pure second derivatives @



u from

above. Following [CNS], who followed [LiP2], we make use of the function

w D e

ˇjruj



where ˇ is a constant that will be chosen later. Suppose this is maximized, among

all unit  2 R

;x2 ,at D 

;xD x

. Rotating coordinates, we can

assume







u.x



is in diagonal form and 

D .1;0;:::;0/.Set

D @

u,sowetake

(15.16) w D e

ˇjruj

D .ru/u

We now derive some identities and inequalities valid on all of .

15. Monge–Ampere equations 285

Differentiating (15.15), we obtain

(15.17)

u D @

f.x;u; ru/;

D g

u/.@

u/ C @

where .g

/ is the inverse matrix to .g

/ D .@

u/, as above. Also, a calculation

gives

(15.18)

1

w D .log /

u C u

1

u/;

1

w D w

2

w/.@

w/ C .log /

u/.@

C .log /

u/ C u

1

 u

2

u/.@

u/:

Forming w

1

w and using (15.17) to rewrite the term u

1

,we

obtain

(15.19)

1

 u

.log /

u/.@

u/ C .log /

C g

u/.@

u/  u

1

u/.@

u/ C @

Now we have .log /

D ˇp

and .log /

D ˇı

, and hence

(15.20) .log /

u/.@

u/ D ˇı

u/ D ˇu:

Let us assume the following bounds hold on f.x;u;p/:

(15.21) j.rf /.x; u;p/j; j.@

f /.x; u;p/j:

Using the ﬁrst identity in (15.17), we have

(15.22)

.log /

u C @

 f

1

w/u

 C



1 Cj@

C ˇ.1 Cj@

uj/



;

with C D C



; kruk

./



Now, let us look at x

, where, recall, e

ˇjruj

u is maximal, among all values

of e

ˇjru.x/j



u.x/.Ifx

2  (i.e., x

… @), then @

w.x

/ D 0 and the left

side of (15.19)is 0 at x

. Furthermore, due to the diagonal nature of .g

/ at

, we easily verify that g



 g



, and hence

(15.23) u

1

u/.@

u/  g

u/.@

u/;

286 14. Nonlinear Elliptic Equations

at x

. Thus the evaluation of (15.19)atx

implies the estimate

(15.24) 0  ˇ.@

u/.u/   C



1 Cj@

C ˇ.1 Cj@

uj/



if x

… @. Hence, with X D @

u.x

(15.25) .ˇ  C

 ˇC

.1 C X/ C ;

where C

and C

depend on  and kruk

, but not on ˇ.Takingˇ large, we

obtain a bound on X:

(15.26) @

u.x

/  C



; kruk

./



if x

… @:

On the other hand, if sup w is achieved on @,wehave

sup

x;



u.x/jsup

@

uj exp



ˇkruk



This establishes the following.

Lemma 15.4. If u 2 C

./ \ C

./ solves (15.15) and the hypotheses above

hold, then

(15.27) sup



ujC



; kruk

./



1 C sup

@

To estimate @

u at a boundary point y 2 @, suppose coordinates are rotated

so that .y/ is parallel to the x

-axis. Pick vector ﬁelds Y

, tangent to @,sothat

.y/ D @

;1 j  n  1. Then we easily get

(15.28) j@

u.y/jjY

'.y/jCC jru.y/j;1 j; k  n  1:

In fact, for later reference, we note the following. Suppose Y

is the vector ﬁeld

tangent to @, equal to @

at y, and obtained by parallel transport along geodesics

emanating from y.IfY

D b

,then

(15.29)

u.y/ D @

u.y/ C



.y/



u.y/

D @

u.y/ C





u.y/;

where r

is the standard ﬂat connection on R

.Ifr is the Levi–Civita connection

on @,wehaver

D 0 at y, hence r

D

II.@



at y,where



DN is the outward-pointing normal and

II is the second fundamental form

of @;see 4 of Appendix C. Hence

(15.30) @

u.y/ D Y

u.y/ C

II.@



u.y/; 1  j; k  n 1:

15. Monge–Ampere equations 287

Later it will be important to note that strong convexity of @ implies positive

deﬁniteness of

II.

We next need to estimate @

u.y/; 1  k  n  1.IfY

D b

.x/ @

,then

D Y

u satisﬁes the equation

(15.31) g

 f

D A.x/ C g

.x/;

where

(15.32)

A.x/ D 2@

C f

.x/ D .@

and v

@

D Y

'. This follows by multiplying the ﬁrst identity in (15.17)byb

and summing over `; one also makes use of the identity g

u D ı

We ﬁrst derive a boundary gradient estimate for v

D Y

u when (15.15) takes

the simpler form

(15.33) log det H.u/  f.x;u/ D 0; u

@

D 'I

that is, ru is not an argument of f . Here, we follow [Au]. We assume ' 2

./,set

(15.34) w

D Y

.u  '/ D v

 Y

then let ˛ and ˇ be real numbers, to be ﬁxed below, and set

(15.35) ew

D w

C ˛h C ˇ.u  '/:

Here, h 2 C

./ is picked to vanish on @ and satisfy a strong convexity con-

dition:

(15.36) .@

h/  I; h

@

D 0:

The hypothesis that

 is strongly convex is equivalent to the existence of such a

function.

Now, a calculation using (15.31) (and noting that in this case f

D 0)gives

(15.37) g

D A.x/ C nˇ C g

.x/; ew

@

D 0;

where A.x/ is as in (15.32) (with the last term equal to zero), and

(15.38)

.x/ D B

.x/  @

' C ˛@

h  ˇ@