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

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218 14. Nonlinear Elliptic Equations

where G.x; u;p;/satisﬁes the ellipticity hypothesis (10.29), and

(10.57) @

G.x; u;p;/ 0:

Then, given u



2 C

./ \ C./,

(10.58) H.D

/  H.D

/ on ; u

 u

on @ H) u

 u

on :

Proof. Same as Proposition 10.3. For the relevant maximum principle, replace

Proposition 2.1 of Chap. 5 by Proposition 2.6 of that chapter.

To continue our analysis of the PDE (10.50), Proposition 10.8 applies to give

(10.53) ) (10.54), provided V

.u/  0. Consequently, we achieve a bound on

kruk

./

, and hence also on kuk

./

, provided V.u/ satisﬁes the hypotheses

stated in (10.52)and(10.55).

It remains to apply the DeGiorgi–Nash–Moser theory. In the simpliﬁed case

where

 D T

n1

 Œ0; 1, we obtain (10.18), this time by regarding (10.51)

as a nonhomogeneous PDE for w

, of the form (9.67), with one term @

namely @

.u/.TheL

-estimate we have on u is more than enough to apply

Theorem 9.7, so we again have (10.18)–(10.19). Next, the argument (10.21)–

(10.23) goes through, so we again have (10.20) and the Morrey space inclusions

(10.21)–(10.22). Hence the hypothesis (8.60) of Theorem 8.4 holds, with r>0

and  D 1. Theorem 8.4 yields

(10.59) kuk

./

 K

;

and a modiﬁcation of the argument parallel to the use of (10.27) works for

  R

The estimates above work for

(10.60) 

.ru/@

u  V

.u/ C .1 /u D 0; u

@

D ';

for all  2 Œ0; 1. Also, each linearized operator is seen to be invertible, provided

.u/  0. Thus all the ingredients needed to use the method of continuity are in

place. We have the following existence result.

Proposition 10.9. Let   R

be any bounded domain with smooth boundary.

If the PDE

(10.61)

.ru/@

u  V

.u/ D 0; u D ' on @;

is uniformly elliptic, and if V

.u/ satisﬁes

(10.62) jV

.u/jA

C A

juj;V

.u/  0;

then (10.61) has a unique solution u 2 C

./, given ' 2 C

.@/.

10. The Dirichlet problem for quasi-linear elliptic equations 219

Consider the case V.u/ D Au

. This satisﬁes (10.62)ifA  0 but not if

A<0. The case A<0corresponds to a repulsive force (away from u D 0)that

increases linearly with distance. The physical basis for the failure of (10.61)to

have a solution is that if u.x/ takes a large enough value, the repulsive force due

to the potential V cannot be matched by the elastic force of the membrane. If

.p/ is independent of p and 2A < 0 is an eigenvalue of the linear operator

, then certainly (10.61) is not solvable.

On the other hand, if V.u/ D Au

with 0>A>`

,where`

is less

than the smallest eigenvalue of all operators

with coefﬁcients satis-

fying (10.38), then one can still hope to establish solvability for (10.61), in the

uniformly elliptic case. We will not pursue the details on such existence results.

We now consider more general equations, of the form

(10.63) H.D

u/ D

.ru/@

u C g.x; u; ru/ D 0; u

@

D ':

Consider the family

(10.64) H



u/ D

.ru/@

u Cg.x;u; ru/ D 0; u

@

D ':

We will prove the following:

Proposition 10.10. Assume that the equation (10.63) satisﬁes the ellipticity con-

dition (10.3) and that @

g.x; u;p/  0.Let  R

be a bounded domain with

smooth boundary, and let ' 2 C

.@/ be given. Assume that, for  2 Œ0; 1, any

solution u D u



to (10.64) has an a priori bound in C

./.Then(10.63) has a

solution u 2 C

./.

Proof. For w

D @

u, we have, in place of (10.14),

(10.65)

.ru/@

D@

g.x; u; ru/:

The C

-bound on u yields an L

-bound on g.x; u; ru/, so, as in the proof of

Proposition 10.9, we can use Theorem 9.7 and proceed from there to obtain high-

order Sobolev estimates on solutions to (10.64).

Thus the largest interval J in Œ0; 1 that contains  D 0 andsuchthat(10.64)

is solvable for all  2 J is closed. The hypothesis @

g  0 implies that the

linearized equation at  D 

is uniquely solvable, so, as in Lemma 10.1, J is

open in Œ0; 1, and the proposition is proved.

A simple example of (10.63) is the equation for a surface z D u.x/ of given

constant mean curvature H :

(10.66) hrui

3

hrui

u D

u.ru; ru/

C nH D 0; u D ' on @;

220 14. Nonlinear Elliptic Equations

which is of the form (10.63), with F.p/ D



1 Cjpj



1=2

and g.x; u;p/ D nH .

Note that members of the family (10.64) are all of the same type in this case,

namely equations for surfaces with mean curvature H. We see that Proposition

10.3 applies to this equation. This implies uniqueness of solutions to (10.66),

provided they exist, and also gives a tool to estimate L

-norms, at least in some

cases, by using equations of graphs of spheres of radius 1=H as candidates to

bound u from above and below. We can also use such functions to construct barri-

ers, replacing the linear functions used in the proof of Proposition 7.5. This change

means that the class of domains and boundary data for which upper and lower bar-

riers can be constructed is different when H ¤ 0 than it is in the minimal surface

case H D 0.

Note that if u solves (10.66), then w

D @

u solves a PDE of the form

(10.14). Thus the maximum principle yields kruk

./

D sup

@

jru.y/j. Con-

sequently, we have the solvability of (10.66) whenever we can construct barriers

to prove the boundary gradient estimate.

The methods for constructing barriers described above do not exhaust the re-

sults one can obtain on boundary gradient estimates, which have been pushed

quite far. We mention a result of H. Jenkins and J. Serrin. They have shown that the

Dirichlet problem (10.66) for surfaces of constant mean curvature H is solvable

for arbitrary ' 2 C

.@/ if and only if the mean curvature ~.y/ of @  R

satisﬁes

(10.67) ~.y/ 

n  1

jH j; 8y 2 @:

In the special case n D 2; H D 0, this implies Proposition 7.3 in this chapter.

See [GT] and [Se2] for proofs of this and extensions, including variable mean

curvature H.x/, as well as extensive general discussions of boundary gradient

estimates. We will have a little more practice constructing barriers and deducing

boundary gradient estimates in  13 and 15 of this chapter. See the proofs of

Lemma 13.12 and of the estimate (15.54).

Results discussed above extend to more general second-order, scalar,

quasi-linear PDE. In particular, Proposition 10.10 can be extended to all equations

of the form

(10.68)

.x; u; ru/@

u C b.x;u; ru/ D 0; u

@

D ':

Let ' 2 C

.@/ be given. As long as it can be shown that, for each  2 Œ0; 1,a

solution to

(10.69)

.x; u; ru/@

u C b.x;u; ru/ D 0; u

@

D ';

has an a priori bound in C

./,then(10.68) has a solution u 2 C

./.This

result, due to O. Ladyzhenskaya and N. Ural’tseva, is proved in [GT] and [LU].

These references, as well as [Se2], also discuss conditions under which one can

Exercises 221

establish a boundary gradient estimate for solutions to such PDE, and when one

can pass from that to a C

./-estimate on solutions. The DeGiorgi–Nash–Moser

estimates are still a major analytical tool in the proof of this general result, but

further work is required beyond what was used to prove Proposition 10.10.

Exercises

1. Carry out the construction of barriers for the equation of a surface of constant mean cur-

vature mentioned below (10.66) and thus obtain some existence results for this equation.

Compare these results with the result of Jenkins and Serrin, stated in (10.67).

Exercises 2–4 deal with quasi-linear elliptic equations of the form

(10.70)

.x; u/@

u D 0 on ; u

@

D ':

Assume there are positive functions A

such that

.u/jj



.x; u/



 A

.u/jj

2. Fix ' 2 C

.@/. Consider the operator ˆ.u/ D v, the solution to

.x; u/@

v D 0; v

@

D ':

Show that, for some r>0,

ˆ W C.

/ ! C

./;

continuously. Use the Schauder ﬁxed-point theorem to deduce that ˆ has a ﬁxed point

in fu 2 C.

/ W sup jujsup j'jg \ C

./.

3. Show that this ﬁxed point lies in C

./.

4. Examine whether solutions to (10.70) are unique.

5. Extend results on (10.1) to the case

(10.71)

.x; ru/ D 0; u

@

D ';

arising from the search for critical points of I.u/ D



F.x;ru/dx, generalizing the

case considered in (10.2).

In Exercises 6–9, we consider a PDE of the form

(10.72)

.x; u; ru/ C b.x;u/ D 0 on :

We assume a

and b are smooth in their arguments and

.x; u;p/jC.u/hpi; jr

.x; u;p/jC.u/:

We make the ellipticity hypothesis

.x; u;p/



 A.u/jj

;A.u/>0:

222 14. Nonlinear Elliptic Equations

6. Show that if u 2 H

./ \ L

./ solves (10.72), then u solves a PDE of the form

.x/@

u C@

.x; u/ C b.x;u/ D 0;

with

2 L

;

.x/



 Ajj

(Hint: Start with

.x; u;p/D a

.x; u;0/C

.x; u;p/p

;

.x; u;p/D

.x; u;sp/ds:/

7. Deduce that if u 2 H

./ \ L

./ solves (10.72), then u is H¨older continuous on

the interior of .

8. If  is a smooth, bounded region in R

and u 2 H

./ \L

./ satisﬁes (10.72)and

@

D ' 2 C

.@/, show that u is H¨older continuous on  and that ru 2 M

./,

for some q>n.

9. If u 2 C

./ satisﬁes (10.72), show that u

D @

u satisﬁes

.x; u; ru/@

C @



.x; u; ru/u



.x; u; ru/ C b

.x; u/u

C b

.x; u/ D 0:

Discuss obtaining estimates on u in C

1Cr

./, given estimates on u in C

./.

11. Direct methods in the calculus of variations

We study the existence of minima (or other stationary points) of functionals of the

form

(11.1) I.u/ D



F.x;u; ru/dV.x/;

on some set of functions, such as fu 2 B W u D g on @g,whereB is a suitable

Banach space of functions on , possibly taking values in R

,andg is a given

smooth function on @. We assume

 is a compact Riemannian manifold with

boundary and

(11.2) F W R

 .R

˝ T



/ ! R is continuous:

Let us begin with a fairly direct generalization of the hypotheses (1.3)–(1.8)

made in  1.Thus,let

(11.3) V Dfu 2 H

.; R

/ W u D g on @g:

11. Direct methods in the calculus of variations 223

For now, we assume that, for each x 2 ,

(11.4) F.x;; / W R

 .R

˝ T



/ ! R is convex;

where the domain has its natural linear structure. We also assume

(11.5) A

jj

 B

jujC

 F.x;u;/;

for some positive constants A

,and

(11.6) jF.x;u;/ F.x;v;/jC



ju vjCj  j



jjCjjC1



These hypotheses will be relaxed below.

Proposition 11.1. Assume  is connected, with nonempty boundary. Assume

I.u/<1 for some u 2 V . Under the hypotheses (11.2)–(11.6), I has a min-

imum on V .

Proof. As in the situation dealt with in Proposition 1.2, we see that I W V ! R is

Lipschitz continuous, bounded below, and convex. Thus, if ˛

D inf

I.u/,then

(11.7) K

Dfu 2 V W ˛

 I.u/  ˛

C "g

is, for each " 2 .0; 1, a nonempty, closed, convex subset of V . Hence K

weakly compact in H

.; R

/. Hence

">0

D K

¤;,andinf I.u/ is

assumed on K

We will state a rather general result whose proof is given by the argument

above.

Proposition 11.2. Let V be a closed, convex subset of a reﬂexive Banach space

W , and let ˆ W V ! R be a continuous map, satisfying:

inf

ˆ D ˛

2 .1; 1/;(11.8)

9 b>˛

such that ˆ

1



Œ˛

;b



is bounded in W;(11.9)

8 y 2 .˛

;b; ˆ

1



Œ˛

;y



is convex.(11.10)

Then there exists v 2 V such that ˆ.v/ D ˛

As above, the proof comes down to the observation that, for 0<"

b  ˛

is a nested family of subsets of W that are compact when W has

the weak topology. This result encompasses such generalizations of Proposition

11.1 as the following. Given p 2 .1; 1/; g 2 C

.@; R

/,let

(11.11) V Dfu 2 H

1;p

.; R

/ W u D g on @g:

224 14. Nonlinear Elliptic Equations

We continue to assume (11.4), but replace (11.5)and(11.6)by

(11.12) A

jj

 B

jujC

 F.x;u;/;

for some positive A

,and

(11.13) jF.x;u;/ F.x;v;/jC



ju vjCj  j



jjCjjC1



p1

Then we have the following:

Proposition 11.3. Assume  is connected, with nonempty boundary. Take p 2

.1; 1/, and assume I.u/<1 for some u 2 V . Under the hypotheses (11.2),

(11.4), and (11.11)–(11.13), I has a minumum on V .

It is useful to extend Propositions 11.1 and 11.3, replacing (11.4) by a hypoth-

esis of convexity only in the last set of variables.

Proposition 11.4. Make the hypotheses of Proposition 11.1, or more generally of

Proposition 11.3, but weaken (11.4) to the hypothesis that

(11.14) F.x;u; / W R

˝ T



 ! R is convex,

for each .x; u/ 2

  R

.ThenI has a minimum on V .

Proof. Let ˛

D inf

I.u/. The hypothesis (11.12) plus Poincar´e’s inequality

imply that ˛

> 1 and that

(11.15) B Dfu 2 V W I.u/  ˛

C 1g is bounded in H

1;p

.; R

Pick u

2 B so that I.u

/ ! ˛

. Passing to a subsequence, we can assume

(11.16) u

! u weakly in H

1;p

.; R

Hence u

! u strongly in L

.; R

/. We want to show that

(11.17) I.u/ D ˛

To this end, set

(11.18) ˆ.u;v/ D



F.x;u;v/dV.x/:

With v

Dru

,wehave

(11.19) ˆ.u

/ ! ˛

Also v

! v Dru weakly in L

.; R

˝ T



11. Direct methods in the calculus of variations 225

We can conclude that I.u/  ˛

, and hence (11.17) holds if we show that

(11.20) ˆ.u;v/ ˛

Now, by hypothesis (11.13)wehave

(11.21)

jˆ.u

/  ˆ.u;v

/jC



 uj



jC1



p1

dV.x/

 C

 uk

./

;

(11.22) ˆ.u;v

/ ! ˛

This time, by (11.5), (11.6), and (11.14) we have that, for each " 2 .0; 1,

(11.23) K

Dfw 2 L

.; R

˝ T



/ W ˆ.u;w/  ˛

C "g

is a closed, convex subset of L

.; R

˝ T



/. Hence K

is weakly compact,

provided it is nonempty. Furthermore, by (11.22), v

2 K

with "

! 0,sowe

have v 2 K

. This implies (11.20), so Proposition 11.4 is proved.

The following extension of Proposition 11.4 applies to certain constrained

minimization problems.

Proposition 11.5. Let p 2 .1; 1/, and let F.x;u;/ satisfy the hypotheses of

Proposition 11.4. Then, if S is any subset of V (given by (11.11)) that is closed in

the weak topology of H

1;p

.; R

/, it follows that I

has a minimum in S.

Proof. Let ˛

D inf

I.u/,andtakeu

2 S; I.u

/ ! ˛

.Since(11.15) holds,

we can take a subsequence u

! u weakly in H

1;p

.; R

/,sou 2 S.We

want to show that I.u/ D ˛

. Indeed, if we form ˆ.u;v/ as in (11.18), then the

argumentinvolving (11.19)–(11.23)continues to hold, and our assertion is proved.

For example, if X  R

is a closed subset, we could take

(11.24) S Dfu 2 V W u.x/ 2 X for a.e. x 2 g;

and Proposition 11.5 applies. As a speciﬁc example, X could be a compact Rie-

mannian manifold, isometrically imbedded in R

, and we could take p D 2;

F.x;u; ru/ Djruj

. The resulting minimum of I.u/ is a harmonic map of 

into X.Ifu W  ! X is a harmonic map, it satisﬁes the PDE

(11.25) u  .u/.ru; ru/ D 0;

226 14. Nonlinear Elliptic Equations

where .u/.ru; ru/ is a certain quadratic form in ru.See 2 of Chap. 15 for a

derivation.

A generalization of the notion of harmonic map arises in the study of “liquid

crystals.” One takes

(11.26) F.x;u; ru/ D a

jruj

.div u/

.u curl u/

ju curl uj

;

where the coefﬁcients a

are positive constants, and then one minimizes the func-

tional



F.x;u; ru/dV.x/over a set S of the form (11.24), with X DS

R

namely, over

(11.27) S Dfu 2 H

.; R

/ Wju.x/jD1 a.e. on ; u D g on @g:

In this case, F.x;u;/has the form

F.x;u;/ D

j;˛

j˛

.u/

j˛

j;˛

.u/  a

>0;

where each coefﬁcient b

j˛

.u/ is a polynomial of degree 2 in u. Clearly, this func-

tion is convex in . The function F.x;u;/does not satisfy (11.6); hence, in going

through the argument establishing Proposition 11.4, we would need to replace the

p D 2 case of (11.22)by

(11.28) jˆ.u

/  ˆ.u;v

/jC



 ujjv

dV .x/:

The following result covers integrands of the form (11.26), as well as many

others. It assumes a slightly bigger lower bound on F than the previous results,

but it greatly relaxes the hypotheses on how rapidly F can vary.

Theorem 11.6. Assume  is connected, with nonempty boundary. Take p 2

.1; 1/, and set

V Dfu 2 H

1;p

.; R

/ W u D g on @g:

Assume I.u/<1 for some u 2 V . Assume that F.x;u;/ is smooth in

its arguments and satisﬁes the convexity condition (11.14)in and the lower

bound

(11.29) A

jj

 F.x;u;/;

for some A

>0.ThenI has a minimum on V .

Also, if S is a subset of V that is closed in the weak topology of H

1;p

.; R

then I

has a minimum in S.

11. Direct methods in the calculus of variations 227

Proof. Clearly, ˛

D inf

I.u/  0. With B as in (11.15), pick u

2 B \ S so

that

(11.30) I.u

/ ! ˛

; u

! u weakly in H

1;p

.; R

Passing to a subsequence, we can assume u

! u a.e. on . We need to show

that

(11.31)



F.x;u; ru/dV  ˛

By Egorov’s theorem, we can pick measurable sets E



 E

C1

 in ,of

measure <2



, such that u

! u uniformly on  nE



. We can also arrange that

(11.32) ju.x/jCjru.x/jC  2



; for x 2  n E



Now, we have

(11.33)

nE



F.x;u; ru/dV D

nE



F.x;u

; ru

/dV

nE





F.x;u

; ru/  F.x;u

; ru



nE





F.x;u; ru/  F.x;u

; ru/



dV:

To estimate the second integral on the right side of (11.33), we use the convexity

hypothesis to write

(11.34) F.x;u

; ru/  F.x;u

; ru

/  D



F.x;u

; ru/  .ru ru

Now, for each ,

(11.35) D



F.x;u

; ru/ ! D



F.x;u; ru/; uniformly on  n E



;

while ru ru

! 0 weakly in L

.; R

/,so

(11.36) lim

j !1

nE





F.x;u

; ru/  F.x;u

; ru



dV D 0:

Estimating the last integral in (11.33) is easy, since

(11.37) F.x;u; ru/  F.x;u

; ru/ ! 0; uniformly on  n E

