Methods on Nonlinear Elliptic Equations (Aims Series on Differential Equations & Dynamical Systems)

Tf(o) =

Jn

f(x)O(x)dx.

It is easy to verify that is a continuous linear functional on D(Q) and is hence a distribution. For any distribution p, if there is an f E Lloc(dl) such that p(q5) =Tf(0), VO E D(,2), then we say that p is (or can be realized as) a locally summable function and identify it as f. An interesting example of a distribution that is not a locally summable function is the well-known Dirac delta function. Let x° be a point in Q. For any 0 E D(.fl), the delta function at x° can be defined as bz-M = OW).

Hence it is a distribution. However, one can show that such a delta function is not locally summable. It is not a function at all. This kind of "function" has

8

1 Introduction to Sobolev Spaces

been used widely and so successfully by physicists and engineers, who often simply view bxo as 8

(x)

J0, forx$xo

loo, forx=x.

Surprisingly, such a delta function is the derivative of some function in the following distributional sense. To explain this, let 12 = (-1,1), and let

f(x) - (0, forx<0

Sl 1, for x > 0.

Then, we have

f(x)cb'(x)dx = - f0 l O'(x)dx = 0(0) = bo(O) 1

Comparing this with the definition of weak derivatives, we may regard bo(x) as f'(x) in the sense of distributions.

1.2 Sobolev Spaces Given a function f E LP(2), we would like to solve the partial differential equation

du = f (x) in the sense of weak derivatives. Naturally, we would seek solutions u, such that du are in 11(12). More generally, we would start from the collections of all distributions whose second weak derivatives are in LP(12). In a variational approach, as we have seen in the introduction, to seek critical points of the functional

fn 1pul2dx- ifJ F(x,u)dx, a natural set of functions to start with is the collection of distributions whose first weak derivatives are in L2 (.2). More generally, we have:

Definition 1.2.1 The Sobolev space Wk,p(Q) (k > 0 and p > 1) is the collection of all distributions u on f2 such that for all multi-index a with jal < k, Dau can be realized as an LP function on Si. Furthermore, W',P(fl) is a Banach space with the norm 1/P Hul l

(1.1
f ID"u(x)Ipdx n

1.2 Sobolev Spaces

9

In the special case when p = 2, it is also a Hilbert space and we usually denote it by Hk(fl).

Definition 1.2.2 WW''(fl) is the closure of Co (fl) in Wk,P(Q). Roughly speaking, WW'P(fl) is the space of functions whose up to (k - 1)th order derivatives vanish on the boundary.

Example 1.2.1 Let fl = (-1,1). Consider the function f(x) = IxIO. For

1

0 < Q < 1, it is obviously not differentiable at the origin. However for any 1
To see this, we first calculate

fx,(x) =

Qxi xI2_1',

for x # 0,

and hence

Iof(x)I = IxIR

(ls)

.

Fix a small e > 0. Then for any 0 E D(Q), by integration by parts, we have

f

f.,(x)O(x)dx = - f

f (x) Ox. (x)dx +

f

fOvids,

is an inward-normal vector on 8B6(0). Now, under the condition that /3 > 1 - n/p, fx, is in LP(12) C L' (Q), and

where v = (vi, v2,

,

f q5vidSl < Cc--'+O-+O,

as e-0.

BB,,(O)

It follows that

f

lxlg¢'(x)dx

-fa

a2'$O(x)dx. I

Therefore, the weak first partial derivatives of IxI' are '3xi

Moreover, from (1.9) one can see that IvfI is in LP(f2) if and only if

> 1-n

10

1 Introduction to Sobolev Spaces

1.3 Approximation by Smooth Functions While working in Sobolev spaces, for instance, in proving inequalities, it may feel inconvenient and cumbersome to manage a weak derivative directly based on its definition. To get around this, we will show in this section that any function in a Sobolev space can be approached by a sequence of smooth functions. In other words, the smooth functions are dense in Sobolev spaces. Based on this, when deriving many useful properties of Sobolev spaces, we can just work on smooth functions and then take limits. At the end of the section, for more convenient application of the approximation results, we prove an Operator Completion Theorem and an Extension Theorem. Both theorems will be used frequently in the next few sections.

The idea in approximation is based on mollifiers . Let {ce1-r-

j(x) =

0

if1x(<1 if1xI>1.

One can verify that

7(x) E Co (Bi(0)) Choose the constant c, such that JRn,7(x)dx = 1.

For each e > 0, define

3E(X) = 0A E Obviously,

jJ(x)dx = 1, He > 0. R^

One can also verify that jE E Co (BE(0)), and lim

= J 0 for x # 0 oo for x = 0.

The above observations suggest that the limit of jj(x) may be viewed as a delta function, and from the well-known property of the delta function, we would naturally expect that for any continuous function f (x) and for any point x E 1l, (JEf)(x)

f jE(x - y)f(y)dy-+f(x),

as e-r0.

(1.10)

n

More generally, if f (x) is only in L"(fl), we would expect (1.10) to hold almost everywhere. We call j6(x) a mollifier and (JE f)(x) the mollification of f(x). We will show that for each e > 0, JE f is a C°° function, and as e-40,

1.3 Approximation by Smooth Functions

11

JEf-*f in Wk'P Actually, notice that (JEf)(x) = J

B

(s)nn

9E(x - y)f(y)dy,

and hence in order for J,f(x) to approximate f(x) well, we need B,(x) to be

completely contained in JZ to ensure that JB,(:)nn

j,(x - y)dy = 1

(an important property of the delta function). Equivalently, we need x to be in the interior of .fl. For this reason, we first prove a local approximation

theorem.

Theorem 1.3.1 (Local Approximation by Smooth Functions).

For any f E Wk'P(s7), JE f E C°°(R") and JEf i f in W`oP(Jl) as e- O. To extend this result to the entire .2, we will choose infinitely many open sets Oi, i = 1, 2, - , each of which has a positive distance to the boundary of B and whose union is the whole Q. Based on the above theorem, we are able to approximate a Wk'P(q) function on each Oi by a sequence of smooth functions. Combining this with a partition of unity, and a cut-off function if JZ is unbounded, we will then prove: - -

Theorem 1.3.2 (Global Approximation by Smooth Functions). For any f E Wc'P(fl), there exists a sequence of functions {fm} C C°°(0) f1Wk'P(,fl) such that fm -* f in Wk'P(Q) as m oo.

Theorem 1.3.3 (Global Approximation by Smooth Functions up to the Boundary).

Assume that Bl is bounded with C' boundary 80. Then for any f e Wk'P(fl), there exists a sequence of functions {f,,,} C C°°(fl) = C°°(0) n W',P(dl) such that fm -> f in Wk'P(JZ) as moo.

When .2 is the entire space R", the approximation by C°° or by Co functions are essentially the same. We have

Theorem 1.3.4 Wk'P(R") = Wk,,(R"). In other words, for any f E W',P(R"), there exists a sequence of functions {fm} C Co (R"), such that

fm 4f,

as m->oo;

in Wk'P(R").

12

1 Introduction to Sobolev Spaces

Proof of Theorem 1.3.1 We prove the theorem in three steps. In step 1, we show that JEf E C°°(R") and IIJcfIILP(ra) ! IIf IILP(a).

From the definition of JEf (x), we can see that it is well defined for all x E R'°, and it vanishes if x is of e distance away from Q. Here and in the following, for simplicity of argument, we extend f to be zero outside of Q.

In step 2, we prove that if f is in L1(Q),

(JEf)-if in Ltoc(Q). We first verify this for continuous functions and then approximate LP functions by continuous functions. In step 3, we reach the conclusion of the Theorem. For each f E W"P(Q)

and Ial < k, D' f is in LP(Q). Then from the result in Step 2, we have

JE(D"f)-rD"f in Lloc(r). Hence, what we need to verify is

D"(JEf)(x) - JE(D"f)(x). As the reader will notice, the arguments in the last two steps only work in any compact subset of ,(l. Step 1.

Let e; = (0, , 0) be the unit vector in the xi direction. , 0, 1, 0, Fix e > 0 and x E R. By the definition of JEf we have, for Ihi < e, -

(JEf)(x +hei) - (J
- JB2,(x)n.ft JE(x + hei - y) - 7,(x - y) f(y)dy h

h

(1.11)

Since as h-+0,

7E(x+hei-y)-?E(x-y) h

a?E(x-y) axi

uniformly for all y E B2E(x) fl Sl, we can pass the limit through the integral sign in (1.11) to obtain a(Jcf)(x)

axi Similarly, we have

=

r a7E(x - y)f(y)dy

n

axi

1.3 Approximation by Smooth Functions

13

D'(JFf)(x) = f DX'jE(x - y)f (y)dy. is infinitely differentiable, we conclude that JEf is also

Noticing that

infinitely differentiable.

Then we derive IIJEfIILP(n) <- IIfIILP(n).

(1.12)

By the Holder inequality, we have

I(J
<_

(n in

j°(x-y)j'(x-y)f(y)dyl

f njc(x-y)dy) \ fnj<(x-y)If(y)I°dy I

r

P

I

\fL(x)

j6(x-y)If(y)Ipdy).

It follows that

In

n

so

(JB'(X)

jE(x-y)lf(y)IPdy)dx

fR^If(y)IP(fR =

f

If(y)Pdy.

Notice that here we have extended f to be zero outside .fl. This verifies (1.12). Step 2.

We prove that for any compact subset K of Q, IIJEf - f IILP(K)#0,

as e-40.

We first show this for a continuous function f. By writing

(JJf)(x) - f(x) =

f

B (o)

jE(y)If(x - y) - f(x)]dy,

we have

I(JEf)(x)-f(x)I

<xEmaIx<EIf(x-y)-f(x)I JB.(o)jE(y)dy Kyl

<xEmax If(x-y)-f(x)I.

(1.13)

14

1 Introduction to Sobolev Spaces

Due to the continuity off and the compactness of K, the last term in the above inequality tends to zero uniformly as e-40. This verifies (1.13) for continuous functions.

For any fin LP(?), and given any 5 > 0, choose a a continuous function g, such that -

If - 91ILP(n) <

(1.14)

This can be derived from the well-known fact that any LP function can be approximated by a simple function of the form Ej=1 ajXj(x), where Xi is the characteristic function of some measurable set Aj; and each simple function can be approximated by a continuous function. For the continuous function g, (1.13) implies that for sufficiently small e, we have IIJE9 - 911LP(K) <

It follows from this and (1.14) that -JE911LP(K)+IIJE9-911LP(K)+119-fIILP(K) IIJEf - fIILP(K) 5 IIJEf <211f-911LP(K)+II49-911Lr(K)

2 3+3 =6. This proves (1.13). Step 3.

Now assume that f E Wk,p(Q). Then for any a with Ial < k, we have D' f E LP(.fl). We show that

D"(JEf)-3D"f

in L2(K).

(1.15)

By the result in Step 2, we have

JE(D"f)-3D"f

in LP(K).

What remains to verify is that for sufficiently small e,

D'(JEf)(x) = JE(D"f)(x), Vx E K. To see this, we fix a point x in K. Choose e < 1dist(K,8,2) so that any point y in the ball BE(x) is then in the interior of £7. Consequently,

1.3 Approximation by Smooth Functions

D"(JEf)(x) =

15

y)f(y)dy

Jn

Dxj.(x - y)f(y)dy BE(s)

_ (-1)101 fB (s) Dyj<(x-y)f(y)dy j6(x - y)Daf(y)dy = J j. (x - y)D&f (y)dy. Here we have used the fact that jE (x - y) and all its derivatives are supported in BE(x) C Si. This completes the proof of the Theorem 1.3.1.

The Proof of Theorem 1.3.2 Step 1. For a Bounded Region fl. Let

fli = {x E fl I dist(x, UQ) > l },

i=1,2,3,-

Write Oi = fli+3 \ fli+r. Choose some open set Oo CC fl so that fl = U0_0O;. Choose a smooth partition of unity {?7i}°=-o associated with the open sets {Oi}mo,

< 7)i(x) < 1, ni E CQ (Oi)

E -o qi(x) = 1, x E Si. 00

Given any function f E Wk,p(Q), obviously nif E Wk-p(Q) and supp(nif) C Oi.

Fix a 6 > 0. Choose ei > 0 so small that fi

JE; (ni f) satisfies

11Ifi-11ifllWk.a(n) 2+Tiri-01 1' i = 1, 2, . .. l suPPfi C (fli+a \ fli),

1

(1.16) .

Set g = E°Oo fi. Then g E C°°(fl) because each fi is in C°°(0), and for each open set 0 cc fl, there are at most finitely many nonzero terms in the sum. To see that g E Wk,p(fl), we write 00

g - f - E(A - nif)i=0

It follows from (1.16) that

16

1 Introduction to Sobolev Spaces

m

0o

1

= S.

2i+1

From here we can see that the series E2'=0(fi -rti f) converges in W k'P (.f2) (see

Exercise 1.3.1 below); hence, (g - f) E W'"P(Q) and therefore g E Wk,P(fl). Moreover, from the above inequality, we have 119 - f llwk.P(o) < S.

Since b > 0 is any number, we complete Step 1.

Remark 1.3.1 In the above argument, neither E fi nor > r)i f converge, but their difference converges. Although each partial sum of E fi is in Co ([2), the infinite series E fi does not converge to g in Wk,P(fl). Then how did we prove that g E Wk=P(Q)? We showed that the difference (g- f) is in Wk,P(12).

Exercise 1.3.1

.

Let X be a Banach space and vi E X. Show that the series E°° vi converges in X if E°° IIvill < oo. Hint: Show that the partial sum is a Cauchy sequence. Step 2. For Unbounded Region S?.

Given any S > 0, since f E Wk-P(d2), there exist R > 0, such that Ilfllwk.P(o\Ba_3(o)) <- S.

(1.17)

Choose a cut-off function rp E C°° (R") satisfying (x)

1, x E BR-2(0), 0,x ER \BR(0); and IDa0(x)I < 1 Vx E R^,dlal < k.

Then by (1.17), it is easy to verify that there exists a constant C, such that I10f - f llwk.P(n) < CS.

(1.18)

Now in the bounded domain ,f2 fl BR(0), by the argument in Step 1, there is a function g E C' (S? fl BR(0)), such that 119 - f Ilwk.P(nnBr,(o)) <6-

Obviously, the function ¢g is in C°°($2), and by (1.18) and (1.19), IIog - Ill

II09 -Of ll wk,P(o) + Ilmf - f IIwk.P(n) < C4119 - f llwk,P(onBR(o)) + CS < (Cl + C)S.

(1.19)

1.3 Approximation by Smooth Functions

17

This completes the proof of the Theorem 1.3.2.

The Proof of Theorem 1.3.3 We will continue to use mollifiers to approximate a function. As compared

to the proofs of the previous theorem, the main difficulty here is that for a point on the boundary, there is no room to mollify. To circumvent this, we cover ,fl with finitely many open sets, and on each set that covers the boundary

layer, we will translate the function a little bit inward so that there is room to mollify within ,fl. Then we will again use the partition of unity to complete the proof.

Step 1. Approximating in a Small Set Covering 8P.

Let x° be any point on 80. Since 80 is C1, we can make a Cl change of coordinates locally, so that in the new coordinates system (x1, can express, for a sufficiently small r > 0, Br(x°) n n = {x E Br(x°)I xn >

O(xlr..

,

we

. x.°-1)}

with some C1 function 0. Set

D = n n Br/2(x°). Shift every point x E D in x. direction ac units, define x` = x + aee,,. and

DE={x`IxED}. This D` is obtained by shifting D toward the inside of fl by ac units. Choose a sufficiently large, so that the ball B°(x) lies in fl n Br(x°) for all x E D` and for all small e > 0. There is now room to mollify a given Wk,P function f on D6 within .fl. More precisely, we first translate f a distance e in the x direction to become f`(x) = f(xC), then mollify it. We claim that

J fC-+f

in Wk,P(D).

Actually, for any multi-index lal < k, we have IID°(JEfE) - D°fIILP(D) <_ IID°(JEf`) -

D°f`IILP(D) + IID°fC - D°f IILP(D).

A similar argument as in the proof of Theorem 1.3.1 implies that the first term on the right hand side goes to zero as e- O, while the second term also vanishes in the process due to the continuity of the translation in the LP norm.

18

1 Introduction to Sobolev Spaces

Step 2. Applying the Partition of Unity. Since 8Q is compact, we can find finitely many such sets D, call them Di, i = 1, 2, - , N. the union of which covers 8111. Given 6 > 0, from the argument in Step 1, for each Di, there exists gi E C°O(Di), such that - -

(1.20)

1191 - fllwk,P(D,) <_ J.

Choose an open set Do CC S2 such that S1 C U;v 0Di, and select a function go E C`°(DO) such that (1.21)

1190 - f II Wk.P(D0) <_ 6.

Let {ni} be a smooth partition of unity subordinated to the open sets

{Di};'_o. Define N

9=Eni9i i=0

Then obviously g c CO°(Q), and f = >1 o nif Similar to the proof of Theorem 1.3.1, it follows from (1.20) and (1.21) that, for each lal < k N

IID'g-DafIILP(n)

>IID°(nigi)-Da(nif)IILP(D.) i=0 N

G C> 119i - f II W'.P(Di) = C(N + 1)6. i=O

This completes the proof of the Theorem 1.3.3.

The Proof of Theorem 1.3.4 Let ¢(r) be a Co cut-off function such that

for0
1,

Or) =

between 0 and 1

,

for 1 < r < 2;

forr>2.

1 0,

Then by a direct computation, one can show that 110(

RI)f(x) -f(x)IIwk.P(Rn)-+O,

as R-soo.

(1.22)

Thus, there exists a sequence of numbers {R,,,} with R,,,-*oo, such that for 9m(x)

we have

0(

In 1V

W,

1.3 Approximation by Smooth Functions 119m-fIIWk,P(Rn) <_

1.

19 (1.23)

On the other hand, from the Approximation Theorem 1.3.3, for each fixed in,

as e-40,

J,(gm)-*gm,

in Wk,p(R")

Hence there exist cm, such that for fm we have fm - 9miI Wk,p(Rn)

7n .

(1.24)

Obviously, each function fm is in Cp (Rn), and by (1.23) and (1.24), Ilfm

fIIWk,P(Rn) <

Ilfm-gmIIWk,P(Rn)+Ilsm-flIWk.P(RT) <

-->O, as M-400.

This completes the proof of Theorem 1.3.4.

We have now proved all four Approximation Theorems, which show that smooth functions are dense in Sobolev spaces Wk,p. In other words, Wk,' is the completion of Ck under the norm II' IIWk,r. Later, particularly in the next three sections, when we derive various inequalities in Sobolev spaces, we can first work on smooth functions and then extend them to whole Sobolev spaces. In order to make such extensions more convenient (i.e., avoid going through the approximation process in each particular space), we prove the following Operator Completion Theorem in general Banach spaces.

Theorem 1.3.5 Let D be a dense linear subspace of a nonmed space X. Let Y be a Banach space. Assume

T:D'-4Y is a bounded linear map. Then there exists an extension T of T from D to the whole space X, such that T is a bounded linear operator from X to Y, IITII = IITII

and Tx = Tx Vx E D.

Proof. Given any element xp E X, since D is dense in X, there exists a sequence {x1} E D, such that xt->xo,

It follows that

as 1-+00.

20

1 Introduction to Sobolev Spaces as i, j-+oo.

IITxi - Txi II <_ IITII Ilxi - xi IH0,

This implies that {Txi} is a Cauchy sequence in Y. Since Y is a Banach space, {Txi} converges to some element yo in Y. Let

Txo = yo.

(1.25)

To see that (1.25) is well defined, suppose there is another sequence {x'} that converges to xo in X and Tx;->yl in Y. Then Ilyi

you = urn IITx( -TxiII S

nil = 0.

Now, for x c D, define Tx = Tx; and for other x E X, define Tx by (1.25). Obviously, T is a linear operator. Moreover

IITxoII = lim ITx:Ij < ilim IITIIIIxiIl = IITIIIIxoll. Hence T is also bounded. This completes the proof.

1-1

As a direct application of this Operator Completion Theorem, we prove the following Extension Theorem. Theorem 1.3.6 Assume that S2 is bounded and OS? is G. Then for any open W0.P(0) set 0 D (2, there exists a bounded linear operator E : Wk,P(S2) --> such that Eu = it a. e. in Q.

Proof. To define the extension operator E, we first work on functions it E Ck(Q). Then we can apply the Operator Completion Theorem to extend E to Wk.P(S2), since C'(.(2) is dense in Wk,P(0) (see Theorem 1.3.3). From this density, it is easy to show that, for it E Wk,P(0),

E(u)(x) = u(x) almost everywhere on d2 based on

E(u)(x) =u(x) on ,2 Vu E Ck(S2). Now we prove the theorem for u E Ck(Q). We define E(u) in the following two steps. Step 1. The special case when S2 is a half ball

Br(0):=Ix

0}.

Assume a E Ck(B, (0)). We extend it to be a Ck function on the whole ball Br(0). Define

1.3 Approximation by Smooth Functions

21

xn > 0 u(x,xn)_ u(x',xn), E 0aiu(x',-Aixn),xn<0. To guarantee that all partial derivatives asu(x', xn) up to order k are contin-

8n x

uous across the hyper plane xn = 0, we first pick A1, such that

0
Y, a1A =1,

7=0,1,...

k

i=0

, ak. Since the coefficient determinant det(A) is not to determine ao, a1, zero, the system has a unique solution. One can easily verify that, for such Ai and ai, the extended function v. is in Ck(B,.(0)). Step 2. Reduce the general domains to the case of the half ball covered in Step 1 via domain transformations and a partition of unity. Let 0 be an open set containing 11. Given any x° E 8f1, there exists a neighborhood U of x° and a Ck transformation 0 : U-rRn which satisfies that 4'(x°) = 0, q5(U fl 00) lies on the hyper plane xn = 0, and ¢(U n fl) is

contained in R+. Then there exists an r° > 0, such that Bro(0) CC 4'(U), Dyo :_ ¢-1(Bro(0)) is an open set containing x°, and 95 E Ck(Dyo). Choose r° sufficiently small so that Dz0 C O. All such Dyo and U forms an open covering of U, hence there exists a finite sub-covering

Do := U, Dl, ..., D. and a corresponding partition of unity ,rlm,

770,771,...

such that

77iEC0 (Di) i=0,1,--- ,m and

E17i(x) = 1 V X E Q. i=o

Let 4'i be the mapping associated with Di as described above, and let iii(y) = u(¢ 1(y) be the function defined on the half ball Br (0) = Wi (Di n fl)

,

i = 1, 2, - - m. -

From Step 1, each ui(y) can be extended as a Ck function ui(y) onto the whole ball Br, (0).

1 Introduction to Sobolev Spaces

22

We can now define the extension of it from .f2 to its neighborhood 0 as E(u)

_

rji(x)ui(6i(x)) + rfou(x).

i.1

Obviously, E(u) E Co (O), and

«llullwk,ylal' Moreover, for any x E .(2, we have

E(u)(x)

- i=1 1Ji(x)tii(01(x)) +'7o(x)u(x) _i=1Eij (x)u'i(0i(x)) +no(x)u(x) m

>n

E??i(x)ti'(q

- i=1

1(0i(x))) +7]o(x)u(x) = Y:gi(x)u(x) +77o(x)u(x) i=1

(t) 71'(x) = n(x). This completes the proof of the Extension Theorem.

Remark 1.3.2 Notice that the Extension Theorem actually implies an improved version of the Approximation Theorem under stronger assumption on 80 (be Ck instead of C1). From the Extension Theorem, one can immediately derive:

Corollary 1.3.1 Assume that .f2 is bounded and 80 is Ck. Let 0 be an open set containing D. Let 0E :_ {x E Rn I dist(x, 0) < e}. Then the linear operator JE(E(u)) : Wk,P(n)-4Co (O<) is bounded, and for each it E W?O, (Q), we have II JE(E(u)) - uIIwk,Pliz>-*O ,

as - 0.

Here, as compared to the previous approximation theorems, the improvement is that one can write out the explicit form of the approximation.

1.4 Sobolev Embeddings When we seek weak solutions of partial differential equations, we start with functions in a Sobolev space W°,P. Naturally, we would like to know whether

1.4 Sobolev Embeddings

23

or not the functions in this space also automatically belong to some other spaces. The following theorem answers the question and at the same time provides inequalities among the relevant norms.

Theorem 1.4.1 (General Sobolev Inequalities). Assume .(2 is bounded and has a Cl boundary. Let it E Wk,P(S2). (i) If k < p, then it E L7(f2) with 4 = n - f-a and there exists a constant C such that IIuIIL9(a) <-

(ii) If k > p, then it E

Ck-[pl

GIIuIIWi,P(n)

r'7(Sl), and there exists a constant C, such

that 1lullck_Iyi_1.7(71)
where

ifP is not an integer any t[plpositive number < 1, if v is an integer . P,

Here, [b] is the integer part of the number b.

The proof of the Theorem is based upon several simpler theorems. We first consider the functions in W1"P(S2). From the definition, apparently these functions belong to LQ(S2) for 1 < q < p. Naturally, one would expect more,

and what is more meaningful is to find out how large this q can be. And to control the LQ norm for larger q by W1'P norm-the norm of the derivatives P

IIDUIILP(n) = (J IDulPdx)

-would suppose to be more useful in practice. For simplicity, we start with the smooth functions with compact supports

in R. We would like to know for what value of q can we establish the inequality IIullLq(Rn) <-CIIDullLP(Rn),

(1.26)

with constant C independent of it e Co (R"). Now suppose (1.26) holds. Then it must also be true for the re-scaled function of u: U,\(X) = u(Ax),

that is IluAIILq(Rn) < CIIDuAIILP(Rn).

By substitution, we have

(1.27)

24

1 Introduction to Sobolev Spaces

Iu(y)I9dy, and

flua(x)IQdx =

R^

f

P

IDu),lPdx =

J RR'

T T_ R^

IDu(y)IPdy.

It follows from (1.27) that IItIILq(Rn) <_ Car P+4IIDuIILP(Rn).

Therefore, for C to be independent of u, it is necessary that the power of A here be zero, that is, np

4=n-p It turns out that this condition is also sufficient, as will be stated in the following theorem. Here for q to be positive, we must require p < n.

Theorem 1.4.2 (Gagliardo-Nirenberg-Sobolev Inequality).

Assume that 1 < p < n. Then there exists a constant C = C(p, n), such that IIUIILP*(Rn) <_ CIIDuIIL=(Rn),

UE

Co(R"),

(1.28)

where p* =

Since any function in W1'P(Rn) can be approached by a sequence of functions in Co(R"), we derive immediately:

Corollary 1.4.1 Inequality (1.28) holds for all functions u in W',P(Rn). For functions in W',P(fl), we can extend them to be W',P(R") functions by the Extension Theorem and arrive at:

Theorem 1.4.3 Assume that I? is a bounded, open subset of R" with Cr boundary. Suppose that 1 < p < n and u E Wr°P(I1). Then u is in LP*(Q) and there exists a constant C = C(p, it, 0), such that IIUIILP*(Q) <_ CIIuIIw=,P(n)

(1.29)

In the limiting case as p-->n, p* := isioc. Then one may suspect that

u c- L°° when p = n. Unfortunately, this it = 1, from

u(x) = we derive immediately that

true only in dimension one. For

J y u'(x)dx,

1.4 Sobolev Embeddings Iu(x)I <_

25

Iv'(x)Idx.

J

However, for it > 1, it is false. One counter example is u = loglog(1 + I ) on I? = B1(O). It belongs to W',n(f2), but not to L°°(f2). This is a delicate situation, and we will deal with it later. Naturally, for p > n, one would expect W1'' to embed into better spaces. To get some rough idea what these spaces might be, let us first consider the simplest case when it = 1 and p > 1. Obviously, for any x, y E R1 with x < y, I

I

we have

u(y) - u(x) = J v u'(t) dt, x

and consequently, by the Holder inequality, bu(y) - u(x)I < rs iv,'(t)ldt < x

(

V

u'(t)Ipdt I p /

x

1

\Jx

dt I

It follows that Iu(y) - u(x) I

Iy-xI1 P

< (f

Iu'(t)IPdt I 00

Taking the supremum over all pairs x, y in R1, the left hand side of the above

inequality is the norm in the Holder space C°'7(Rr) with y = 1 - p. This is indeed true in general, and we have:

Theorem 1.4.4 (Morrey's Inequality). Assume it < p < oo. Then there exists a constant C = C(p, n) such that IIUIICu.7(R^) < CIIuIIWi.P(Rn),

u E C'(R")

where y = 1 -

To establish an inequality like (1.28), we follow two basic principles. First, we consider the special case where ,f2 = Rn. Instead of dealing with functions in Wk,p, we reduce the proof to functions with enough smoothness (which is just Theorem 1.4.2).

Secondly, we deal with general domains by extending functions u E Whp(.Q) to Whp(R") via the Extension Theorem. Here, one sees that Theorem 1.4.2 is the `key,' and inequality (1.28) provides the foundation for the proof of the Sobolev embedding. Often, the proof of (1.28) is called a `hard analysis', and the steps leading from (1.28) to (1.29) are called `soft analysis'. In the following, we will show the `soft' parts first and then the `hard' ones. First, we will assume that Theorem 1.4.2 is true

1 Introduction to Sobolev Spaces

26

and derive Corollary 1.4.1 and Theorem 1.4.3. Then, we will prove Theorem 1.4.2.

The Proof of Corollary 1.4.1. Given any function u E Whp(R°), by the Approximation Theorem, there exists a sequence {uk} c Co (R") such that IIu-ukIIwi.P(Rn) -> 0 ask -> oc. Applying Theorem 1.4.2, we obtain IIu=-njIILP*(Rn)SCIIRi-ujIIw=.P(Rn)->0,

asi,,j

oo.

Thus {uk} also converges to u in Lp*(R"). Consequently, we arrive at IIuIILP'(R') =limIIUkIILP'(Rt) 5limCllukIIW ,P(Rr)

This completes the proof of the Corollary.

The Proof of Theorem 1.4.3. Now for functions in W1'p(.Q), to apply inequality (1.27), we first extend them to be functions with compact supports in R. More precisely, let 0 be an open set that covers I?. By the Extension Theorem (Theorem 1.3.6), for every u in W1'P(.Q), there exists a function u in Wd"3(O), such that

u = u, almost everywhere in [2; moreover, there exists a constant Cl = Cr (p, n, .f2, O), such that IIRIIWI,P(O) 5 C]IIu1IW1.P(n).

(1.30)

Now we can apply the Gagliardo-Nirenberg-Sobolev inequality to u to derive IIkIILP'(U) < IIRIILP'(0) 5 CIIuIIw1.P(o) 5 CC,IIUIIWI.P(n).

This completes the proof of the Theorem.

The Proof of Theorem 1.4.2. We first establish the inequality for p = 1, i.e., we prove

\JRn

LRn

IDulda.

(1.31)

Then we will apply (1.31) to Iul' for a properly chosen y > 1 to extend the inequality to the case when p > 1. We need

1.4 Sobolev Embeddings

27

Lemma 1.4.1 (General Holder Inequality). Assume that u; E LP' (Q) for i = 1, 2, ... , m. and

1

1

1

P1

P2

Pm

Then

Ja

Iu1u2... umldx <

11

IJ

(1.32)

/

n

The proof can be obtained by applying induction to the usual Holder inequality for two functions. Now we are ready to prove the Theorem. Step 1. The case p = I.

To better illustrate the idea, we first derive inequality (1.31) for n = 2; that is, we prove

L Iu(x)I2dx < (1 IDuIdx)2.

(1.33)

Since u has a compact support, we have (/r=, 8u

u(x) -

J-cot `ylrx2)dyl

It follows that In(x)I <_

J

I Du(y1,x2)Idy1.

Similarly, we have In(x)I <_

l

IDu(x1,y2)Idy2.

The above two inequalities together imply that

Iu(x)I2<

IDu(yl,x2)Idyl

IDu(xl,y2)Idy2

Now, integrating both sides of the above inequality with respect to x1 and x2 from -co to oo, we arrive at (1.33). Then we deal with the general situation when n > 2. We write

u(x) = Consequently,

8u (xl...

a

xi-1, yi, xi+l, ... , xn)dyi, i = 1, 2, .

n.

1 Introduction to Sobolev Spaces

28


Iu(x)I

:yi:"-

00

)Idyi, i=1,2,...n.

And it follows that

Iv(x)I=, < jj

I

(f

yi,

IDu(xi,..

i=1

xn)ldyi

Integrating both sides with respect to x1 and applying the general Holder

inequality (1.32), we obtain

f

w

m

(J

\

f00m

r

Duldyl I /

\

IDuldyi

Q1( J

Inl=,dxi < (f IDuldy1)

(U2

- dx1

r: IDuldxldyi

.J m

-i

)

Then integrate the above inequality with respect to x2. We have 00

f r

IuIldxldx2 <_

00

f oc

IDuldxldy2

n

T_1-1

\C IDuI dy1/

x

r ,!

11 i=3

n

(1: -oo

IDuldxldyil

/

1 dx2, J

Again, applying the General Holder Inequality, we arrive at

f J- nl='dxidx2 00

00

(f \J_

00

00

C

x i=3

00

I Dul dytdx2)

:CC

JDudxidx2dyi

I

l

Duldxidy2)

n

Continuing this way by integrating with respect to x3,

--

, xn_1, we deduce

IuI='dxi...dxn_i

<

r rte... ...1

x I\\J

x

\

Duldyidx2...dxn-i)

co

... I oo

I Dul dxldy2dx3 ... dxn-1 00

l

Duldxi ... dxn_2dyn_il

f

(r IDuldx) -' \IR'

1.4 Sobolev Embeddings

29

Finally, integrating both sides with respect to x and applying the general Holder inequality, we obtain

IuI-1 dx <

(L IDuldx)

This verifies (1.31).

Exercise 1.4.1 Write your own proof with all details for the cases n = 3 and n = 4. Step 2. The Case p > 1.

Applying (1.31) to the function luI with y > 1 to be chosen later, and by the Holder inequality, we have ^_,

(fR^uI

dx)

<

f^ ID(Gur")Idx = y f IuPry-1IDuJdx R R^ (7-1)(,.-1)


J U.

Jul ^-1 dx

/

//

J R^

IDuI

Y+^-1 dx

It follows that

(

Jul

R^

^

Now choose y, so that

dx)


I Dul ,++.-1 dx) Rn

= p, that is yn y+n-1 p(n - 1)

n-p we obtain

n-v

rf

l

r

1

\JR^ l uI Ct Du&Pdx) dx) ^n This completes the proof of the Theorem. I

The Proof of Theorem 1.4.4 (Morrey's inequality). We will establish two inequalities sup Jul <_ Cllullw1,r(R^),

(1.34)

R^ and

sup lu(x) -u(y)I < CIIDuIILp(R.,).

x#Y Ix - yll-'i/P

-

(1.35)

1 Introduction to Sobolev Spaces

30

Both of them can be derived from the following °B,(=)IyDx(I?I1dy,

n(x)Idy

IBr(x)IJBr(x)lu(y)

(1.36)

where IB,(x)I is the volume of B,(x). We will carry the proof out in three steps. In Step 1, we prove (1.36), and in Step 2 and Step 3, we verify (1.34) and (1.35), respectively.

Step 1. We start from

u(y) - u(x) =

Jo

where

x

Y_ y

w

f

1 dtu(x + t(y - x))dt =

,

J0

3

Du(x +7w) . wdr,

s = Ix - yl, and hence y = x + sw.

Ix

It follows that

Iu(x+sw) - u(x)I < r8 IDu(x+rw)Idr. 0

Integrating both sides with respect to w on the unit sphere 8B1(0), then converting the integral on the right hand side from polar to rectangular coordinates,, we obtain

Jas1(o)

Iu(x+sw)-u(x)Ido< rs r

IDu(x+Tw)ldodr

as1(o)

o /fs

- Jo

OBI (0)

f

JB (_)

IDu(x+rw)ITn-jdodr

IxIDu(z)I - zI" r dz.

Multiplying both sides by s"-1, integrating with respect to s from 0 to r, and taking into account that the integrand on the right hand side is non-negative, we arrive at

sr(s) Iu(y)

u(x)Idy

n Js,(s)

yu )Idy.

xD

I

This verifies (1.36).

Step 2. For each fixed x E R", we have Iu(x)I =

IB1(x)I.1a1(-)

IBr(x)I

=I1+I2.

u(x)Idy I

L1(=) Iu(x) -u(y)Idy+

IBr(x)I JB1(=)

Iu(y)Idy (1.37)

1.4 Sobolev Embeddings

31

By (1.36) and the Holder inequality, we deduce ]Du(y)

Il
CIBi(x) x - yn-1

dy dy

IDulpdy)1/p

f


(J

dy) PP'

)

1/p

< C1 (fR^ IDniPdy)

(1.38)

Here we have used the condition that p > n, so that

(s-1 P

p-1

< u and hence

the integral

dy

al(x) Ix - yI

-1 p

is finite.

Also it is obvious that (1.39)

72 < CIIUIILP(Rn).

Now (1.34) is an immediate consequence of (1.37), (1.38), and (1.39).

Step 3. For any pair of fixed points x and y in R", let r = Ix - yI and D = Br(x) fl Br(y). Then

lu(x)-u(y)I

fDIu(x)-u(z)ldz+IDI4Iu(z)-u(y)ldz.

D

(1.40)

Again by (1.36), we have

IDI

fD lu(x) - u(z)ldz < IBCx)I f

.(=)

/

lu(x) -u(z)ldz \ 1/p

f f IDuiPdzI =

dz

(Lr(x) (1.41)

Cr1-n/PIIDuMILP(Rn).

Similarly, 1

IDI

o lu(z) - u(y)ldz < Cr1-n/PIIDuIILP(Rn).

Now (1.40), (1.41), and (1.42) yield

Iu(x) - u(y)l <- CIx -

yl1-n/PIIDuiiLP(R^)

.

This implies (1.35) and thus completes the proof of the Theorem.

(1.42)

32

1 Introduction to Sobolev Spaces

The Proof of Theorem 1.4.1 (the General Sobolev Inequality). (i) Assume that u E Wk,'(fl) with k < -'. We want to show that u E P

Lq(fl) and IIUIIL9(n) <_ CII

(1.43)

ullwk,P(a)

with 1 = v - nkk. This can be done by applying Theorem 1.4.3 successively on

the integer k. Again denote p* = p, then P. = n n For Ial < k-1, D°u E W1'P(Q). By Theorem 1.4.3, we have D°u E LP'(Q), thus u E Wk-"P* (.2) and IIUIIWk-1.P'(Q) < C1IIullWk,P(n).

((1), we have u E Wk-2''" (fl) and

Applying Theorem 1.4.3 again to Wk-1

IIUIIwk-?'P"(n) <_ C2IInllWk-1.P'(n) <- C2C1IIullWk,P(n),

where p" = 'P* n_p., or 1

1

1

1

1

1

2

n

p

n

1

Continuing this way k times, we arrive at (1.43).

(ii) Now assume that k > n. Recall that in the Morrey's inequality P

IIuIIcan(n) <_ CIIuIlw1.P(n),

(1.44)

we require that p > n. However, in our situation, this condition is not necessarily met. To remedy this, we can use the result in the previous step. We can first decrease the order of differentiation to increase the power of summability. More precisely, we will try to find a smallest integer in, such that Wk_-.q(fl),

Wk.P(t2) . with q > n. That is, we want

1

np

n - rap and equivalently,

>n,

n m>--1. P

Obviously, the smallest such integer in is

m = [P]

1.5 Compact Embedding

33

For this choice of m, we can apply Morrey's inequality (1.44) to D°u, with

any lal < k- m- 1 to obtain IID'ulloo,Y(o)
Or equivalently, Ilull Ck-`n1-l.w(Q) < CIIuIIWk,P(n).

Here, when

n p

is not an integer, Y=1--n + n q p

While

n p

n P

n

is an integer, we have m = P, and in this case, q can be any number

> n, which implies that y can be any positive number < 1. This completes the proof of the Theorem.

1.5 Compact Embedding In the previous section, we proved that W1'P(Q) is embedded into LP (,fl) with p* = Obviously, for q < p', the embedding of W1-P(11) into Lq(.f1) is still true, if the region ,fl is bounded. Actually, due to the strict inequality on the exponent, one can expect more, as stated below. Theorem 1.5.1 (Rellich-Kondrachov Compact Embedding). Assume that .fl is a bounded open subset in R' with C1 boundary 811. Suppose 1 < p < n. Then for each 1 < q < p*, W1-P(11) is compactly embedded into Lq(11):

W1.P(11) yy Lq(f2), in the sense that

i) there is a constant C, such that IIuIILq(n) <-

C1IuI1WI.P(n)

,

V U E W1'P(11);

(1.45)

and

ii) every bounded sequence in W1'P(Q) possesses a convergent subsequence in D(11).

Proof. The first part-inequality (1.45)- is included in the general Embedding Theorem. What we need to show is that if {uk} is a bounded sequence in W"P(Q), then it possesses a convergent subsequence in Lq(12). This can be derived immediately from the following:

34

1 Introduction to Sobolev Spaces

Lemma 1.5.1 Every bounded sequence in W1''(fl) possesses a convergent subsequence in Lt(fl). We postpone the proof of the Lemma for a moment and see how it implies the Theorem. In fact, assume that {uk} is a bounded sequence in W',P(.fl). Then there exists a subsequence (still denoted by {uk}) which converges weakly to an element u, in Wr"P(Q). By the Sobolev embedding, {uk} is bounded in LP'(dl).

On the other hand, it is also bounded in Wl-l(,fl), since p > 1 and 12 is bounded. Now, by Lemma 1.5.1, there is a subsequence (still denoted by {uk}) that converges strongly to uo in L1(Q). Applying the Holder inequality link - UOIILc(n) <- IIuk - noliL=(n) Ilnk - UoII LPB(n)+

we conclude immediately that {uk} converges strongly to uo in LQ(fl). This proves the Theorem. We now come back to prove the Lemma. Let {uk} be a bounded sequence in W1r(0). We will show the strong convergence of this sequence in three steps with the help of a family of mollifiers uk(x) = f JE(Y)uk(x - y)dy

n

First, we show that Uk- Uk in L'(tl) as c-*O, uniformly in k.

(1.46)

Then, for each fixed c > 0, we prove that there is a subsequence of {uk} which converges uniformly

.

(1.47)

Finally, corresponding to the above convergent sequence {4}, we extract diagonally a subsequence of {uk} which converges strongly in L1(.fl). Based on the Extension Theorem, we may assume, without loss of generality, that 17 = R', the sequence of functions {uk} all have compact support

in a bounded open set G C R", and IIwkiiW=1(G)
.

Since every Writ function can be approached by a sequence of smooth

functions, we may also assume that each uk is smooth. Step 1. From the property of mollifiers, we have

1.5 Compact Embedding

4 (x) - Uk (x)

35

? (y) Iuk (x - Ey) - 'uk (x)I dy i(Y)1

fR1 (0)

=

f

1

dtuk(x - Ety)dtdy Ifr1

eJR,(0)i(y)f Duk(x-ety)dtydy. Integrating with respect to x and changing the order of integration, we

obtain 1

Iuk -ukIIL(G)

E fa,(0)i(y)f fGIDuk(x - Ety)Idxdtdy e

fG IDuk(z)Idz <

(1.48)

It follows that

Iluk - ukIIL=(G)-)0, as c-40, uniformly ink.

(1.49)

Step 2.Nowfix ane>0.Then

f

a

j`(x-y)Iuk(y)Idy

(y)

IhcIILw(Rn)IIukIIL1(G) <

E

<00.

(1.50)

Similarly, IDuk(x)I <

E n+l

< Co'

(1.51)

(1.50) and (1.51) imply that, for each fixed c > 0, the sequence {uk} is uniformly bounded and equi-continuous. Therefore, by the Arzela-Ascoli Theorem (see the Appendix), it possesses a subsequence (still denoted by {uk}) which converges uniformly on G, in particular lim IIuj - uiIILI(G) = 0.

j,i- 00

(1.52)

,1/k, - - successively, and denote Step 3. Now choose e to be 1,1/2,1/3, the corresponding subsequence that satisfies (1.52) by 1/k 1/k 1/k v,kl ' v'k2 1 I'k3

36

1 Introduction to Sobolev Spaces

for k = 1.2, 3,

.

Pick the diagonal subsequence from the above:

{u;!° C {uk}. Then select the corresponding subsequence from {uk}: U11,U22,U33,+

'

This is our desired subsequence, because as i, jioo,

luii-ujjIILI(o) 5 luizui 'IIL=(G)+Ilui -Ujj'IIL (G)+IIu7T-UjjIIL (G)-r0, due to the fact that each norm on the right hand side -r0. This completes the proof of the Lemma and hence the Theorem.

1.6 Other Basic Inequalities 1.6.1 Poincare's Inequality For functions that vanish on the boundary, we have

Theorem 1.6.1 (Poincare's Inequality I). Assume .fl is bounded. Suppose u E WJ'p(Q) for some 1 < p< oo. Then IIUIILP(n) <_ CIIDuIILP(n).

(1.53)

Remark 1.6.1 i) Now based on this inequality, one can take an equivalent norm of Wo'1'(S7) as IIuIIwo,P(.a) = IIDuIILP(np

ii) Just for this theorem, one can easily prove it by using the Sobolev inequality IIuIILP <_ IoUIIL4 withp= a and the Holder inequality. However, the one we present in the following is a unified proof that works for both this and the next theorem. Proof. For convenience, we abbreviate IIUIILP(n) as IIuIIp. Suppose inequality (1.53) does not hold, then there exists a sequence {uk} C Wo'p(Q), such that

IIDukIIp = 1, while llukllp+oo, as k-+oo. Since Co (Q) is dense in Wo'p(Q), we may assume that {uk} C Co (fl). Let vk =

Uk

IIukllp .

Then

1.6 Other Basic Inequalities

37

IIvkIIP=1, and IIDvkIIP-30.

Consequently, {vk} is bounded in W'-P(S2) and hence possesses a subsequence (still denoted by {vk}) that converges weakly to some vo E W1"P(.(2). From the compact embedding results in the previous section, {vk} converges strongly in LP(S2) to v0, and therefore

IIv0IIP=I.

(1.54)

On the other hand, for each ¢ E Co (.2),

L v ,z,dx

k-+. n

vk¢y,dx = - kl. If vk,s: dx = 0. JR

It follows that Dvo (x) = 0, a.e.

Thus vo is a constant. Taking into account that vk E Co (S2), we must have vo - 0. This contradicts with (1.54) and therefore completes the proof of the Theorem.

For functions in W1'P(fl), which may not be zero on the the boundary, we have another version of Poincare's inequality.

Theorem 1.6.2 (Poincare Inequality II). Let 12 be a bounded, connected, and open subset in R' with C' boundary.

Let u be the average of u on ,f2. Assume 1 < p < co. Then there exists a constant C = C(n,p, ,f2), such that

IIn-uIIP
(1.55)

The proof of this Theorem is similar to the previous one. Instead of letting we choose vk

I

ukIIP'

Uk - itk

Ilnk -kIIP

Vk

Remark 1.6.2 The connectedness of .2 is essential in this version of the inequality. A simple counter example is when n = 1, ,2 = [0, 1] U [2, 3], and

u(x)

1 for x E [0,1] 1

forxE[2,3].

1 Introduction to Sobolev Spaces

38

1.6.2 The Classical Hardy-Littlewood-Sobolev Inequality Theorem 1.6.3 (Hardy-Littlewood-Sobolev Inequality).

Let0 1 such that 1

A

1

-+-+-=2. r n s Assume that f E Lr(Rn) and 9 E L'(R"). Then Rnf(x)Ix-yl-a9(y)dxdyCC(n,s,A)IIfII,IIgIIe

JR^

(1.56)

where

C n, s A) = n (

B" A/n

(I - A)rs

)

A/n

((1-1/r)

Ann

+

A/n

A/n

(I-1/s)

with IB"I being the volume of the unit ball in Rn, and where IIfllr := II!IILr(R0)-

Proof. (Adapted from Lieb and Loss's book [LL] with minor modifications.)

Without loss of generality, we may assume that both f and g are nonnegative and IIfIIr = 1 = II9I1s. Let

XG(x) =

1 ifxEG 0 if x O G

be the characteristic function of the set G. Then one can see obviously that f(x) =

f

X{f>a}(x) da

(1.57)

X{g>b} (x) db

(1.58)

D

g(x) = I D

IxI

\ = X 0f c

1X{Is


To see the last identity, one may first write

IxI ,= f 0X{Irh-a>e}(x)de, and then let 6 = c

-A.

(1.59)

1.6 Other Basic Inequalities

39

Substituting (1.57), (1.58), and (1.59) into the left hand side of (1.56), we have

I A

f(x)Ix

JR^ JR^

f f mf mf f

ff f 0

0

=X

yI

R^ R^

0

;,g(y) da dy =

c A-1X{f>a}(x)X{IxIb}(y)dxdydedbda

c a-1I(a, b, c) dcdb da.

(1.60)

a

Let

u(c) = IBnlcn,

the volume of the ball of radius c, and let v(a) =

f

R^

X{f>a}(x)dx, w(b) = f X{g>b}(y)dy,

the measure of the sets {x I f(x) > a} and {y I g(y) > b}, respectively. Then we can express the norms as M

oo

If IIr = r f ar-'v(a) do = 1 and 11911 = s f a

b's-1w(b) db = 1.

(1.61)

o

It is easy to see that

I(a,b,c)
f^ X{IxIb}(y)dxdy

R^ R

u(c)X{g>b} (y) dy = u(c)w(b)

Similarly, one can show that 1(a, b, c) is bounded above by other pairs and arrive at I(a,b,c) <min{u(c)w(b),u(c)v(a),v(a)w(b)}. (1.62) We integrate with respect to c first. By (1.62), we have 0f.

c A-1I(a, b, c) dc

<

f

c a-1w(b)u(c) dc+

=w(b)lB"I

f

C-A-1 w(b)v(a) dc

u(c)>v(a)

u(c)
f

(v a)lIB^0'1^

oe

c a-1}n do+w(b)v(a) f

c a-1 do

o

B"I,\"w(b)v(a)'-A/n =

I

+

a

" nIBnIA/n X(a

A)

w(b)v(a) 1 A/ "

IB

A

]A/"w(b)v(a)r-

a/ (1.63)

40

1 Introduction to Sobolev Spaces

Exchanging v(a) with w(b) in the second line of (1.63), we also obtain

1.

c A-1I(a, b, c) dc

0

1

<

c a-1v(a)u(c) dc+

u(c)<w(b)

n

r

Ju(c)>-(b))

c a-lw(b)v(a) do

n ,\/n v(a)w(b)1-a/n

(1.64)

In view of (1.61), we split the b-integral into two parts, one from 0 to a'/s and the other from a''/s to oc. By virtue of (1.60), (1.63), and (1.64), we derive

I< n

n"

A

Ir0

B"ja/nx

v(a)1

dbda+ 1 v(a)

o

w(b)dbda } (1.65)

o JJJ

To estimate the first integral in (1.65), we use Holder inequality with

m=(s-1)(1-A/7e) .

w(b)1-a/nb'nb-"` db

(a

10a,/

<

1-a/n

/a

{

w(b)bs-1 db I

/

0

w(b)b'_1

db I

/

\0

b-pan/a db)

J

o

1-a/n .


A/n

a'/a

(1 / I

A/n A

f

A)) n - s(n - A)/

ar-1; (1.66)

because mn/A < 1 and 1

1

A

r

S

n

It follows that the first integral in (1.65) is bounded above by A

a/n

\ n - s(n - A))

pp

\J0

l' v(a)ar-1 da/ (I

pco w(b)b8

=

1 dbl

1-a/n

1 / A/n v A/n TS

1 - 1/T J

To estimate the second integral in (1.65), we first rewrite it as b,/*

0o

10

w(b)1 0

v(a)1_a/n

da A,

(1.67)

1.6 Other Basic Inequalities

41

then an analogous computation shows that it is bounded above by 1

/ x/n \ a/"

rs

(1.68)

1 - 1/s /l

Now the desired Hardy-Littlewood-Sobolev inequality follows directly from (1.65), (1.67), and (1.68).

Theorem 1.6.4 (An Equivalent Form of the Hardy-Littlewood-Sobolev inequality)

Let g E L W+ p (R") forn'a < p < oo. Define

T9(x) = JR. It - yI"-"9(y)dy. IIT9IIp <_ C(n,p,a)IIgI1.

(1.69)

Proof. By the classical Hardy-Littlewood-Sobolev inequality, we have < f,Tg >=< C(n, s, a) II f1IT IIg1Ia

,

> is the L2 inner product. where < Consequently, IIT9IIp=

sup
where 1 + 1 = 1 r

+ 1 =

a

8

Solving for s, we arrive at

s=

np

n+ap 0

This completes the proof of the Theorem.

Remark 1.6.3 To see the relation between inequality (1.69) and the Sobolev inequality, let's rewrite it as IIT9II

with

CII9II9

1< q:=n+aP

< a

(1.70)

42

1 Introduction to Sobolev Spaces

Let it = T9. Then one can show that (see [CLO]),

(-L)au=9. Now inequality (1.70) becomes the Sobolev one

II1IIn"a, <-CII(-,L)ttII9

2

Existence of Weak Solutions

2.1

Second Order Elliptic Operators

2.2

Weak Solutions

2.3

Method of Linear Functional Analysis

2.4

2.3.1

Linear Equations

2.3.2

Some Basic Principles in Functional Analysis

2.3.3

Existence of Weak Solutions

Variational Methods 2.4.1

Semi-linear Equations

2.4.2

2.4.5

Calculus of Variations Existence of Minimizers Existence of Minimizers under Constrains Mini-max Critical Points

2.4.6

Existence of a Mini-max via the Mountain Pass Theorem

2.4.3 2.4.4

Many physical models come with natural variational structures where one can minimize a functional, usually an energy E(u). Then the minimizers are weak solutions of the related partial differential equation E'(u) = 0.

For example, for an open bounded domain 17 C R^, and for f E Lz(fl), the functional E(u) = 2 IVul'dx - J f udx

Jn

n

44

2 Existence of Weak Solutions

possesses a minimizer among all u E Ho (fl) (try to show this by using the knowledge from the previous chapter). As we will see in this chapter, the minimizer is a weak solution of the Dirichlet boundary value problem

'Lu=f(x),xEI? U(X) = 0,

x E 8,f2.

In practice, it is often much easier to obtain a weak solution than a classical

one. One of the seven millennium problems posed by the Clay Institute of Mathematical Sciences is to show that some suitable weak solutions to the Navier-Stokes equation with good initial values are in fact classical solutions for all time. In this chapter, we consider the existence of weak solutions to second order elliptic equations, both linear and semi-linear. For linear equations, we will use the Lax-Milgram Theorem to show the existence of weak solutions. For semi-linear equations, we will introduce variational methods. We will consider the corresponding functionals in proper Hilbert spaces and seek weak solutions of the equations associated with the critical points of the functionals. We will use examples to show how to find a minimizer, a minimizer under constraint, and a mini-max critical point. The well-known Mountain Pass Theorem is introduced and applied to show the existence of such a mini-max. To show that a weak solution possesses the desired differentiability so that

it is actually a classical one is called the regularity method, which will be studied in Chapter 3.

2.1 Second Order Elliptic Operators Let ,fl be an open bounded set in R. Let aij (x), bi (x), and c(x) be bounded functions on 0 with aij(x) = aji(x). Consider the second order partial differential operator L either in the divergence form [om

Lu = -

/

(ai9(x)ux;)x,i +Lbi(x)ux. +C(x)u, 1,j=1

(2.1)

i=1

or in the non-divergence form

n

Lu = -

bi(x)uz, + c(x)u.

ail (x)us.., + i.j=1

(2.2)

i=1

We say that L is uniformly elliptic if there exists a constant 6 > 0, such that

2.2 Weak Solutions

45

n

E

j > bj£12

i,9=1

for almost every x E 2 and for all E Rn. In the simplest case when ai5 (x) = bi bi (x) 0, and c(x) 0, L reduces to -0. And, as we will see later, the solutions of the general second order elliptic equation Lu = 0 share many similar properties with the harmonic functions.

2.2 Weak Solutions Let the operator L be in the divergence form as defined in (2.1). Consider the second order elliptic equation with Dirichlet boundary condition

JLu =.f,xEQ,

(2.3)

U-0, xE8f2.

When f = f (x), the equation is linear; and when f = f (x, u), semi-linear. Assume that f (x) is in L2 (2), or for a given solution u, f (x, u) is in L2(2). Multiplying both sides of the equation by a test function v E Co (.2), and integrating by parts on ,fl, we have aii(x)uy:vz; +I: bi(x)us;v + c(x)uv I dx = Jn fvdx.

Jn

(2.4)

J

i_1

9=l

There are no boundary terms because both u and v vanish on 499. One can see that the integrals in (2.4) are well defined if u, v, and their first derivatives are square integrable. Write Hl(S2) = W1,2(2), and let H01(d2) be the completion

of Cp (,fl) in the norm of H'(2): 1/2

hull =

Dub2

LJa(I

+ IuI2)dx]

Then (2.4) remains true for any v E Ho (,2). One can easily see that Hp (f2) is also the most appropriate space for u. On the other hand, if u E Hu (0) satisfies (2.4) and is second order differentiable, then through integration by parts, we have

(Lu - f)vdx = 0, Vv E Ha (.2). 12

This implies that

Lu=f, VxEQ,

and therefore u is a classical solution of (2.3). The above observation naturally leads to the following:

46

2 Existence of Weak Solutions

Definition 2.2.1 The weak solution of problem (2.3) is a function u E Ho (.2) that satisfies (2.4) for all v E HH(Q).

Remark 2.2.1 Actually, here the condition on f can be relaxed a little. By the Sobolev embedding

H'(r) - L.'-2 (9), we see that v is in L ^a (,2), and hence by duality, f only needs to be in L +_ (2). In the semi-linear case, if f (x, u) = up for any p < generally,

nu

- 222 , or more

z

f(x,u) < Cr + C2IuI

then for anyu E H1(fl), f(x,u) is in L+(.2).

2.3 Methods of Linear Functional Analysis 2.3.1 Linear Equations For the Dirichlet problem with linear equation

Lu=f(x),xE.Q xEafl, u=0,

(2.5)

to find its weak solutions, we can apply representation type theorems in Linear Functional Analysis, such as the Riesz Representation Theorem or the LaxMilgram Theorem. To this end, we introduce the bilinear form r

n

B[u,v] =J

n

aij(x)ux,v., +bi(x)uz,v+c(x)uv I dx i,j=1

%=1

1

defined on Ho(2), which is the left hand side of (2.4) in the definition of the weak solutions, while the right hand side of (2.4) may be regarded as a linear functional on Ho (2):

< f,v>:= J fvdx. Now, to find a weak solution of (2.5), it is equivalent to show that there exists a function u E HO '(9), such that B[u, v] =< f, v >, V v E Hp (fl).

(2.6)

This can be realized by representation type theorems in Functional Analysis.

2.3 Methods of Linear Functional Analysis

47

2.3.2 Some Basic Principles in Functional Analysis Here we list briefly some basic definitions and principles of functional analysis, which will be used to obtain the existence of weak solutions. For more details, please see, for example, [Schl].

Definition 2.3.1 A Banach space X is a complete, normed linear space with norm 11

-

11 that satisfies

(1) Ilu+vII < IIuII + 1lvII, Vu,v E X, (ii) IIAull _ JAIVIuII, Vu, v E X,AE R,

(iii) hull = 0 if and only if u = 0. The examples of Banach spaces are: (i) LP(fl) with norm JuIIyn(n), (ii) Sobolev spaces Wk,P(fl) with norm lluIIwk,v(n), and (iii) Holder spaces C°''r(Q) with norm lluhlcon(n).

Definition 2.3.2 A Hilbert space H is a Banach space endowed with an inner product (., .) which generates the norm (u,u)112

IIuIJ

with the following properties (i) (u, v) = (v, u), Vu, v E H,

(ii) the mapping u H (u, v) is linear for each v E H, (iii) (u, u) > 0, Vu E H, and (iv) (u, u) = 0 if and only if u = 0. Examples of Hilbert spaces are (i) L2(fl) with inner product u(x)v(x)dx,

(u,v) =

J

and

(ii) Sobolev spaces H1(0) or Ho (fl) with inner product (u, v) =

J [u(x)v(x) + Du(x) . Dv(x)]dx.

Definition 2.3.3 (i) A mapping A : X-->Y is a linear operator if A[au + bv] = aA[u] + bA[u] V u, v E X, a, b E R1.

2 Existence of Weak Solutions

48

(ii) A linear operator A : X-+Y is bounded provided IIAII :=

sup

IIAully < oo.

IuIIx<_1

(iii) A bounded linear operator f : X-+R1 is called a bounded linear functional on X. The collection of all bounded linear functionals on X, denoted by X*, is called the dual space of X. We use

:=f[u] to denote the pairing of X* and X.

Lemma 2.3.1 (Projection). Let M be a closed subspace of H. Then for every it E H, there is a unique v E M, such that (u - v, w) = 0, Vw E M. (2.7) In other words,

H=M®M1

with

M1 := {u E H I u1M}.

Here v is the projection of it on M. The reader may find the proof of the Lemma in the book [Sch] (Theorem 1-3) or as given in the following exercise.

Exercise 2.3.1 i) Given any it E H and u 0 M, there exist a vo E M such that

Ilvo -nil = vnf IIv - uII. Hint: Show that a minimizing sequence is a Cauchy sequence. ii) Show that

(u-vo,w)=0, VwEM. Based on this Projection Lemma, one can prove the following:

Theorem 2.3.1 (Riesz Representation Theorem). Let H be a real Hilbert space with inner product and H* be its dual space. Then H* can be canonically identified with H; more precisely, for each u* E H*, there exists a unique element it E H, such that

< u*, v >= (u, v), Vv E H. The mapping u* -+ it is a linear isomorphism from H* onto H.

2.3 Methods of Linear Tinctional Analysis

49

This is a well-known theorem in functional analysis. The reader may see the book [Schl] for its proof or do it as the following exercise.

Exercise 2.3.2 Prove the Riesz Representation Theorem in 4 steps. Let T be a linear operator from H to Rl and T # 0. Show that i) K(T) := {u E H I Tu = 0} is closed. ii) H = K(T) (D K(T)1. iii) The codimension of K(T) is 1. iv) There exists v E K(T)1, such that To = 1. Then

Tv=(u,v), VuEH. From Riesz Representation Theorem, one can derive the following:

Theorem 2.3.2 (Lax-Milgram Theorem). Let H be a real Hilbert space with norm 11

-

11. Let

B:HxH-rR1

be a bilinear mapping. Assume that there exist constants M, m > 0, such that IB[u,vlj < MIlullIIvII, Vu,v E H rnIIuII2 < B[u,u], Vu E H.

Then for each bounded linear functional f on H, there exists a unique element u E H, such that B[u, v] =< f, v >, Vv E H. Proof. 1. If B[u,v] is symmetric, i.e., B[u, v] = B[v,u], then the conditions

(i) and (ii) ensure that B[u,v] can be made an inner product in H, and the conclusion of the theorem follows directly from the Riesz Representation Theorem.

2. In the case B[u, v] may not be symmetric, we proceed as follows.

On the one hand, by the Riesz Representation Theorem, for a given bounded linear functional f on H, there is an f E H, such that

=(f,v), VvEH.

(2.8)

On the other hand, for each fixed element is E H, by condition (i), B[u, ] is also a bounded linear functional on H, and hence there exist a u E H, such that B[u, v] = (u, v), Vv E H. (2.9)

50

2 Existence of Weak Solutions

Denote this mapping from it to ii by A, i.e.,

A : H-+H, Au = w.

(2.10)

From (2.8), (2.9), and (2.10), it suffices to show that for each f E H, there exists a unique element it E H, such that Au

f.

We carry this out in two parts. In part (a) we show that A is a bounded linear operator, and in part (b) we prove that it is one-to-one and onto. (a) For any u1, u2 E H, any al, a2 E Rl, and each v E H, we have

(A(alul + a2u2), v) = B[(alul + a2u2), v] = a1 B [ul, v] + a2B [u2, v]

= a,(Aul,v) + a2(Au2,v) = (aiAul +a2Au2,v) Hence

A(alul + a2u2) = a1Au1 + a2Au2. A is linear. Moreover, by condition (i), for any it e H, IIAuII2 = (Au, Au) = B[u, Au] < MIIuII IIAuII,

and consequently IlAull 5 MIIuII Vu E H.

(2.11)

This verifies that A is bounded. (b) To see that A is one-to-one, we apply condition (ii): MIIuII2
and it follows that mllull s IlAull.

(2.12)

Now if Au = Av, then from (2.12), milu - vil <- II Au - AvII.

This implies it = v. Hence A is one-to-one.

From (2.12), one can also see that R(A), the range of A in H, is closed. In fact, assume that vk E R(A) and vk-*v in H. Then for each vk, there is a uk, such that Auk = vk. Inequality (2.12) implies that

Ilui-u;11 «IA-vi11,

2.3 Methods of Linear Functional Analysis

51

i.e., Uk is a Cauchy sequence in H, and hence uk-- U for some u E H. Now, by (2.11), Auk-*Au and v = Au E H. Therefore, R(A) is closed.

To show that R(A) = H, we need the Projection Lemma 2.3.1. For any u E H, by the Projection Lemma, there exists a v E R(A), such that

d=(u-v,A(u-v))=B[(u-v), (u - v)]?mllu-vu12. Consequently, u = v E R(A) and hence H C R(A). Therefore R(A) = H. This completes the proof of the Theorem.

0

2.3.3 Existence of Weak Solutions We next explore in what situations our second order elliptic operator L would satisfy the conditions (i) and (ii) requested by the Lax-Milgram Theorem. Let H = Ho (R) be our Hilbert space with inner product

(u,v):=J (DuDv+uv)dx. n

By Poincare's inequality, we can use the equivalent inner product (u, v)H :=

Jn

DuDvdx

and the corresponding norm IIUIIH :=

J IDuI2dx.

For any v E H, by the Sobolev Embedding, v is in L-2 (.(l). And by virtue of the Holder inequality,

:=JHf(x)v(x)dx<(Llf(x)t +2dx)

(flv(x)l-=dx)

and hence we only need to assume that f E L +2 ((l). In the simplest case when L = -0, we have

B[u, v] = 1 Du - Dvdx. JJJn

Obviously, condition (i) is met: IB[u,v]l < IIUIIHIIVIIH,

52

2 Existence of Weak Solutions

Condition (ii) is also satisfied: B[u,w] = IIUIIH.

Now applying the Lax-Milgram Theorem, we conclude that there exists a unique u E H such that

B[u,v]=, VvEH. We have actually proved the following theorem:

Theorem 2.3.3 For every f (x) in L +2 ((Z), the Dirichlet problem An = f (x), x E ,fl, u(x) = 0, x E 8fl has a unique weak solution u in Ho(.fl). In general,

( E a;j(x)us;vxj + Eb;(x)us;v + c(x)uv)

B[u,v] =

Jn

tij=1

dx.

i=1

Under the assumption that laijIILO(n), IIbiIIL-(n), IICIIL-(n) < C,

and by the Holder inequality, one can easily verify that IB[u,v]I <3CIIuIIHIIvIIH.

Hence condition (i) is always satisfied. However, condition (ii) may not automatically hold. It depends on the sign of c(x) and the L°° norm of bi(x) and c(x). First, from the uniform ellipticity condition, we have

f

n i,j=1

aij(x)us;ux;dx > 6 f IDuI2dX n

= 6IIuII12r,

(2.13)

for some 6 > 0.

From here one can see that if bi(x) - 0 and c(x) > 0, then condition (ii) is met. Actually, the condition on c(x) can be relaxed a little; for instance, if

c(x) > -6/(2C)

(2.14)

with 6 as in (2.13), and where C is the constant in the Poincare inequality, then condition (ii) holds. Hence we have

2.4 Variational Methods

Theorem 2.3.4 Assume that bi(x) every f E L (1), the problem

53

0 and c(x) satisfies (2.1¢). Then for

Lu=f(x),xE1? {u(x)=0, xEOQ has a unique weak solution in Ho(fl).

2.4 Variational Methods 2.4.1 Semi-linear Equations To find the weak solutions of semi-linear equations

Lu = f (x, u) the Lax-Milgram Theorem can no longer be applied. We will use variational

methods to obtain the existence. Roughly speaking, we will associate the equation with the functional

J(u) = 2

n

(Lu . u)dx -

n

F(x, u)dx

where F(x U) =

f(x s)ds

J0 and L is a self-adjoint operator (with bi(x) _- 0). We show that the critical point of J(u) is a weak solution of our problem. Then we will focus on how to seek critical points in various situations.

2.4.2 Calculus of Variations For a continuously differentiable function g defined on Rn, a critical point is a point where Dg vanishes. The simplest sort of critical points are global or local maxima or minima. Others are saddle points. To seek weak solutions of partial differential equations, we generalize this concept to infinite dimensional spaces. To illustrate the idea, let us start from a simple example:

Du = f (x) , x E fl; , x E 8n.

U(X) = 0

To find weak solutions of the problem, we study the functional

J(u) = 1 2

Jn

f (x)udx

IDu12dx n

(2.15)

2 Existence of Weak Solutions

54

in HH(fl).

Now suppose there is some function it that happens to be a minimizer of J(-) in Ho (fl); we will demonstrate that it is actually a weak solution of our problem.

Let v be any function in Ho (Q). Consider the real-valued function

g(t) = J(u+tv), t E R. Since it is a minimizer of J(-), the function g(t) has a minimum at t = 0; and we therefore must have i.e.,

g '(O) = 0,

dtJ(u+tv) It=o= 0.

Here, explicitly,

J(u+tv)=21f ID(u+tv)12dx-J R

f (x) (u + tv)dx,

R

and Wt

J(u+tv) =

J

D(u+tv) - Dvdx -

JR

f(x)vdx.

Consequently, g'(0) = 0 yields

fn Du - Dvdx - fn f (x)vdx = 0, Vv E Ho (,(2).

(2.16)

This implies that it is a weak solution of problem (2.15). Furthermore, if it is also second order differentiable, then through integration by parts, we obtain JR [-Au

- f (x)] vdx = 0.

Since this is true for any v E Ho (Q), we conclude

-Au - f (x) = 0. Therefore, u is a classical solution of the problem. From the above argument, the reader can see that in order to be a weak solution of the problem, here it need not be a minimum; it can be a maximum, or a saddle point of the functional, or generally any point that satisfies

dtJ(u+tv) It=o More precisely, we have the following:

2.4 Variational Methods

55

Definition 2.4.1 Let J be a functional on a Banach space X. i) We say that J is Frechet differentiable at it E X if there exists a continuous linear map L = L(u) from X to X* satisfying: For any e > 0, there is a 5 = S(e, u), such that

J(u+v) - J(u)- < L(u),v > 1 < eIIvIIx whenever jjvjjx <J1 where

< L(u), v >:= L(u) (v)

is the duality between X and X*. The mapping L(u) is usually denoted by J'(u). ii) A critical point of J is a point at which J'(u) = 0; that is,

< J'(u),v >= 0 Yv E X. iii) We call J'(u) = 0 the Euler-Lagrange equation of the functional J.

Remark 2.4.1 One can verify that, if J is Frechet differentiable at it, then

limoJ(u+tv) -J(u) <J'(u),v>= t-+ t

=

dtJ(u+tv)

It=o.

In the previous subsection we observed that the critical points of the functional

J(u) = 1 2

Jn

IDuI2dx - L f (x)udx n

are weak solutions of the Dirichlet problem (2.15). Our next question is this: Does the functional J actually possess a critical point? We will show that there exists a minimizer of J. To this end, we verify that J possesses the following three properties: i) boundedness from below, ii) coercivity , and iii) weak lower semi-continuity. i) Boundedness from Below.

First we prove that J is bounded from below in H Ho (dl) if f E L2 (Q). Again, for simplicity, we use the equivalent norm in H:

2 Existence of Weak Solutions

56

Dul2dx.

IIuIIH

fn

By the Poincare and Holder inequalities, we have

J(u) ?

2IIUIIH

- CIIUIIHIIfIIL2(Q) 2

= 2 (IIuIIH-CIIfIIL2(a))2-

-2

If 11L2(f2)

2

(2.17)

2 IIfIIL2(n)

Therefore, the functional J is bounded from below. ii) Coercivity. However, a functional that is bounded from below does not guarantee that it has a minimum. A simple counterexample is

g(x)=1+x2, xERr. Obviously, the infimum of g(x) is 0, but it can never be attained. If we take a minimizing sequence {xk} such that g(xk)->0, then xk goes away to infinity. This suggests that in order to prevent a minimizing sequence from "leaking" to infinity, we need to have some sort of "coercive" condition on our functional J. That is, if a sequence {uk} goes to infinity, i.e., if IIuIIH--boo, then J(uk) must also become unbounded. Therefore, a minimizing sequence would be retained in a bounded set. And this is indeed true for our functional J. Actually, from the second part of (2.17), one can see that if IIukIIH-*00, then

This implies that any minimizing sequence must be bounded in H. iii) Weak Lower Semi-continuity. If H is a finite dimensional space, then a bounded minimizing sequence {uk} would possesses a convergent subsequence,

and the limit would be a minimizer of J. This is no longer true in infinite dimensional spaces. We therefore turn our attention to weak topology.

Definition 2.4.2 Let X be a real Banach space. If a sequence NO C X satisfies

< f, uk > -r < f, uo > as k-roo for every bounded linear functional f on X, we say that {uk} converges weakly

to uo in X and write uk

U0.

Definition 2.4.3 A Banach space is called reflexive if (X*)* = X, where X* is the dual of X.

2.4 Variational Methods

57

Theorem 2.4.1 (Weak Compactness.) Let X be a reflexive Banach space. Assume that the sequence {uk} is bounded in X. Then there exists a subsequence of {uk}, which converges weakly in X.

Let us now return to our Hilbert space H := Ho (,2). By the Riesz Representation Theorem, for every linear functional f on H, there is a unique v E H, such that < f, u >= (v, u), Vu E H. It follows that H* = H, and therefore H is reflexive. By the Coercivity and Weak Compactness Theorem, now a minimizing sequence {uk} is bounded and hence possesses a subsequence {uk, } that converges weakly to an element U. in H: (uku v)-Y(uo, v), Vv E H.

Naturally, we wish the functional J we consider to be continuous with respect to this weak convergence, that is,

llim. J(uk.) = J(uo), then uo would be our desired minimizer. However, this is not the case with our

functional, nor is it true for most other functionals of interest. Fortunately, we do not really need such a continuity but instead need only a weaker one:

J(uo) < l minf J(uk,). Since on the other hand, by the definition of a minimizing sequence, we obviously have

J(u0) >

J(uk,),

and hence

J(uo) = lief J(uk; ) uo is therefore a minimizer.

Definition 2.4.4 We say that a functional J() is weakly lower semicontinuous on a Banach space X, if for every weakly convergent sequence

uk - U. ,

in X,

we have

ki.f J(u0) < liM

J(uk)

Now we show that our functional J is weakly lower semi-continuous in H. Assume that {uk} C H, and Uk - uo

in H.

58

2 Existence of Weak Solutions

Since for each fixed f E L +2- (12), fn fudx is a linear functional on H, it follows immediately

f

r fukdx-+

fuodx,

as k->oo.

(2.18)

Jn

Jn

From the algebraic inequality a2 + b2 > 2ab, we have

IDuk12dx>J IDu0I2dx+2J Du,-(Duk-Duo)dx. Jf n

n

n

Here the second term on the right goes to zero due to the weak convergence Uk -' uo. Therefore liminf fn IDukI2dx > Jn IDuo12dx. k-4w

(2.19)

Now (2.18) and (2.19) imply the weakly lower semi-continuity. So far, we have proved the following:

Theorem 2.4.2 Assume that ,12 is a bounded domain with smooth boundary. Then for every f e L +_ (d2) with n > 2, the functional

J(u) = 2 L IDuI2dx - f fudx n

n

possesses a minimum uo in Ho (fl), which is a weak solution of the boundary value problem

'Lu=f(x),xE12 1U(X) = 0,

x E 4912.

2.4.4 Existence of Minimizers Under Constraints Now we consider the semi-linear Dirichlet problem

Du = uj' 1u, x E 57, u(x) = 0,

(2.20)

x E 49n,

with 1
2.4 Variational Methods

J(u) =

1

n

2

59

- p+1 Q IuI"+idx. 1

IDuI2dx

It is easy to verify that dt

J(u + tv) It=o=

r

Jn (Du Dv - luIp-'uv) dx.

Therefore, a critical point of the functional J in H := Ho (B) is a weak solution of (2.20)

However, this functional is not bounded from below. To see this, fix an element u E H and consider

IDuI2dx- v+r

t2

J(tu) =

_2

p+1Jn Iulp+idx.

12

Noticing p + 1 > 2, we find that J(tu)-> - oo,

as t-+oo.

Therefore, there is no minimizer of J(u). For this reason, instead of J, we will consider another functional

I(u) := 2

n

IDuI2dx,

in the constrain set

M:= {uEHIG(u):=f IuIp+'dx=1}. We seek minimizers of I in M. Let {uk} C M be a minimizing sequence, i.e.,

kli mI(uk) =

inf 1(u) := in.

It follows that fn IDuk12dx is bounded, hence {uk} is bounded in H. By the Weak Compactness Theorem, there exists a subsequence (for simplicity, we still denote it by {uk}), which converges weakly to up in H. The weak lower semi-continuity of the functional I yields

I(u0)
kim

On the other hand, since p+ 1 < ding Theorem

Zu

(2.21)

n-2 , by the Compact Sobolev Embed-

H1(Q) y,i V+'(Q), we find that uk converges strongly to uo in Lp+i (12), and hence

60

2 Existence of Weak Solutions

fo

lu°In+'dx = 1.

That is, u° E M and therefore I(u°) > M. Together with (2.21), we obtain

I(u°) = M.

We have now shown the existence of a minimizer u° of I in M. To link u° to a weak solution of (2.20), we derive the corresponding Euler-Lagrange equation for this minimizer under the constraint.

Theorem 2.4.3 (Lagrange Multiplier). let it be a minimizer of I in M, i.e.,

I(u) = min I(v). vEM

Then there exists a real number A such that

I'(u) = AG'(u), or

=A, VvEH. Before proving the theorem, let us first recall the Lagrange Multiplier for

functions defined on R". Let f(x) and g(x) be smooth functions in R". It is well known that the critical points (in particular, the minima) x° of f(x) under the constraint g(x) = c satisfy

Df (x°) = .Dg(x°)

(2.22)

for some constant A. Geometrically, Df(x°) is a vector in R. It can be decomposed as the sum of two perpendicular vectors:

Df (x°) = Df (x°) IN +Df (x°) IT,

where the two terms on the right hand side are the projection of Df(x°) onto the normal and tangent spaces of the level set g(x) = c at point x°. By definition, x° being a critical point of f(x) on the level set g(x) = c means that the tangential projection Df(x°) IT= 0. While the normal projection Df(x°) IN is parallel to Dg(x°), we therefore have (2.22). Heuristically, we may generalize this perception into infinite dimensional space H. At a fixed point u E H, I'(u) is a linear functional on H. By the Riesz Representation

2.4 Variational Methods

61

Theorem, it can be identified as a point (or a vector) in H, denoted by I'(u), such that < I'(u), v >= (I'(u), v), Vv E H.

Similarly for G'(u), we have G'(u) E H, and it is a normal vector of M at point u. At a minimum it of I on the hyper surface {w E H I G(w) = 1}, I;(u must be perpendicular to the tangent space at point it, and hence '-(u) = AG'(u). We now prove this rigorously.

The Proof of Theorem 2.4.3. Recall that if it is the minimum of J in the whole space H, then, for any v E H, we have dt7(u+tv)

t=o=0.

(2.23)

Now it is only the minimum on M; we can no longer use (2.23) because a+tv cannot be kept on M, and we cannot guarantee that

G(u + tv) := 12 Ju+tvlP+1dx = 1 for all small t. To remedy this, we consider g(t, s) := G(u + tv + sw).

For each small t, we try to show that there is an s = s(t), such that G(u + tv + s(t)w) = 1. Then we can calculate the variation of J(u + tv + s(t)w) as t--30. In fact, 9

(0, 0) =< G'(u), w >= (p + 1)

n

JuIv-ruwdx.

Since it E M, fn luIP+idx = 1, there exists in (may just take w as u), such that the right hand side of the above is not zero. Hence, according to the Implicit Function Theorem, there exists a Cr function s : R1-*R1, such that

s(0) = 0,

and g(t, s(t)) = 1,

for sufficiently small t. Now u+tv+s(t)w is on M, and hence at the minimum it of J on M, we must have

dtI(u+tv+s(t)w) It=o=+s'(0)=0.

(2.24)

2 Existence of Weak Solutions

62

On the other hand, we have 0)

° = dt (0' °) = at (0, + as (°' °)s (°) = < G'(u), v > + < G'(u), w > s'(0). Consequently

< G'(u),v >

s (°)

< G'(u), w >

Substituting this into (2.24) denoted by A

< G'(v), w >'

we arrive at

=A, Vv E H. This completes the proof of the Theorem. Now let's come back to the weak solution of problem (2.20). Our minimizer

ua of I under the constraint G(u) = 1 satisfies

=A, VvEH; that is,

n

Duo Dvdx = A J ju0I n

'uovdx, Vv E H.

(2.25)

One can see that A > 0. Let u = auo with A/ap-r = 1. This is possible since p > 1. Then it is easy to verify that

r

n

Du Dvdx = L vjP-luvdx. n

Now u is the desired weak solution of (2.20) in H.

Exercise 2.4.1 Show that A > 0 in (2.25).

2.4.5 Mini-max Critical Points Besides local or global minima and maxima, there are other types of critical points: saddle points or mini-max points. For a simple example, let's consider the function f (x, y) = x2 - y2 from R2 to R'. (x, y) = (0, 0) is a critical point of f, i.e., D f (0, 0) = 0. However, it is neither a local maximum nor a local minimum. The graph of z = f (x, y) in R3 looks like a horse saddle. If we move along the graph in the direction of the x-axis, (0,0) is the minimum, while if we

2.4 Variational Methods

63

move along the graph in the direction of the y-axis, (0,0) is the maximum. For this reason, we also call (0,0) a mini-max critical point of f(x, y). The way to locate or to show the existence of such a mini-max point is called a mini-max variational method. To illustrate the main idea, let's continue with this simple example. Suppose we don't know whether f (x, y) possesses a mini-max point, but we do detect a "mountain range" on the graph near the xz plane. To show

the existence of a mini-max point, we pick two points on the xy plane, for instance, P = (1, 1) and Q = (-1, -1) on two sides of the "mountain range". Let y(s) be a continuous curve linking the two points P and Q with 7(0) = P and 7(1) = Q. To go from (P, f (P)) to (Q, f (Q)) on the graph, one needs first to climb over the "mountain range" near the xz plane, then descend to the point (Q, f (Q)). Hence on this path, there is a highest point. We then take the infimum among the highest points on all possible paths linking P and Q. More precisely, we define

c= inf max f(7(s)), 7EP sE[0,11

where

l = {ry ry = -y(s) is continuous on [0,1], and 7(0) =P,-y(l) = Q}.

We then try to show that there exists a point Po at which f attains this mini-max value c, and more importantly, D f (Po) = 0. For a functional J defined on an infinite dimensional space X, we will apply a similar idea to seek its mini-max critical points. Unlike finite dimensional space, in an infinite dimensional space, a bounded sequence may not converge. Hence we require that functional J satisfy some compactness condition, which is the Palais-Smale condition.

Definition 2.4.5 We say that the functional J satisfies the Palais-Smale condition (in short, (PS)), if any sequence {uk} E X for which J(Uk) is bounded and J'(uk)-*O as k->oo possesses a convergent subsequence. Under this compactness condition, Ambrosetti and Rabinowitz [AR] established the following well-known theorem on the existence of a mini-max critical point.

Theorem 2.4.4 (Mountain Pass Theorem). Let X be a real Banach space and J E C1(X, R1). Suppose J satisfies (PS), J(0) = 0, (Ji) there exist constants p, a > 0 such that J Ion (o)> a, and (J2) there is an e E X \ B,,(0), such that J(e) < 0. Then J possesses a critical value c > a which can be characterized as

c = inf

max

iEP uEi([0,1])

J(u)

2 Existence of Weak Solutions

64

where

r = (-y e C([o,11,X) I -y(0) = 0,7(1) = e}.

Here, a critical value c means a value of the functional which is attained by a critical point u, i.e., J'(u) = 0 and J(u) = c. Heuristically, the Theorem says if the point 0 is located in a valley surrounded by a mountain range, and if at the other side of the range, there is another low point e; then there must be a mountain pass from 0 to e which contains a mini-max critical point of J. In the figure below, on the graph of J, the path connecting the two low points (0, J(0)) and (e, J(e)) contains a highest point S, which is the lowest as compared to the highest points on the nearby paths. This S is a saddle point, the mini-max critical point of J.

Fig. 2.1. Mountain Pass

The proof of the Theorem is mainly based on a deformation theorem which

exploits the change of topology of the level sets of J when the value of J goes through a critical value. To illustrate this, let's come back to the simple example f (x, y) = x2 - y2. Denote the level set off by

2.4 Variational Methods

fa

65

((x,y) E Rz I f(x,y) < c}.

One can see that 0 is a critical value and for any a > 0, the level sets fa and f-a has different topology. fa is connected while f-a is not. If there is no critical value between the two level sets, say fa and fb with 0 < a < b, then one can continuously deform the set fb into fa. One convenient way is to let the points in the set fb "flow" along the direction of -Df into fa; correspondingly, the points on the graph "flow downhill" into a lower level. More precisely and generally, we have

Theorem 2.4.5 (Deformation Theorem). Let X be a real Banach space. As-

sume that J E C1(X,R1) and satisfies the (PS). Suppose that c is not a critical value of J.

Then for each sufficiently small e > 0, there exists a constant 0 < 6 < e and a continuous mapping n(t,u) from [0,1] x X to X, such that

(i)tl(0,u)=u, VuEX, (ii)r7(1,u)=u, VuOJ-1[c-E,c+e], (iii) J(q(t, u)) < J(u), Vu E X, t E [0,1], (iv) 70, Ja+6) C J0_6.

Roughly speaking, the theorem says that if c is not a critical value of J, then one can continuously deform a higher level set Ja}6 into a lower level J,_b by "flowing downhill". Condition (ii) is to ensure that after the deformation, the two points 0 and e as in the Mountain Pass Theorem remain fixed.

Proof of the Deformation Theorem. The main idea is to solve an ODE initial value problem (with respect to t) for each u E X, which roughly looks like the following:

dt(t,u) = -J'(r!(t,u)),

'7(0, u) = U.

(2.26)

That is, let the "flow" go along the direction of -J'(,q(t, u)) so that the value of J(r)(t,u) would decrease as t increases. However, we need to modify the

right hand side of the equation a little so that the flow will satisfy other desired conditions.

Step 1. We first choose a small e > 0, so that the flow does not proceed too slowly in Ja+E \ J. This is actually guaranteed by the (PS). We claim that there exist constants 0 < a, e < 1, such that IIJ'(u)II ? a, Vu E JJ+E \ J0-E.

(2.27)

Otherwise, there would exist sequences ak-*0, Ek--+O and elements Uk E Ja+Ek \

J._,,, with IIJ'(uk)II 5 ak. Then by virtue of the (PS) condition, there is a

2 Existence of Weak Solutions

66

subsequence {uk, } that converges to an element ua EX. Since J E C' (X, Rl ), we must have J(uo) = c and J'(uo) = 0.

This is a contradiction with the assumption that c is not a critical value of J. Therefore (2.27) holds. Step 2. To satisfy condition (ii), i.e., to make the flow stay still in the set

M :_ {u E X I J(u) < c - E or J(u)

c + E},

we could multiply the right hand side of equation (2.26) by dist(rj, M), but this would make the flow go too slowly near M. To modify further, we choose 0 < 6 < E, such that (2.28)

and let

N:={uEX I c-S<J(u)
Now for it E X, define dist(u, M)

g(u)

dist(u, M) + dist(u, N)'

and let

h(s) :=

1 if0<s<1 s ifs > 1.

We finally modify the -J'(q) as F(71) := -g(77)h(IIJ'(q)II)J'(77)

The introduction of the factor h() is to ensure the boundedness of Step 3. Now for each fixed it E X, we consider the modified ODE problem

rdt (t, u) = F(rl(t, u)) l i(0, u) = u. One can verify that F is bounded and Lipschitz continuous on bounded sets; and therefore by the well-known ODE theory, there exists a unique solution g(t, u) for all time t > 0. Condition (i) is satisfied automatically. Ftom the definition of g

g(u)=0VuEM; hence condition (ii) is also satisfied. To verify (iii) and (iv), we compute

2.4 Variational Methods dtJ(n(t,u)) _
67

dt (t, u) >

_ < J'(r)(t, u)), F(,q(t, u)) >

=-9(n(t,u))h(IIJ'(n(t,u))IUIIJ'(n(t,u))112.

-

(2.29)

In the above equality, the right hand side is obviously non-positive, and hence J(rl(t,u)) is non-increasing. This verifies (iii). To see (iv), it suffices to show that for each u in N = JJ+a \ J.-a, we have n(1, u) E J.-6

-

Notice that for n E N, g(rl) _ 1, and (2.29) becomes dt J(n(t, u)) = -h(II J'(n(t, U)) I U II

j, (?7(t, u)112.

By the definition of h and (2.27), we have,

atJ(n(t,u)) _ {-IIJ'(n(t,u))II-Z -aa2 if IIJ'(n(t,u))II > 1. In any case, we derive

J(71(1, u)) <J(rl(0,u))-a2
The Proof of the Mountain Pass Theorem. First, due to the definition, we have c < oo. To see this, we take a particular

path y(t) = to and consider the function

9(t) = J(y(t)). It is continuous on the closed interval [0,1] and hence bounded from above. Therefore

c < max J(u) = max g(t) < oo. uEy([0,1]) tE[0,1] On the other hand, for any y c F y([o,1]) n aBp(o) # 0.

Hence by condition (J1),

68

2 Existence of Weak Solutions

max J(u) >

nE7([0,1])

inf

J(v) > a.

,

It follows that c > a. Suppose the number c so defined is not a critical value. We will use the Deformation Theorem to continuously deform a path y E P down to the level set JJ_5 to derive a contradiction. More precisely, choose the e in the Deformation Theorem to be z and 0 < 8 < c. FYom the definition of c, we can pick a path 7 E F, such that ,

max

uEy([0,11)

J(u)
7([0,1]) C Jc+a Now by the Deformation Theorem, this path can be continuously deformed down to the level set JJ_a, that is n(1, 7Q0,1]) C JJ-6,

or in order words, max

J(u) < c - 6.

(2.30)

uE+7(1,7(10, 1]))

On the other hand, since 0 and e are in J,_E, by (ii) of the Deformation Theorem,

77(1,0)=0 andq(1,e)=e. This means that 7](1, 7([0,1])) is also a path in F, and therefore we must have

max

J(u) > c.

uE77(1 ,7([0,1]))

This contradicts (2.30) and hence completes the proof of the Theorem.

2.4.6 Existence of a Mini-max via the Mountain Pass Theorem Now we apply the Mountain Pass Theorem to seek weak solutions of the semi-linear elliptic problem

Lu=f(x,u), xEQ, U(X) = 0,

x E 8f1.

(2.31)

Although the method we illustrate here could be applied to more general second-order, uniformly-elliptic equations, we prefer this simple model to

better illustrate the main ideas.

2.4 Variational Methods

69

We assume that f (x, u) satisfies (fl) f (x, s) E C((1 x R, Rl), (f2) there exists constants ci, c2 > 0, such that If (x, S) I < C1 + C2I SIP,

with 0 < p < n±Z if n > 2. If n = 1, (f2) can be dropped; if it = 2, it suffices

that If(x,s)I < cleo(s) with O(s)/s2-- 0 as IsIioc. (f 3)f(x, s) = o(I3I) as s-*O, and

(f4) there are constants p > 2 and r > 0 such that for IsI > r, 0 < pF(x,s) < sf(x,s),

where

s

F(x, s) =

J0

f (x, t)dt.

Remark 2.4.2 A simple example of such function f (x, u) is IuV 'u.

Theorem 2.4.6 (Rabinowitz [Raj) If f satisfies condition (fl) - (f4), then problem (2.31) possesses a nontrivial weak solution. To find weak solutions of the problem, we seek mini-max critical points of the functional

J(u) = 2 L J IDuI2dx

F(x,u)dx

on the Hilbert space H := Ho (11). We verify that J satisfies the conditions in the Mountain Pass Theorem.

We first need to show that J E C'(H, R1). Since we have verified that the first term

L is continuously differentiable, we need only check the second term

I(u) := J F(x,u)dx. n

Proposition 2.4.1 (Rabinowitz (Ra]) Let S? C R" be a bounded domain and let g satisfy (gl) g E C(.(1 x R1, R'), and (g2) there are constants r, s > 1 and a1, a2 > 0 such that

2 Existence of Weak Solutions

70

Ig(x, f)I <_ a1 + a2M 'ls

Then the map u(x)->g(x, u(x)) belongs to C(L'(.Q), Ls(0)). Proof. By (92),

f

n

Ig(x,u(x))I'dx <

n

(a1+a2IuI'/8)'dx


9(x,') : L'(0)-sL'(Q). We now verify the continuity of the map. Fix u E L'(,f2). Given any E > 0, we want to show that there is a b > 0, such that,

whenever II0IILr(n) <6, we have II9(-,u+0)-9(-,u)IIL'(Q) <E.

(2.32)

Let 01 :_ {x E .f2 I I0(x)I < m}

and

122:=f2\01. Let I, = J

n`

J9(x,u(x) + 0(x)) - g(x,u(x))I'dx, i=1,2.

By the continuity of g(x, ), for any ij > 0, there exists m > 0, such that 9(x, u(x) + 0(x)) - g(x, u(x))I <- 7), Vx E 01.

It follows that I1 <- I01Ir19 <_ IOIne,

where I12I is the volume of Q. For the given E > 0, we choose r) and then in,

so that I0I7s <

(Es

2

and therefore ll

((

r1

Now we fix this in, and estimate 12. By (g2), we have

2

e

(2.33)

2.4 Variational Methods

I2<

f

(cl+c2(IuIr+IOfr))dx
n,

71

IuI'dx+C2II Ilir(o), (2.34)

for some constant cl and c2. Moreover, as II0IILr(n) < 6,

6r >

In

(2.35)

I¢rrdx > mrIQ2I.

?

Since it E Lr(,f2), we can make ffla JuIrdx as small as we wish by letting 10221 be small. Now by (2.35), we can choose 6 so small, such that 1021 is sufficiently small, and hence the right hand side of (2.34) is less than (2)8. Consequently, by (2.33), for such a small 6,

I1+I2 <-

\2/e+\2/e.

This verifies (2.32) and therefore completes the proof of the Proposition.

Proposition 2.4.2 (Rabinowitz (Ra]) Assume that f(x,s) satisfies (ft) and

(f2). Letn>3. ThenI eC'(H,R/) and < 1'(U), V >= / f (x, u)vdx, Vv E H. n is weakly continuous and I'() is compact, i.e.,

Moreover,

I(uk)-rI(u) and I'(uk)-*I'(u), whenever uk - it in H.

Proof. We will first show that I is Frechet differentiable on H and then prove that F(u) is continuous. 1. Let u,v E H. We want to show that given any c > 0, there exists a

6 = 6(e, u), such that

Q(u, v) := II(u+v) -I(U) -

r

Jn

f (x, u)vdxI < ellvii

(2.36)

whenever iIvII < J. Here evil denotes the H'(12) norm of v. In fact, Q(u,v)

IF(x,u(x) + v(x)) - F(x,u(x)) - f(x,u(x))v(x)ldx

a

=

Jn

If(x,u(x) + e(x)) - f(x,u(x))I.Iv(x)Idx

If(-,u+e)- f(, u)I1L

(.R)HullL

,(f2JKIIvII.

(Q)

(2.37)

72

2 Existence of Weak Solutions

Here we have applied the Mean Value Theorem (with Ii;(x)I < Iv(x)I), the Holder inequality, and the Sobolev inequality IIvIIL

W)

< KIIvII-

In Proposition 2.4.1, let r = n^2^2 and s = +2. Then by (f2), we deduce that the map 'a(x)-+f(x, u(x))

is continuous from L- (2) to L +2 (fl). Hence for any given e > 0, we can choose sufficiently small S > 0, such that whenever IIvII < 8, we have < KS, and hence IIeiiL; (Q) < KS, and therefore IIV IIL "z (fl)

(0) <

K.

It follows from (2.37) that

Q(u,v) < e , whenever IIvII <J. This verifies (2.36) and hence I(.) is differentiable, and

=J f(x, u(x))v(x)dx.

(2.38)

n

2. To verify the continuity of I'(.), we show that for each fixed u,

III'(u+0)-I'(u)II-+o, as II4II-o.

(2.39)

By definition,

III'(u+4))-f(u)11= sup Ilvll<1

sup

J [f(x,u(x) + 0(x)) - f(x,u(x)))v(x)dx

IIv<1 0

sup

< KIIf(',u+4)) -

f(',u)IIL

In1KIIvii (2.40)

Here we have applied (2.38), the Holder inequality, and the Sobolev inequality Now (2.39) follows again from Proposition 2.4.1. Therefore, is continuous

3. Finally, we verify the weak continuity of I(u) and the compactness 0

I'(u). Assume that

2.4 Variational Methods

73

{uk} C H, and Uk -' u in H. We show that

I(uk)iI(u) as k->oo. In fact,

II(uk)-I(U)1<- f IF(x, uk(x)) - F(x, u(x))Idx < Jn

If(x,fk(x))I - luk(x) -u(x)Idx -UIILa(Q) (n)Iluk -u1IL4(n)

<

(2.41)

Here we have applied the Mean Value Theorem and the Holder inequality with 6k(x) between uk(x) and u(x), and

r

2n

r

2n

p(n_2), 4r_1_1=2n-p(u

2)

It is easy to see that 2n

4
III'(uk)-I'(u)II

(Q).

(2.42)

Since p <, we have <=2. Then by the Compact Embedding, H -4 -4 Ln+2 (fl), we derive

uk-+u strongly in Ln

(fl).

Consequently, form Proposition 2.4.1,

f(-,uk)--f(',u)strongly inL +2(Q). Now, together with (2.42), we arrive at

I'(uk)->I'(u) as k-+oo. This completes the proof of the proposition.

0

2 Existence of Weak Solutions

74

Now let's come back to Problem (2.31) and prove Theorem 2.4.6. We will

verify that J(.) satisfies the conditions in the Mountain Pass Theorem and thus possesses a non-trivial mini-max critical point, which is a weak solution of (2.31).

First we verify the (PS) condition. Assume that {uk} is a sequence in H satisfying IJ(uk)I < M and J'(uk)-30. We show that {uk} possesses a convergent subsequence.

By (fy), we have

AM+ IIJ''(uk)IIIIukII > ppJ(Uk)- < J'(uk),uk >

µ 2

- 1)

IDuk I2dx +

fn

J

[f(x, uk)uk - jLF(x, Uk)1 dx

IDukI2dx+J 2

> (2

1)

= CoIIUk

II2-Ca,

Jn

[f(x, uk)uk- AF(x, Uk)] dx Iukfr)I
.f2

IDukl2dx

- (air +a27-p+')If2I (2.43)

for some positive constant co and Co. Since IIJ'(uk)II is bounded, (2.43) implies

that {uk} must be bounded in H. Hence there exists a subsequence (still denoted by {uk}), which converges weakly to some element up in H.

Let A : H-*H* be the duality map between H and its dual space H*. Then for any it, v E H,

< Au, v >= J Du Dvdx. n

Consequently,

A-'J'(u) = u - A-'f (-, u).

(2.44)

From Proposition 2.4.2,

f (, uk)-> f (., u) strongly in H*. It follows that

uk = A-'J'(uk)+A-'f(.,uk)

as k-roo.

This verifies the (PS). To see there is a "mountain range" surrounding the origin, we estimate JJ F(x, u)dx. By (f3), for any given e > 0, there is a 6 > 0, such that, for all x E ,fl,

IF(x, s)I < elsl2, whenever Isl < 5.

(2.45)

2.4 Variational Methods

75

While by (f2), there is a constant M = M(6), such that for all x E S2, IF(x, s)I < MIsIr+r

(2.46)

Combining (2.45) and (2.46), we have, IF(x, s)I < EISI2 +MIsI1+l,

for all x E £2 and for all s E R.

It follows, via the Poincare and the Sobolev inequality, that

f F(x,u)dxI < E

L u2dx+M f

lul"+1dX < C(E+MIIuIIp-1)IIuII2 (2.47)

And consequently,

J(u) > [2 - C(E+MIIuhIp-1)]

(2.48)

IIuPI2.

Choose c = &C. For this E, we fix M; then choose IIull = p small, so that

CMPP-r < $. Then from (2.48), we deduce JIaB,(o) ?

P2

To see the existence of a point e E H, such that J(e) < 0, we fix any u $ 0 in H, and consider IDuI2dx -

J(tu) = 2

n in

J

F(x,tu)dx.

(2.49)

By (f4), we have for IsI > r, d

ds)){>0

ifs
In any case, we deduce, for IsI > r,

F(x, s) > a3sI"; and it follows that

F(x,s) > a3IsV" -a4-

(2.50)

Combining (2.49) and (2.50), and taking into account that p > 2, we conclude

that

J(tu)i - co, as t-*co. Now J satisfies all the conditions in the Mountain Pass Theorem, hence it possesses a mini-max critical point u,, which is a weak solution of (2.31). Moreover, J(uo) > 0, therefore it is a nontrivial solution. This completes the proof of the Theorem 2.4.6.

3

Regularity of Solutions

3.1

W2,r a priori Estimates 3.1.1

3.1.2 3,2

Newtonian Potentials Uniform Elliptic Equations

W2.v Regularity of Solutions 3.2.1

3.2.2 3.2.3

The Case p > 2 The Case 1 < p < 2 Other Useful Results Concerning the Existence, Uniqueness, and

Regularity 3.3

Regularity Lifting 3.3.1

3.3.2 3.3.3

3.3.4 3:3.5

Bootstraps Regularity Lifting by Contracting Operators Applications to PDEs Applications to Integral Equations Regularity Lifting by Combinations of Contracting and Shrinking

Operators 3.3.6 3.3.7

Applications to Integral Equations Applications to Fully Nonlinear Systems of Wolff Type

In the previous chapter, we used functional analysis, mainly the calculus of variations, to seek the existence of weak solutions for second order linear or semi-linear elliptic equations. These weak solutions we obtained were in the Sobolev space W1,2 and hence, by Sobolev embedding, in LP space for p < 2. Usually, the weak solutions are easier to obtain than the classical ones, and this is particularly true for non-linear equations. However, in practice, we

78

3 Regularity of Solutions

are often required to find classical solutions. Therefore, one would naturally want to know whether these weak solutions are actually differentiable, so that they can become classical ones. These questions will be answered here. In this chapter, we will introduce methods to show that, in most cases, a weak solution is in fact smooth. This is called the regularity argument. Very often, the regularity is equivalent to the a priori, estimate of solutions. To see the difference between the regularity and the a priori estimate, we take the equation

-Au = f (x) for example. Roughly speaking, the W2,P regularity theory implies that If u E W1"p is a weak solution and if f E LP, then u E W2,p.

While the W2'P a priori estimate says If u c Wo'p n W2,p is a weak solution and if f E LP, then IIUIIW=.n G CIIfIILP.

In the a priori estimate, we presumed that it is in W2'P. It might seem a bit surprising at this moment that the a priori estimates turn out to be powerful tools in deriving regularities. The readers will see in Section 3.2 how the a priori estimates and uniqueness of solutions lead to the regularity. Let I? be an open bounded set in Rn. Let aid (x), bi(x), and c(x) be bounded functions on .fl with aij (x) = aji(x). Consider the second order partial differential equation n

n

Lu :_ - E ai1(x)us;x; + E bi(x)uy, + c(x)u = f(x). i,)=1

(3.1)

i=1

We assume L is uniformly elliptic, that is, there exists a constant S > 0, such that n

E atj(x)ei£9 > ERn. In Section 3.1, we establish W2,p a priori estimate for the solutions of 6

for all

(3.1). We show that if the function f (x) is in L"(92), and it c W2'p(.f2) is a strong solution of (3.1), then there exists a constant C, such that IIuIIW?..P <- C(IIuIILP + IflILY).

We will first establish this for a Newtonian potential, then use the method of "frozen coefficients" to generalize it to uniformly elliptic equations.

3.1 W2'" a priori Estimates

79

In Section 3.2, we establish a W2,P regularity. We show that if f (x) is in LP(Q), then any weak solution u of (3.1) is in W2'P(.fl). In Section 3.3, we present two Regularity Lifting Theorems. The first The-

orem uses an operator which is contracting in both spaces. It is a simple method to boost the regularity of solutions and has been used extensively in various forms in the authors' previous works. The essence of the approach is well known in the analysis community, while the version here contains some new developments. It is much more general and very easy to use. We hope that the method will be helpful to both experts and non-experts in the field. The second Theorem employs an operator which is contracting in one space and shrinking in the other space. It is a brand new idea, and as we believe, it will find broad applications in a variety of nonlinear problems. We will use examples to show how these Theorems can be applied to systems of PDEs and integral equations including a fully nonlinear integral system of Wolff type.

3.1 W2" a priori Estimates We establish W2 'P estimate for the solutions of

Lu = f (x)

,

x E Q.

First we start with the Newtonian potential.

3.1.1 Newtonian Potentials Let fl be a bounded domain of R". Let 1

P(x) =

n - 3,

(n - 2)w"-1 IxIn_2

be the fundamental solution of the Laplace equation, where w"_1 is the area of the (n - 1)-dimensional unit sphere. The Newtonian potential is known as

w(x) = L P(x - y)f(y)dy. n

We prove the following:

Theorem 3.1.1 Let f C LP(.fl) for 1 < p < oo, and let to be the Newtonian potential of f. Then w e W2,P(fl) and

Lw = f (x), a.e. x E 0,

(3.3)

IID2WIIL1(n) <- Of 11L=(12)-

(3.4)

and

3 Regularity of Solutions

80

Write w = Nf, where N is obviously a linear operator. For fixed i, define the linear operator T as

Tf=D.,Nf=Di,R

nr(x-y)f(y)dy.

To prove Theorem 3.1.1, it is equivalent to show that

T : LP(Q) -* LP(Q) is a bounded linear operator .

(3.5)

Our proof can actually be applied to more general operators. To this end; we introduce the concept of weak type and strong type operators. Define

uf(t)=1 ix E12IIf

(x)I>t}I.

For p > 1, a weak LP space Lj(fl) is the collection of functions f that satisfy

IlfllL.(f) = sup{ttf(t)tP, Vt > o} < oo. An operator T : LP (Q) --* Lq(fl) is of strong type (p, q) if IITfIIL4(Q)

CIIfIILP(r),Vf E LP(.(2).

T is of weak type (p, q) if IITfIILw(.a)

CIIfIILP(n),Vf E L'(9)-

Outline of Proof of (3.5). We decompose the proof into the proofs of the following four lemmas. First, we use Fourier transform to show

Lemma 3.1.1 T : L2(S2) -4 L2(fl) is a bounded linear operator. i.e. T is of strong type (2,2). Secondly, we use Calderon-Zygmund's Decomposition Lemma to prove

Lemma 3.1.2 T is of weak type (1,1). Thirdly, we employ the Marcinkiewicz Interpolation Theorem to derive

Lemma 3.1.3 T is of strong type (r, r) for any 1 < r < 2. Finally, by duality, we conclude

Lemma 3.1.4 T is of strong type (p, p), for 1 < p < eo.

3.1 W2iD a priori. Estimates

81

The Proof of Lemma 3.1.1. First we consider f E Co (fl) C Co (R"). Then obviously w e C°°(R") and satisfies

Lw=f(x)

Let

VxER°.

,

1

f(f) _ (2rr)^.1R.

' f (x)dx

be the Fourier transform of f, where G x,e >_ E k

is the inner product of R", and i = vf--I. We use the following simple property of the transform

-i kf (0 , and Djk f (0

Dkf

and the well-known Plancherel Theorem: IIfIIL2(Rn) = IIfIIL=(R-).

It follows from (3.6) and (3.7),

f If(x)I2dx= S1

lf(x)I2dx

pR

= fRn Inw(x)I2dx fR Iow(e)I2de = fRn

Ef n

k1 R^

Iw(f)I2dd

n

IDkjw(e)I2d, k j_1 R^ R' n k,j=1

f R^ IDkjw(x)I2dx

ID2wl2dx.

Consequently,

3 Regularity of Solutions

82

IITfIIL=lal <_ IIfIIL2(Q)

,

of E 00'(S?)-

(3.8)

Now to verify (3.8) for any f E L2(.f2), we simply pick a sequence { fk} c

Co ((l) that converges to f in L2(fl), and take the limit. This proves the Lemma.

11

The Proof of Lemma 3.1.2. We need the following, well-known Calderdn-Zygmund Decomposition Lemma (See its proof in Appendix A.4).

Lemma 3.1.5 For f E Ll (R"), fixed a > 0, 3 E, G such that

(i)R"=EuG, EnG=O;

(ii) If (x) I< a, a. e. x c E; (iii) G = U Qk, {Qk}: disjoint cubes s.t. k=1

a<

IQ kI

If(x)Idx<2"a. Qk

For any f E L1(Q), to apply the Calderon-Zygmund Decomposition Lemma, we first extend f to vanish outside 1?. For any given a > 0, fix a large cube Qo in R", such that f@oIf(x)Idx
wrf(o)

I{x E R" I ITf(x)I ? a}I <

CIIfIIL=lal

a

Split the function f into the "good" part g and "bad" part b: f = g + b, where f(x) g(x)

@k f @ k f (x)dx

for x E E

for x E Qk, k = 1, 2, .. .

Since the operator T is linear, T f = Tg + Tb; and therefore /2Tf (a) C fLTg(a) + ATb(C ).

We will estimate PTg(2) and µTb(2) separately. The estimate of the first one is easy, because g E L2. To estimate µTb(z ), we divide R" into two parts: G` and E' := R" \ G' (See below for the precise definition of G'). We will show

that

3.1 W2'1 a priori Estimates

83

(a) IG`I 5 CIIf IIL=(a), and

a f.,. ITb(x)Idx <-

}1 <_

(b) I{x E E" I ITb(x)I ?

IIfIIL'(n)

These will imply the desired estimate for pm(2 ). Obviously, from the definition of g, we have Ig(x) I < 2"a , almost everywhere

(3.10)

and

b(x)=O for x E E, and f

= 0 for k = 1, 2,

.

(3.11)

Qk

We first estimate jaTg. By Lemma 3.1.1 and (3.10), we derive ATg(

,a

4

) :S

2

r J

9 2 (x)dx

<

2n+2

a

J 9(x) Idx <

2n+2

a

f If (X) IdX.

(3.12)

We then estimate PTe. Let bk(x)

J b(x) for x E Qk

l0

elsewhere .

Then

Tb =

Tbk. k=1

For each fixed k, let {bkm} C Co (Qk) be a sequence converging to bk in

L2(Q) satisfying bkm(x)dx =

bk(x)dx = 0.

(3.13)

JQk

ek f.

From the expression

Tbkm(x) = f D:j F(x - y)bkm(Y)dy, Qk

one can see that due to the singularity of D;jT (x - y) in Qk and the fact that bkm may not be bounded in Qk, one can only estimate Tbkm(X) when x is of positive distance away from Qk. For this reason, we cover Qk by a bigger ball Bk which has the same center as Qk, and the radius of the ball 6k is the same as the diameter of Qk. We now estimate the integral in the complement of Bk:

3 Regularity of Solutions

84

f

R^\Bk

ITbkml(x)dx =

f

11 Dijr(x - y)bkm(y)dyl dx

Qo\Bk

=j

Qk I

Qo\Bk

f [Dijr(x-y)-Dijr(x-y)]bkm(y)dyldx Qk

1



6k


(3.14)

Qk

where y is the center of the cube Qk. One small trick here is to add a term (which is 0 by (3.13)):

f

- y)bkm(y)dy

Qkk

to produce a helpful factor Sk by applying the mean value theorem to the

difference:

Dijr(x - y) - Dij r(x - y) _ (y - y) - D(Dij r)(x - ) < bk I D(Dij) (x - ) I Now letting m-foc in (3.14), we obtain

f

R^\Bk

Let

ITbk(x)ldx<
f bk(y)Idy Qk

00

G' = U Bk

and E* = R" \ G*.

k=1

It follows that

ITbkldx
f' ITb(x)ldx
:5 CE

f

k=1 Qk

Ibk(y)Idy
ITbkldx

k=1 R^\Bk

k=1 R^\G'

f

R^

If(x)ldx.

(3.15)

Obviously,

pTb(2) < IG*I + I{x E E' I Tb(x) > ')1. By (iii) in the Calderon-Zygmund Decomposition Lemma, we have

(3.16)

3.1 W2'' a priori Estimates 0o

k =1

f

00

co

Qkl-<

IG"I=yIBkI=C

°-E f If(x)Idx=C f a a k=1 Qk

k=1

85

If(x)Idx. (3.17)

Write

Then by (3.15), we derive 2


(3.18)

Now the desired inequality (3.9) is a direct consequence of (3.12), (3.16), (3.17, and (3.18). This completes the proof of the Lemma.

The proof of Lemma 3.1.3. In the previous lemmas, we have shown that the operator T is of weak type (1, 1) and strong type (2, 2) (of course also weak type (2, 2)). Now Lemma 3.1.3 is a direct consequence of the Marcinkiewicz interpolation theorem in the following restricted form:

Lemma 3.1.6 Let T be a linear operator from LP (fl) nLq(tl) into itself with 15 p < q < oo. If T is of weak type (p, p) and weak type (q, q), then for any p < r < q, T is of strong type (r, r). More precisely, if there exist constants BP and Bq, such that

prf(t)C

(BPItfIIP)P and

tTf(t)<(BgltfII)

VfELP(Q)nLq(Q),

then I

ITfIIr
where

,

of E LP(Q) nLq(9),

1

0

1-0

r

p

q

and C depends only on p, q, and r. The proof of this lemma can be found in Appendix A.4.

The proof of Lemma 3.1.4. From the previous lemma, we know that, for any 1 < r < 2, we have IIT9IILt(n) <-CrIIgIILr(n) Let

< f,9 >=

fn f(x)g(x)dx

be the duality between f and g. Then it is easy to verify that

(3.19)

86

3 Regularity of Solutions

< g, T f >_< Tg, f > . i.e. I

Given any 2 < p < oo, let r = p ,

(3.20)

+ r = 1. Obviously, 1 < r < 2. It

follows from (3.19) and (3.20) that IITfIILP =

= sup

sup

II90L-1

1911L'=1

<

IIfIILPIITSIIL-
SUP 1191ILr=1

This completes the proof of the Lemma.

3.1.2 Uniform Elliptic Equations In this section, we consider general second order uniform elliptic equations with Dirichlet boundary condition

Lu :_ - En=1 aij (x)u,; + E {u(x)=0,

1

bi (x)ua, + c(x)u = f (x), x E 1?

xE8f2. (3.21)

We assume that !? is bounded with C2 ° boundary, ao(x) E Co(t), bi(x) E c(x) E L°°(.(2), and AI6I2 < ai96i6J < A

and A.

Definition 3.1.1 We say that u is a strong solution of Lu = f (x), x E ,f2 if it is twice weakly differentiable in fl and satisfies the equation almost everywhere in ,(2.

Based on the result in the previous section-the estimates on the Newtonian potentials-we will establish a priori estimates on the strong solutions of (3.21).

Theorem 3.1.2 Let it E W2.p(.(2) n W01'2(Q) be a strong solution of (3.21). Assume that f E LP((?). Then IIuIIw=.P(n)

5C(IIuIILP(n)+IIfIILP(Q))

(3.22)

where C is a constant depending on Ilbi(x)IILW(.a), IIc(x)IIL-(n), A, A, n, p, and the domain 0.

3.1 W2'' a priori Estimates

87

gemark 3.1.1 Conditions bi(x),c(x) E L°°(,2)

can be replaced by the weaker ones

bi(x), c(x) E L"(Q)

,

for any p > n.

Proof of Theorem 3.1.2. We divide the proof into two major parts. Part 1. Interior Estimate 11o2UIILP(K) <_ C(IIvUIILP(n) + IIUIILP(n) + IIf I ILP(n))

(3.23)

where K is any compact subset of Q.

Part 2. Boundary Estimate

I IUIIW2,P(n ns)

C(IIVuIILP(n)+IIuIILP(n)+IIf IILP(n))

(3.24)

where S26 = {x E f21 d(x, 8,2) > 6}.

Combining the interior and the boundary estimates, we obtain

IIu1IW2.P(.n) S

C(IIVuIILP(n) +I IUIILP(O) +I If IILP(n)).

(3.25)

Theorem 3.1.2 is then a simple consequence of (3.25) and the following well-known Gargaliado-John-Nirenberg type interpolation estimate

IIVUIILP(n) S CIIuIIi (n)Ilo2ullL (n) S EI102uIILP(n) + 4EIIu11LP(n). Substituting this estimate of I IVUI I LP(n) into our inequality (3.25), we get: IIUIIW2.P(n) <_ CEIIV2uIILP(n) + C(4 IIUIILP(n) + IIIIILP(n))

Choosing e < 2C, we can then absorb the term CEIIV2UIILP(n) on the right hand side by the left hand side and arrive at the conclusion of our theorem IIu11W2,P(n) <_ C(IIUIILP(n)+IIfIILP(n)).

(3.26)

For both the interior and the boundary estimates, the main idea is the well-known method of "frozen coefficients." Locally, near a point x°, we may

88

3 Regularity of Solutions

regard the leading coefficients ai;(x) of the operator approximately as the

constant a;(x°) (as if the functions aij are frozen at point x°), and thus we are able to treat the operator as a constant coefficient one and apply the

potential estimates derived in the previous section. Part 1: Interior Estimates. First we define the cut-off function ¢(s) to be a compact supported C°° function:

(P(s)=1 ifs<1

1¢(s)=0 ifs>2. Then we quantify the `module' continuity of aij with E(a) =

1ai.i(x) -a15(y)I. sup jx-yj<6;x,yeg;1
The function e(6) ',, 0 as b ',, 0, and it measures the `module' continuity of the functions ai;. For any x° E Q25, let bx°I)

and w(x) = 71(x)u(x),

then

ai1(x°)aaax;

=

(aij( )-ai7(x))aaax; +ai5(x)axax. 2

2

= (ai.i(x')-aij(x))aai_x; +r(x)ai;(x)aaax + a2?t

x + a15 (x)u(x) axza,

On a71 + tai7 5 (x)

_xi ax', F

a02ax

(aii (x°) - ail (x))

.

+ r1(x)(bi(x) a2i + Cx)n - J (x))

-

_2771 all alt + ai;(x)u(x)x,ax +2aij(x)__ , ,

= F(x) for x E R.

In the above, we skipped the summation sign E. We abbreviated

ai;

as a and so on. Here we notice that all terms are supported in B23 B23(x°) C .2. With a simple linear transformation, we may assume aij(x°) _ di;. Apparently both to and

F* F:= n(n 1 2)cjn - JRn Ix - 1 are solutions of

yln

_2

F(y)dy

3.1 W2,P a Priori Estimates

89

Du=F. Thus, by uniqueness, w = F * F (see the exercise after the proof of the theorem). Now, following the Newtonian potential estimate (see Theorem 3.3), we obtain: II,72

II V2 WIILP(B26) =

(3.27)

WIILP(an) <_ CIIFIILP(nn) = CIIFIILP(B26)

Calculating term by term, we have c(28)IIo2wlILP(B26)+IIf IILP(B26)+C(IIVBIILP(B,6)+IIuIILP(B26)).

IIFIILP(B26) C

Combining this with the previous inequality (3.27) and choosing 8 so small such that Cc(28) < 1/2, we get 1102 wI I L°(B26)

Ce(26)II V2

5 21I o2

WIILP(B23) + C(I If I

wIILP026) +

ILP(B25) + I IVUI ILP(B,s) +

C(Ilfl!LP(B26) +

IIUIILP(B26))

IIVUIILP(B26) + IItIILP(B26))

This is equivalent to 11 ,72

WIILP(B26) <_

C(IIflILP(B26) + IIVUIILP(B26) + IIUIILP(B26))

Since it = w on B6(x°), we have V2WIILP(B6) II,72UIILP(B6) = II

Consequently, 11 '7'

UIILP(B6) <_ C(IIfllLP(B26) + IIVUIILP(B26) +

IIUI6(B26)).

(3.28)

To extend this estimate from a 8-ball to a compact set, we notice that the collection of balls {Bo(x) I X E .(22a}

forms an open covering of ,(120. Hence there are finitely many balls {Bo(xi) I i = 1, .......rn} that have already covered 1120. We now apply the above estimate (3.28) on each of the balls and sum up to obtain In

II 02 UIILP(Q26)

Ell V2 fhlLP(B6)(x1) i=1 In

p

>C( II J II

I Ioul

I Iul

ILP(B26)(x'))

i=1

< C(II f II LP({;) + I I VUI ILP(Q) + I IBI ILP(n))

(3.29)

90

3 Regularity of Solutions

For any compact subset K of ,fl, let 6 < 1dist(K, 8.(l); then, K C 1225, and we derive IIV2uhILr(K) _C(IIVUIILP(n)+IIrIILP(n)+II.fIIL=(n)).

(3.30)

This establishes the interior estimates.

Part 2. Boundary Estimates The boundary estimates are very similar. Let's just describe how we can

apply almost the same scheme here. For any point x° E 0(, B5(x°) n 8.2 is a C2,a graph for 6 small. With a suitable rotation, we may assume that Bs(x°) n812 is given by the graph of X. = h'(x1, X2,..., xn-1) = h(x ), and 12 is on top of this graph locally. Let

y = i(x) = (x - x°', xn - h(x )), then a/i is a diffeomorphism that maps a neighborhood of x° in (2 onto B,+(0) _ {y c B,(0) I yn > 0}. The equation becomes

ya -1 aii (y)'av:v, + z=1 bi(y)nv:

{u(y)=0, foryEBB,+(0)with

c(y) a =

1(y), for y E Br (0) (3.31)

where the new coefficients are computed from the original ones via the chain rule of differentiation. For example,

aij(y) =

8xt('

If necessary, we make a linear transformation so that ai5(0) = hip. Since a plane is still mapped to a plane, we may assume that equation (3.31) is valid for some smaller r. Applying the method of frozen coefficients (with vl).u(y)) we get: w(y) = 0(2 Aw(y) = F(y) on Br (0). Now let w(y) and F(y) be the odd extension of w(y) and F(y) from B,: (0) to B,.(0), i.e., 'D(y)=w(y1,...,Yn-1,yn)w(i,

Similarly for F(y). Then one can check that:

,, Yyni,Yn),

-Y.), if V, < 0.

3.2 W2'P Regularity of Solutions

91

Lw(y) = F(y) , for y E B,(0). Through the same calculations as in Part 1, we derive the basic interior estimate II 02 uIILP(B,(s°)) < C(II fII LP(B2,(s°)) + I I VUIILP(B,r(z°)) +

< Cl(IIfIILP(B,, (xo)nn) +

IIUIILP(B2r(z°)))

Ilnl ILP(n2r(so)nn))

for any x° on 82 and for some small radius r. The last inequality in the above was due to the symmetric extension of w to w from the half ball to the whole ball.

These balls form a covering of the compact set 81 and thus have a finite covering Br; (x') (i = 1, 2. ..... k). These balls also cover a neighborhood of 8.12

including .2\05 for some 5 small. Summing the estimates up, and following the same steps as we did for the interior estimates, we get:

IIUIIW2.P(n\na) < C(IIouIILP(n) +I U11LP(Q) +I If IILP(n)).

(3.32)

This establishes the boundary estimates and hence completes the proof of the theorem.

Exercise 3.1.1 Assume that B is bounded, and w E WW ''(Q). Show that if

-Aw = F, then w = F * F. Hint: (a) Show that r * F(x) = 0 for x 0 (2. (b) Show that it is true for w E Co (Rn). (c) Show that

A(JEW) = JE(Aw)

and thus

J,w = F * (JF). Let fi0 to derive w = F * F.

3.2 W2" Regularity of Solutions In this section, we will use the a priori estimates established in the previous section to derive the W2'p regularity for the weak solutions. Let L be a second order uniformly elliptic operator in divergence form n

n

Lu = -

bi(x)u1, + c(x)u.

(a,j (x)us,)y. +

i,j=1

i=1

(3.33)

3 Regularity of Solutions

92

Definition 3.2.1 We say that u E WJ'P(fl) is a weak solution of

Lu=f(x),xE0

{u(x)=0, xEafl

(3.34)

if for any t E- W01, (p),

I J

>bi(x)uxiv

Y

aij(x)ux,vi, +

+ c(x)uv dx = J f(x)vdx,

(3.35)

where 1 + q = 1.

Remark 3.2.1 Since Co (.2) is dense in WJ'P((l), we only need to require (3.35) be true for all v E Co (fl); and this is more convenient in many applications.

The main result of the section is this:

Theorem 3.2.1 Assume fl is a bounded domain in R'°. Let L be a uniformly elliptic operator defined in (3.33) with aij(x) Lipschitz continuous and bi(x) and c(x) bounded. Assume that u E WJ'P(Q) is a weak solution of (3.34). If f(x) e LP (S?), then u E W2'P(Q) .

Outline of the Proof. The proof of the theorem is quite complex and will be accomplished in two stages.

In stage 1, we first consider the case when p > 2, because one can rather easily show the uniqueness of the weak solutions by multiplying both sides of the equation by the solution itself and integrating by parts. As the reader will see, this uniqueness plays a key role in deriving the regularity. The proof of the regularity is based on the following fundamental proposition on the existence, uniqueness, and regularity for the solution of the Laplace equation on a unit ball.

Proposition 3.2.1 Assume f E LP(Bl(0)) with p > 2. Then the Dirichlet problem

J u = f (x) , x E Bl (0) l u(x) = 0 ,

x E aBr(0)

(3.36)

has a unique solution u E W2"P(Bt(0)) satisfying IIuIIW=.P(B,) «IIlIILP(B1)-

(3.37)

Regularity of Solutions

3.2

93

We will prove this proposition in subsection 3.2.1, and then use the "frozen coefficients" method to derive the regularity for general operator L.

In stage 2 (subsection 3.2.2), we consider the case 1 < p < 2. The main difficulty lies in the uniqueness of the weak solution. To show the uniqueness, we first prove a W2,' version of the Redholm Alternative for p > 2 based on the regularity result in stage 1, which will be used to deduce the existence of solutions for equation

- E(aii(x)ws:)x; = F(x). ij

Then we use this equation with a proper choice of F(x) to derive a contradiction if there exists a non-trivial solution of equation

- E(aij(x)vs,)s,

= 0.

23

After proving the uniqueness, to derive the regularity, everything else is the same as in stage 1. Remark 3.2.2 Actually, the conditions on the coefficients of L can be weakened. This will use the Regularity Lifting Theorem in Section 3.3, and hence will be elaborated there.

3.2.1 The Case p > 2 We will prove Proposition 3.2.1 and use it to derive the regularity of weak solutions for general operator L. To this end, we need

Lemma 3.2.1 (Better A Priori Estimates at Presence of Uniqueness.) Assume that

i) for solutions of

Lu = f (x) , x E ,2

(3.38)

lull W2(Q)
(3.39)

the a priori estimate

holds, and

ii) (uniqueness) if Lu = 0, then u = 0. Then we have a better a priori estimate for the solution of (3.38) IIuIIwa,P(n)
(3.40)

3 Regularity of Solutions

94

Proof. Suppose the inequality (3.40) is false, then there exists a sequence of functions {fk} with IIfkIILP = 1 and the corresponding sequence of solutions {uk} for Lak = fk(x) , x E l2 such that

IIukIIW2,pioo as k->oo.

It follows from the a priori estimate (3.39) that IIvkIILP-+00

as k-ioo.

Let Uk

and sk

Vk

_

IIvkIILP

fk II1k1ILP'

Then I I vk I I LP = 1 , and II9k iI LP --+.

(3.41)

Lvk=9k(x), xEQ

(3.42)

From the equation

the a priori estimate IIvkII W

<_

c(IIvkjILP + II9kiiLP)

(3.43)

holds. (3.41) and (3.43) imply that {vk} is bounded in W2.n, and hence there exists a subsequence (still denoted by {vk}) that converges weakly to

v in W2,n. By the compact Sobolev embedding, the same sequence converges strongly to v in LP, and hence IIvIILP = 1. From (3.42), we arrive at

Lv=0, xES2. Therefore by the uniqueness assumption, we must have v = 0. This contradicts with the fact IIvIILP = 1 and therefore completes the proof. 0

The Proof of Proposition 3.2.1. For convenience, we abbreviate B,.(0) as B,..

To see the uniqueness, assume that both u and v are solutions of (3.36). Let w = it - v. Then w weakly satisfies

xEBi lw(x)=0,xEBB1 Multiplying both sides of the equation by w and then integrating on B1, we derive

s,

Ipwl2dx = 0.

3.2 W2'P Regularity of Solutions

95

Hence Vw = 0 almost everywhere; this, together with the zero boundary condition, implies w = 0 almost everywhere. For the existence, it is well known that if f is continuous, then G(x,y)f(y)dy

u(x) = J

is the solution. Here

G(x, y) = r(x - y) - h(x, y) is Green's function, in which 1 (n-72T..- Islr , n > 3

r(x)

{2nInlxj

n=2

,

is the fundamental solution of the Laplace equation, as introduced in the definition of Newtonian Potentials, and

h(x,y) =

1

1

n(n - 2)wn (jxI I

- yU"-2

for n > 3

is a harmonic function. The addition of h(x, y) is to ensure that G(x, y) vanishes on the boundary. We have obtained the needed estimate on the first part fs r(x - y)f(y)dy ,

when we worked on the Newtonian Potentials. For the second part, noticing that h(x, y) is smooth when y is away from the boundary (see the exercise below), we start from a smaller ball B1_5 for each 6 > 0. Let U5(X) =

G(x,y)fs(y)dy, J B,

where

, x E Bt-5 Mx) = f(x) elsewhere . 0,

l

Now, by our result for the Newtonian Potentials, D2n5 is in L7(B1). Applying the Poincare inequality, we derive that u6 is also in LU(BI), and hence it is in W2,P(Bl). Moreover, due to uniqueness and Lemma 3.2.1, we have the better a priori estimate IIu6IIW2.P(Bl)

GIIfb IILP(B,).

(3.44)

Pick a sequence {6j} tending to zero, then the corresponding solutions

{ua;} is a Cauchy sequence in W2'P(B1) because

96

3 Regularity of Solutions

I ua;

- us,

CII fa;

- f6, IILP(B,)"0, as ',j-+00-

Let

U, _ lim ug, , in W2'P(Bi) z-*co

Then uo is a solution of (3.36). Moreover, from 3.44, we see that the better a priori estimate (3.37) holds for u°. This completes the proof of Proposition 3.2.1.

Exercise 3.2.1 Show that if I y I < 1 - 6 for some 6 > 0, then there exists a constant co > 0, such that

IxIII2 - yi ? co , Vx E Bi(0).

Proof of Regularity Theorem 3.2.1 for p > 2. The general framework of the proof is similar to that of the W2.P a priori estimate in the previous section. The key difference here is that in the a priori estimate, we pre-assume that it is a strong solution in W2'P, and here we only

assume that it is a weak solution in W1'P. After reading the proof of the a priori estimates, the reader may notice that if one can establish a W2,P regularity in a small neighborhood of each point in 0, then by an entirely similar argument as in obtaining the a priori estimates, one can derive the regularity on the whole of Q. As in the proof of the a priori estimates, we define the cut-off function

ifs<1 (s)- I1, 0, ifs>2. Let it be a WJ'P(fl) weak solution of (3.34). For any E X26 := {x E 12 dist(x, 80) > 26}, let 77(x)

IX

6

and w(x) = r)(x)u(x).

Then w is supported in B25(x°). By (3.35) and a straightforward calculation, one can verify that, for any v E Co (B2s(x°))

JB26(.,) aij(x°)wx.vx;dx= JB2a(x°)

[aij(x')-aij(x)]wsivs,dx+426

where F(x) = f(x) - (aij(x)rlx,u)x, - bi(x)ux, - c(x)u(x).

Fvdx (X')

3.2 W2'p Regularity of Solutions

97

In order words, to is a weak solution of

aij(x°)wx,xj = ([aij(x°) - aij(x)]wx,)xj - F(x) , x E B25(x°)

{ w(x) = 0,

x E 8B28(x°).

(3.45)

In the above, we omitted the summation signs E. We abbreviated E 'j=1 ail as aij and so on. With a change of coordinates, we may assume that aij (x°) = 8ij and write equation (3.45) as ,n,w = ([aij(

) - aij(x)]wx,)Xi - F(x) (3.46)

= [aij(x°) - aij(x)]wx,xj - F(x) where

F(x) = [aij(x)]xjwx, + F(x). For any v E W2,P(B2s(x°)), obviously,

[aij(

)- aij(x)]vx,xj E LP(B25(X'))

Also, one can easily verify that F(x) is in LP(B2s(x°)). By virtue of Proposition 3.2.1, the operator L is invertible. Consider the equation in W2''

v = Kv + L-' ,P

(3.47)

where

Q-1 (Kv) (x) = l [aij (x°) - aij (x)]vs.xj Under the assumption that aij (x) are Lipschitz continuous, one can verify that (see the exercise below), for sufficiently small 6, K is a contracting map from W2,P(B2s(x°)) to itself. Therefore, there exists a unique solution v of equation (3.47). This v is also a weak solution of equation (3.46). Similar to the proof of Proposition 3.2.1, one can show (as an exercise) the uniqueness of the weak solution of (3.46). Therefore, we must have to = v and thus conclude that w is also in W2,p(B26(x°)). This completes stage 1 of the proof of Theorem 3.2.1. 1.

Exercise 3.2.2 Assume that each aij (x) is Lipschitz continuous. Let (Kv) (x) =

1 L

[aij (x°) - aij (x)]vx.xj }

,

x E B2a (X').

Show that, for 6 sufficiently small, the operator K is a contracting map in W ,P(B25(x°)), i.e., there exists a constant 0 < y < 1, such that IIKO-KOIIw2.P(B26(x°)) <-7110

Exercise 3.2.3 Prove the uniqueness of the W1'P(B2b(x°)) weak solution for equation (3.46) when p > 2.

98

3 Regularity of Solutions

3.2.2 The Case 1 < p < 2. Lemma 3.2.2 Let p > 2. If u = 0 whenever Lu = 0 (in the sense of K' (P) weak solution), then for any f E LP(fl), there exists a unique u E WJ"(f2) r W2'p(fl), such that

Lu = f(x) and IIu1Iw2.n(n)
(3.48)

Proof. This is actually a W2'' version of the Fredholm Alternative. From the previous chapter, one can see that for a sufficiently large, by the Lax-Milgram Theorem, for any given g E L2(fl), there exists a unique Wp'2(12) weak solution v of the equation

(L+a)v=g, xEfl. From the Regularity Theorem in this chapter, we have v E W2'2(fl). Therefore, we can write v = (L + a)-'g

where (L+a) ' is a bounded linear operator from L2 (Q) to

W2'2(Q).

Let { fk} C Co' (fl) be a sequence of smooth approximations of f (x). For each k, consider equation

(L + a)w - aw = fk,

(3.49)

or equivalently,

(I - K)w = F, (3.50) where I is the identity operator, K = a(L + a)-', and F = (L + a)-' fk.

One can see that K is a compact operator from W. 1,2(S?) into itself because of the compact embedding of W2'2 into W''2. Now we can apply the FYedholm Alternative: Equation (3.50) possesses a unique solution if and only if the corresponding homogeneous equation (I - K)w = 0 has only trivial solution. The secoc ; half of the above is just the assumption of the theorem, hence. equation (3.50) possesses a unique solution Uk in Wo'2(fl), i.e.

uk - a(L + a)-'uk = (L + a)-' fk.

(3.51)

Furthermore, since both a(L + a) -'uk and (L + a)-' fk are in W2'2(Q), we derive immediately that uk is also in W2'2(Q), or we can deduce this from the Regularity Theorem 3.2.1. Since fk is in LP(fl) for any p, we conclude from Theorem 3.2.1 that uk is in W2'p(fl), and furthermore, due to uniqueness, we have the improved a priori estimate (see Lemma 3.2.1)

3.2 W21P Regularity of Solutions

99

k=1,2,.

IIuk1IW2.P
Moreover, for any integers i and j,

L(uj - uj) = fi (x) - fj (x) , x E Q. It follows that

Ilea - UjIIW-,= < CIIft-f,IILP-+0, asi,y-+oo. This implies that {uk} is a Cauchy sequence in W2' and hence it converges to an element u in W2°P(Q). Now this function u is the desired solution.

proposition 3.2.2 (Uniqueness of Wo1" Weak Solution for 1 < p < oo. ) Let fl be a bounded domain in R". Assume that aij(x) are Lipschitz continuous. If a is a W0i'P((2) weak solution of

(aij(x)ux,)., = 0,

(3.52)

then u = 0 almost everywhere.

Proof. When p > 2, we have proved the uniqueness. Now assume 1 < p < 2. Suppose there exists a non-trivial weak solution u. Let {uk} be a sequence of Co (0) functions such that

Uk-fl in W(, "(.fl)

,

as k-ioo.

For each k, consider the equation IVUklp/q'7U,k

(aij(x)v.,)xj = Fk(x) := 0

1+IVUkl2

(3.53)

where 1+P1 =1. Obviously, for p < 2, q is greater than 2, and Fk(x) is in Lq(Q). Since we already have uniqueness for W.1" weak solution, Lemma 3.2.2 guarantees the existence of a solution vk for equation (3.53). Through integration by parts several times, one can verify that 0

J vk(aij(x)us.).,dx= n

n

1+Ivukl2

(3.54)

Since uk-#u in W''P, it is easy to see that IDUkIPIgvUk

1+Ivuk12

IVUIP/qVU

1+IVUI2

in Lq(fl), and therefore, by (3.54, we must have vu = 0 almost everywhere. Taking into account that u E W0'P(0), we finally deduce that u = 0 almost everywhere. This completes the proof of the proposition.

100

3 Regularity of Solutions

After proving the uniqueness, the rest of the arguments are entirely the same as in stage 1. This completes stage 2 of the proof for Theorem 3.2.1-

3.2.3 Other Useful Results Concerning the Existence, Uniqueness, and Regularity Theorem 3.2.2 Given any pair p, q > 1 and f E L9(Il), if u E WW'P (.fl) is a weak solution of

Lu = f (x) , x E ,(l, then u E WZ^Q(fl) and the a priori estimate

(3.55)

(3.56)

Ilullw2,a(n) <_ C(QJufLq + IfIIL9)

holds.

Proof. Case i) If q < p, then obviously u is also a Wo'4(fl) weak solution, and the results follow from the Regularity Theorem 3.2.1 and the W2'' a priori estimates. Case ii) If q > p, then we use the Sobolev embedding w2,r `--) Wr,pj with

np Pr =

n-p

.

If pl > q, we are done. If pr < q, we use the fact that u is a W1,"' weak solution, and by Regularity, u E W2'"'. Repeating this process, after a few steps, we will arrive at a Pk > q. El From this Theorem, we can derive immediately the equivalence between the uniqueness of weak solutions in any two spaces Wo'p and W0I'4:

Corollary 3.2.1 For any pair p, q > 1, the following are equivalent i) If u (= Wo'"(fl) is a weak solution of Lu = 0, then u = 0. ii) If u e WJ'°(fl) is a weak solution of Lu = 0, then u = 0. Theorem 3.2.3 (W2. p Version of the Fredholm Alternative). Let 1 < p < co. If u = 0 whenever Lu = 0 (in the sense of Wo'"(0) weak solution), then for any f E L"(2), there exist a unique u E Wo'"(,fl) fl W2'"(2), such that

Lu = f(x) and IIuIIwa.P(n) <

CIIfIILP.

For 2 < p < no, we have stated this theorem in Lemma 3.2.2. Now, for 1 < p < 2, after we obtained the W2," regularity in this case, the proof of this theorem is entirely the same as for Lemma 3.2.2.

3.3 Regularity Lifting

101

3.3 Regularity Lifting 3.3.1 Bootstrap Assume that u E H1(fl) is a weak solution of -emu = UP (X) , x E P.

(3.57)

Then by Sobolev Embedding, u E L-% (,(2). If the power p is less than the critical number n±2, then the regularity of u can be enhanced repeatedly through the equation until we reach that u E C'(Q). Finally the "Schauder Estimate" will lift it to C2+°(.2) and hence be a classical solution. We call this the "Bootstrap Method" as will be illustrated below. For each fixed p < n+2, write

n+2 P=,a-2-6 for some 6 > 0. zn

Since it E L^ 2((l), UP

E LP

nRa (Q) = L nn -+2T--Z,---- ]

The equation (3.57) boosts the solution it to W2,q, (?) with 2n

qr

= (n + 2) - 6(n - 2)'

By the Sobolev Embedding

w2,, ({l)"L*(2), we have it E Ls' (fl) with

sl

nql

n-2q1

2n

1

n-21-6

The integrable power of it has been amplified 118a (> 1) times. Now, UP E LQ2 (1?) with

q2=

s1

p

2n

= (1-6)[n+2-6(n-2)]'

Hence, through the equation, we derive it E W2.a2 (0). By Sobolev embedding, if 2Q2 > n, i.e., if

(1 -6)[n+2-6(n- 2)] - 4 <0,

3 Regularity of Solutions

102

then we have either it E L9(Q) for any q or u E C°(fl). We are done. If 2q2 < n, then it E L12 (fl), with 2n 82

(1-6)[n+2-6(n-2)]-4'

It is easy to verify that 82?

2n

1

1

-6)(n-2)1-6-s11-5'

The integrable power of u has again been amplified at least 11,a times. Continuing this process, after finite many steps, we will boost u to L9012) for any q, and hence in C°(Q) for some 0 < a < 1. Finally, by the Schauder estimate, u E C2+°(Q) and is therefore a classical solution. Actually, after repeated application of Schauder estimates, we derive that u E C°°(,fl).

From the above argument, one can see that if p = n±z, the so called "critical power", then 6 = 0, and the integrable power of the solution u cannot be boosted this way. To deal with this situation, we introduce a "Regularity Lifting Method" in the next subsection.

3.3.2 Regularity Lifting by Contracting Operators Here we present a simple method to boost regularity of solutions. It has been used extensively in various forms in the authors' previous works. The essence of the approach is well known in the analysis community. However, the version we present here contains some new developments. It is much more general and is very easy to use. We believe that our method will provide a convenient means of obtaining regularities and will be useful to both experts and novices in the field. Essentially, it is based on the following Regularity Lifting Theorem. Let V be a Hausdorff topological vector space. Suppose there are two

extended norms (i.e,. the norm of an element in V might be infinity) defined on V,

II-IIY:V-4[0,00]. Let

X := {vEV:IIvIIx
Theorem 3.3.1 (Regularity Lifting I). Let T be a contraction map from X into itself and from Y into itself. Assume that f E X and that there exists a function g E Z := X fl Y such that f = T f + g in X. Then f also belongs to Z.

3.3 Regularity Lifting

103

proof. Step 1. Let 112X + 11 II

IF

be a norm on Z. We first show that T : Z -> Z is a contraction. Since T is a contraction on X, there exists a constant 01, 0 < 01 < 1 such that IITh1- Th2IIx <- 01IIh1 - h2llx, V h1, h2 E X.

Similarly, we can find a constant 02, 0 < 02 < 1 such that

IIThr -Th2IIr <- 02IIh1-h2IIr, Vh1,h2 E Y. Let 0 = max{01, 02}. Then, for any hl, h2 E Z,

IIThi - Th2IIz

Th2IIX + IIThi - Th2II;-

<

B111h1-h2IIr+0 IIhi-h2IIr

< 0llhl - h2llz.

Step 2. Since T : Z --s Z is a contraction, given g E Z, we can find a solution h E Z such that h = Th + g. Notice that since T : X -> X is also a contraction and g E Z C X, the solution of the equation x = Tx + g must be unique in X. Because both h and f are solutions of the same equation

x=Tx+g in X, we deduce that f=hEZ. Remark 3.3.1 In practice, we usually choose V to be the space of distributions, and X and Y to be function spaces, for instance, X = 17(Q) and Y = W1-4(Q). We start from a function fin a lower regularity space X; if we can show that T is a contraction from X to itself and from Y to itself, then we can lift the regularity off to be in Z = X fl Y = LP(fl) n W1,9(P).

3.3.3 Applications to PDEs We now explain how the "Regularity Lifting Theorem" proved in the previous subsection can be used to boost the regularity of weak solutions involving the critical exponent: -Au=u142

.

(3.58)

Continue to assume that 1? is a smooth bounded domain in R° with n > 3. Let it E Ho (.fl) be a weak solution of equation (3.58). Then by Sobolev embedding, it E L.22 (f2).

We can split the right hand side of (3.58) into two parts:

3 Regularity of Solutions

104

un =u^-au:=a(x)u. n+2

4

Then obviously, a(x) E L = (0). Hence, more generally, we consider the regularity of the weak solution of the following equation

-Au = a(x)u + b(x).

(3.59)

Theorem 3.3.2 Assume that a(x), b(x) E L= (0). Let is be any weak solution of equation (9.59) in Ho ((2). Then is is in LP(Q) for any 1 < p < oo.

Remark 3.3.2 Even in the best situation when a(x) -- 0, we can only have u E W24_ (.f2) via the W2,P estimate, and hence derive that u E LP(12) for any

p < oo by Sobolev embedding.

Let us now revisit nonlinear equation (3.58). Assume that u is a Ho(Q) weak solution. From the above theorem, we first conclude that u is in LQ(f2) (f2). for any 1 < q < oc. Then by our Regularity Theorem 3.2.1, is is in This implies that u E Cl'°((2) via Sobolev embedding. Finally, by repeated application of Schauder estimates, we derive that u E C°°(.(2). Corollary 3.3.1 If is is a Ho ((2) weak solution of equation (3.58), then u E C°° (Q) and hence is a classical solution.

Proof of Theorem 3.3.2. Let G(x, y) be Green's function of -0 in fl. Define (Tf)(x) = (-L) -'f (x) =

Jn

G(x,y)f(y)dy

Then obviously

Cn

0
Ix-In-2r

and it follows that, for any q E (1, 2), IITf1IL

9(Q) s Ilr*fIIL

IRnI

<-GIIJIIL4cal

Here, we may extend f to be zero outside d2 when necessary. For a positive number A, define

if la(x)I > A aA(x) _ { a(x) 0 otherwise , and aB(x) = a(x) - aA(x).

3.3 Regularity Lifting

105

Let (TAV)(x) = f C(x,y)aA(y)v(y)dy

n

Then equation (3.59) can be written as u(x) = (TAU)(x) + FA(x),

where

FA (x) =

Jn

C(x, y)(as(y)u(y) + b(y))dy

We will show that, for any 1 < p < oc, i) TA is a contracting operator from LP(fl) to LP(fl) for A large, and ii) FA(x) is in LP(Q). Then by the "Regularity Lifting Theorem", we derive immediately that V. E LP(fl).

i) The Estimate of the Operator TA.

For any =2 < p < oo, there is a q, 1 < q < n, such that nq

n-2q

p

and then by the Holder inequality, for any v E LP(fl) we derive IITAVIILP(n)

IIF*aAVIILP(R^) < CIIaAVIILq(n)

CIIaAIILI(,{t)IIvIILP(n).

Since a(x) E L'2 (0), one can choose a large number A, such that IIaAIIL1(0)

-2

and hence arrive at 1 IITAVIILP(n)

2 IIVIILP(n)-

That is, TA : LP(.(l)-+LP(fl) is a contracting operator. ii) The Integrability of FA(x). (a) First consider

C(x,y)b(y)dy

Fa(x):= J

For any =2 < p < oo, choose 1 < q < n, such that p =

.

Extending b(x)

to be zero outside $2, we have IIFAIILP(n) <- IIF * bIILP(R^) 5 CIIbJILq(n)

CIIbIILI(n) < 00.

106

3 Regularity of Solutions

Hence FA (x) e LP (S2).

(b) Then estimate FA(x) =

G(x,y)aB(y)U(y)dy-

J

By the boundedness of aB(x), we have IIFAIILP(n) <- IIUBUIILc(rt) <- CIIuIIL9(n).

Noticing that u E LQ(Q) for any 1 < q < p

=

2n

n-6

2n , and n-2

when q =

2n

n-2

we conclude that, for the following values of p and dimension n,

16, FA(x) E LP (S?), and hence TA is a contracting operator from LP(Q) to LP(Q).

Applying the Regularity Lifting Theorem, we arrive at

ju E LP(Q), for any 1 < p < oo

if 3 < n < 6

luELP(12), forany16.

Now the only thing that prevents us from reaching the full range of p is (0), the estimate of FA (x). However, from the starting point where u E L we have arrived at u E LP(Q) ,

for any l

2n

n-6

.

Now from this point on, through an entirely similar argument as above, we will reach

(uELP(,f2), foranyl10. At each step, we add 4 to the dimension n for which holds

uELP(5), 1
3.3 Regularity Lifting

107

We now use a similar idea and the Regularity Lifting Theorem to weaken the conditions in our Regularity Theorem (See Section 3.2). Let L be a second order uniformly elliptic operator in divergence form n

Lu = -

(aij(x)u:)y; +I: bi(x)ux: +c(x)u. j=1

(3.60)

i=1

We will prove the following:

Theorem 3.3.3 (W2.p Regularity under Weaker Conditions) Let 1? be a bounded domain in R" and 1 < p < oo. Assume that i) aij(x) E W1,n+b(,fl) for some 6 > 0; ii)

( L"(.fl),

bi(x)E

if 1 < p < n

Ln+3(Q), ifp=n LP (0),

ifn
i if 1 < p < n/2 Ln/2+1(12), if p = n/2 L"/2 (,f2), C(x) E

ifn/2
LP((2),

Suppose f E LP(Q) and it is a Wo'P(,f1) weak solution of

Lu = f (x) x E Q.

(3.61)

Then it is in W2,p(Q).

Proof. Write equation (3.61) as - E (ai7(x)u.i),, -

ij=1

- (Ebxu+ i=1

Let

LOu:

-

c(x)u + f(x).

"

(air(x)u=.)z;. i,r=1

By Proposition 3.2.2, we know that equation Lou = 0 has only trivial solutions, and it follows from Theorem 3.2.3 that operator LO is invertible. Denote it by L. 1. Then LO 1 is a bounded linear operator from LP (0) to W2,p(,(2).

For each i and each positive number A, define

108

3 Regularity of Solutions

bi(x)

biA(x)

,

A

if

elsewhere

.

Let

bi(x) - biA(x).

biB(x)

Similarly for CA(x) and CB(x). Now we can rewrite equation (3.61) as U = TAU + FA(U, x)

(3.62)

where TAIL = -Lo 1

\Y

biA(x)Uxj + CA(x)U

l\ and

FA (u, x) = -Lo r

(b2(x)u1 + cB (x)u) + Lo If (x)

We first show that

FA(u -) E W2,p((2), Vu E W"n(Q), f E 11(Q).

(3.63)

Since biB(x) and ciB(x) are bounded, one can easily verify that

(Eb3(x)u.

+ CB(x)U I E Lp(12)

/)

for any U E W',P(fl). This implies (3.63). Then we prove that TA is a contracting operator from W2'p (.f2) to W2,p (12).

In fact, by the Holder inequality and by Sobolev embedding from W2'p to W14, one can see that for any v E W2.p(.f2), IbiADvIILP < C.

if l.

if n

IbiAIILp and

IICAIIL.I2IIVIIW2,r

IICAVIILP
,

if 1

IIcAILn/2+6IIVIIw2,p, ifP=n/2 I CA I I LP I I V I I w2,P ,

(3.65)

if n/2 < P< oo.

Under the conditions of the theorem, the first norms on the right hand side of (3.64) and (3.65), i.e., IIbiAIILn, IICAIILn/2 can be made arbitrarily small if A is sufficiently large. Hence for any e > 0, there exists an A > 0, such that

3.3 Regularity Lifting

109

II b;A(x)vy, +CA(x)VIILP < EIIvIIw2.n , Vv E W2,P(2).

Taking into account that Lo 1 is a bounded operator from LP((?) to W2' (fl), we deduce that TA is a contract mapping from W2'P(.f2) to W2'P(d2). It follows from the Contract Mapping Theorem that there exists v E W2'P(0) such that v = TAV + FA(u,x).

Now u and v satisfy the same equation. By uniqueness, we must have u = v. Therefore we derive that u E

This completes the proof of the

theorem.

3.3.4 Applications to Integral Equations As another application of Regularity Lifting Theorem I, we consider the following integral equation

j

dy , z u(x) = a^ x where a is any real number satisfying 0 < a < n.

E Rn,

(3.66)

It arises as an Euler-Lagrange equation for a functional under a constraint in the context of the Hardy-Littlewood-Sobolev inequalities. It is also closely related to the following family of semi-linear partial differential equations (_Q)a/2u = a(n+a)/(n-a) , x E R" n > 3.

(3.67)

Actually, one can prove (See [CLOD) that the integral equation (3.66) and the PDE (3.67) are equivalent. In the special case a = 2, (3.67) becomes

-tin = u(n+2)/(n-2) x, E Rn. which is the well known Yamabe equation.

In the context of the Hardy-Littlewood-Sobolev inequality, it is natural to start with it c Ln' (R n). We will use the Regularity Lifting Theorem to boost it to LQ(Rn) for any 1 < q < co, and hence in L°°(RT). More generally, we consider the following equation

u(x)= fRnK(y)Iu(y)IP-1u(y)dy J

for some 1 < p < co. We prove

x - yln-a

(3.68)

3 Regularity of Solutions

110

Theorem 3.3.4 Let u be a solution of (3.68). Assume that n u E Lq° (R") for some q° > ,

n-a

(3.69)

IK(y)Iu(y)Ip llady
Then it is in Lq(R") for any 1 < q < co.

Remark 3.3.3 i) If

p> n

-a

,

IK(x)I < M, and u E L a (R"),

(3.71)

then conditions (3.69) and (3.70) are satisfied. ii) The last condition in (3.71) is somewhat sharp in the sense that if it is violated, then equation (3.68) when K(x) - 1 possesses singular solutions such as

c

u(x) =

IxI

Proof of Theorem 3.3.4: Define the linear operator

f K(y)Iv(y)IP-wdy yI"-a JRT

Ix -

For any real number a > 0, define

f u"(x) = u(x) l ua(x) = 0 ,

,

if Iu(x)I > a, or if Ixl > a otherwise

.

Let ub(x) = u(x) - ua(x).

Then since it satisfies equation (3.68), one can verify that ua satisfies the equation U. = Tuaua + g(x)

(3.72)

with the function g(x) =

f JRn

K(y)Iub(y)IP-lub(y)dy

Ix - yI"-o

- ub(x)-

Under the second part of the condition (3.70), it is obvious that g(x) E L°° fl Lq°.

For any q > "n to obtain

we first apply the Hardy-Littlewood-Sobolev inequality

3.3 Regularity Lifting

IITu°tIILq
111

(3.73)

Write H(x) = K(x)Iua(x)Ip-' and apply the Holder inequality to the right hand side of the above inequality +a, ^

<

HU(y)w(y)°°dy}

(R"

.r.dy)

IH(y)I

I

f

\ R^

"9

Iw(y)I

sdy)

/

e

J

^IIwIILa.

(fR^ H(y)I °dy)

(3.74)

Here we have chosen

s_n+aq andr=n+aq n aq It follows from (3.73) and (3.74) that IIT',..wIIt4
(f IK(y)Iua(y)I'-'I° dyIIwIILa.

(3.75)

By (3.70) and (3.75), we deduce, for sufficiently large a, IITu°WIILa 5 2IIrIILa.

(3.76)

Applying (3.76) to both the case q = qo and the case q = po > q0, and by the Contracting Mapping Theorem, we see that the equation

w=T,."w+9(x)

(3.77)

has a unique solution in both Loo and L'° A L9-. From (3.72), ua is a solution of (3.77) in LQ°. Let w be the solution of (3.77) in L" n LQ°; then w is also a solution in Loo. By uniqueness, we must have ua = w c LP, fl LQ° for any pu > n '.. So does u E L"° fl L9-. This completes the proof of the Theorem.

0 3.3.5 Regularity Lifting by Combinations of Contracting and Shrinking Operators In the previous subsections, we introduced a Regularity Lifting Theorem and its applications, in which, in order to lift the regularity of a solution from a

3 Regularity of Solutions

112

lower to a higher space (in terms of regularity), we required that operator T be contracting in both spaces. However; for a nonlinear operator in certain spaces, it is sometimes very difficult, if not impossible, to prove it to be contracting-although one may still be able to show that it is "shrinking". Here we introduce

a more general theorem , namely, one which requires that the operator be contracting in one space but only "shrinking" in the other. We believe this theorem will find increasing application in many other situations in nonlinear analysis. Let V be a Hausdorff topological vector space. Suppose there are two

extended norms (i.e., the norm of an element in V might be infinity) defined on V, II

- IIx,

Let

X:={vE V:IIvIIx
Definition 3.3.1 ( `Y-pair") The pair of spaces (X, Y) described above is called an "XY-pair", if whenever the sequence and IIu,IIy < C will imply u E Y.

C X with u -> u in X

In practice, we usually choose V to be the space of distributions and X, Y to be the function spaces, such as IP spaces, Holder spaces, Sobolev spaces, and so forth. There are many commonly used function spaces that are "XY-pairs," as will be illustrated by some examples at the end of this section.

Theorem 3.3.5 (Regularity Lifting II.) Suppose Banach spaces X, Y are an "XY-pair", both contained in some larger topological space V satisfying properties described above. Let X and T be closed subsets of X and Y respectively. Suppose T : X -> X is a contraction:

IITf - TgQIx <_nnIIf-gIIx, df,gEX and for some 0
(3.78)

and T : 2) -> Y is shrinking: IITgIIy <_ OIIgIIy, Vg E 2J, and for some 0 < 0 < 1.

(3.79)

Define

Sf =Tf +F for some F E tf1T. Moreover, assume that S : X f123-*1

Then there exists a unique solution u of equation

(3.80)

3.3 Regularity Lifting

113

u=Tu+F in 3r, and more importantly,

uEy

proof. Define

fo=F

and

f"=Sf"-r=Tfn-1+F, n=1,2,..

Condition (3.80) guarantees that {fn} C X fl T. Since T a contraction in I, {f"} is a Cauchy sequence in X and

f"-*uEI, asn->oo. Actually, u is the unique solution of equation it = To + F in X. Furthermore, by induction, it is easy to show

IIf"IIY s

1 aIIFIIY

1

IIfolly=IFIIY <_ IIfn.-jIIY <_

IIfnIIY <_ IITfn-iJIY + IIFIIY <_ 0

i BIIFIIr 1

1 1

IIF'IIY,

(1 10IIFIIY) + IIFIIY = 10IIFIIY 1

Hence { f"} is bounded in Y. Now by the property of "XY-pair", we arrive at

uEY. Remark 3.3.4 In some situations, one can choose I = X and 2) = Y. In practice, if one knows that a solution it of it = Su belongs to X (usually with lower regularity), then by Theorem 3.3.5, one can lift the regularity of it up to it E X n Y (with higher regularity).

Remark 3.3.5 "XY-pairs" are quite common, as one can see from the following examples.

X = IY(U) for 1 < p < oo, Y = C°,°(U) for 0 < a < 1, and V is the space of distributions. Here U can be any subset of R° or R" itself.

3 Regularity of Solutions

114

X is a Banach space, Y is a reflexive Banach space, and both are in some bigger topological space, V. Of course, we assume V is Hausdorff and has topology weaker than the topology of X and the weak topology of Y. Then for any un -3 u E X and IlunIb' < C, we have u E Y. Notice that all Hilbert spaces, such as L2, Hl, and H2, are reflexive Banach spaces.

Proof. Case (i): Let

be a sequence such that

u,-4u E L" (U) and IIUnhICo,a(U) < M.

We first consider the restriction on any compact set K C U. Since is uniformly bounded in the Holder space Y, by the Arzela-Ascoli Theorem, there exists a subsequence converging uniformly to some function v. And since

u -4 u in L1(K), we have v = u in LP(K). Moreover, by

Iu(x)I <_ Iu(z) - n.(x)I + Iun(x)I and

u(x)-u(y)1 <- Iu(z)-u(x)I+Iun(x)-un(y)I+Iun(y)-u(y)I, we can choose it large enough, so that

Iu(x)I51+IIunhIL-(U) and

U(X)-u(y)1 <2+suplu"(X)-ua(y)

x - yIIx - yI

It follows that IIUIIcO.a(K) <_ M + 3.

Notice that the upper bound is independent of K, and we arrive at u E Ca-°(U). Therefore,

Case (ii): Suppose u,-+u in X ar is bounded in Y. Since Y is reflexive, there exists a weakly convergent subsequence u., - v for some v c Y. Since the topology of V is weaker than the topology of X and weak topology of Y, we have in V and in V. Since V is Hausdorff,

u=vEY.

Remark 3.3.6 There are counterexamples which fail to be an "XY-pair.' For example,

X =Lr[0,1] and Y=C[0,1].

3.3 Regularity Lifting

115

Define

un(x)=0

for0<x<---, un(x)=1 for-+-<x<1

and linearly on 1

[2

1

1

1

n+it

Then although {un} converges in X and is bounded in Y, it fails to converge

inY=C[0,1]. 3.3.6 Applications to Integral Equations Now, as a simple example to illustrate the idea, we apply the Regularity Lifting Theorem II to the following integral equation

u(x) = L

°u'(y)dy, x E

ix

R'

(3.81)

1

with r > n=a. We will lift the regularity of any given positive solution from Lw(Rn) to C°"1(Rn)-the space of Lipschitz continuous functions. Actually, with greater deliberation, the authors and Chao Ma [CLM] have applied it to obtain Lipschitz continuity for positive solutions of much more complicated fully nonlinear systems as indicated in the next subsection. In the subsection preceding the previous, we used the Regularity Lifting by Contracting Operators to show that every positive solution u of (3.81) is in L7(Rn) for any p > nn-. From here, one can derive immediately that

Theorem 3.3.6 Every positive solution of equation (3.81) is uniformly bounded in Rn.

Proof. Divide the integral into two parts u(x)

= J B, (z) Ix - yIn-°

u' (y)dy +

1

JRn\B, (z) I x - YIn-IX

u' (y)dy

I1+I2. Use the Holder inequality on the first part

r/ Il < I /

1/q

11/p

1n_a)pdy1

u74(y)dyI

[Ll(X)

(3.82)

with 1/p + 1/q = 1. Choosing 1 < p < n=;, both integrals in (3.82) are uniformly bounded.

For the second part, we simply have

116

3 Regularity of Solutions

uT(y)dy <_

12 <_ f

f ur(y)dy < C.

R^\Bi(x)

R

Therefore, u is uniformly bounded. Based on Theorem 3.3.6, we will use Regularity Lifting Theorem II to lift the regularity of u from L°° (R") to C°^r (R"), the space of Lipschitz continuous functions with norm IIVIIC0.1(R^) = IIvIIL-(R^) +Sup =OY{

Iv(x) - v(y)I

Ik - yI

Theorem 3.3.7 Every positive solution u of equation (3.81) is Lipschitz continuous.

Proof. By elementary calculus, one can verify that

r

tO

r0

fst(x) uT

(y)dy t"

.+i

=

L

yl Fn---+l

= (n - CO j Ix

uT (y)dy

1 yluT(y)dy

It follows that the solution of (3.81) only differs by a constant multiple from the solution of the following equation: °°

u(x)

dt

//

=

[

fBt(=) u

(y) dy

t"-.+1.

(3.83)

Hence, for convenience of argument, we prove that every positive solution u of (3.83) is Lipschitz continuous. Let I = {v E X -L°°(R") I IIvIIL- <- 2IIuIIL-) and

2J = (v E Y

C°'r(R) I

IIvIIL- <- 2IIuIILoo).

For every e > 0, define dt

T,v(x) =

fE f 0

and

F(x) =

B,(x)

ff

vT(y)dy

to-a+1

dt uT (y)dy

Then obviously, it is a solution of the equation v = TEv + F.

(3.84)

3.3 Regularity Lifting

117

Write SEv = Tv + F. We will show that for e sufficiently small, (i) T, is a contracting operator from X to X, (ii) TE is a shrinking operator from T to Y,

(iii) FE1n2J,andSE:tnT (i) For any f, g E l and for any x E Rn, we have E

ITEf(x)-Tg(x)I s f

0 E

Jo

f

Bt(x)

If (y) - s(y)Idy

dt to-a+1 dt

fs (_)

7I6r-1(y)IIf(y) - g(y)Idy to-«+1

CpuIILm1IIf-9IILw f EIBt(x)Itnas+l E

= C1IIuIIL;1IIf -9IILm f

dt

a

= C2IIuIILm1 Ea If - 9IILw

Here we have applied the Mean Value Theorem with 6(y) valued between f (y) and g(y), and IBt(x)I denotes the volume of the ball Bt(x).

Choose e sufficiently small such that Ea ` and consequently

IITf-T9IIL- <- 4Ilf-9IILm. Therefore TE is a contracting operator from I to X for such a small E. (ii) Assume v E 2J. Then for any x, z E Rn, ITEv(x) -TEv(=)I = I 10, {fat(=)

ff

vT

(y)dy - fBt(=) vT(y)dy } to

da+1 I

E

=I

0

Bt (x)

-Iv--1(e)IIv(y)-v(y+z-x)Idytnaa+l

//PE

S 0

[vT(y)-vT(y+z-x)]

fBc(x)

Iz - xl f E IBt(x)I to =C1IIuIIL-_'IIvIIco, Iz-xIEa. Again by choosing c sufficiently small, we derive

al +1

3 Regularity of Solutions

118

IT`v(x) _Z`v(z)I <

sup

Combining this with the estimate in (i), we arrive at IITEvIICo. G ZIIvIICo.

, Vv E T.

Hence, TE is a shrinking operator from 2,J to Y.

(iii) To show F is Lipschitz continuous, we split it into two parts: 1

F(x) _ e

B,(x)

dt °° +f na+l t 1 B,(x)

uT(y)dy

dl vT(y)dy

I1(x)+72(x)

t

For the first part, we have

n'(y)dy -

IIi (x)-h(_)I = I

GC f1IIvIIL-t" 1xc

= CiIIuIIL- Ix - zil1-

f

Bt(-)

u.T (y)dy 7 t dt 1

zltn dtt+1 6°-11.

(3.85)

Here we have used the fact that

the volume of [(Bt(x) \ Bt(z)) U (Bt(z) \ Bt(x))] G Ctn'-1Ix - zI. It follows from (3.85) that sup s p

(3.86)

Ix - z

For the second part, we use a different approach. Write 6 = I x - zl. Then, 12(x) - 12(z)

J

l fBra+s>(=) u'T(y)dy to

=1

+s

fB

(z)

uT(y)dy

a+l - 12(z) 6)n-a - I2(z)

as+l (1 +

< 12(z)[(1 + 6)n ° - 1]

< u(z)(n - a)(1 + £)"-°-16

G CIInIIL-6.

Note that here we have applied the Mean Value Theorem with tween 0 and 6.

valued be-

3.3 Regularity Lifting

119

The same inequality holds for 12(z) -12(x). Therefore, sup 112(x) -12(x)I G CIIuIIL-. X 96Z Ix - ZI

(3.87)

Also from the definition of F(x), we immediately have (3.88)

IIFIILOO G- IIUIIL°°

Now (3.86), (3.87), and (3.88) imply that F(x) is Lipschitz continuous, and this together with (3.88) imply

Finally, to see that SE maps X n 2 into itself, we only need to verify that if IIvIIL°o G 2IIuIIL-, then IITEvIILo G- IINIIL°°-

(3.89)

In fact,

dt

[`

TEV(x)

(tJ)dJtn-a+1

Jo Ja,(x)v

G_ IIvIILW f EIBt(x)Itndt

G CIIuIIL-E°.

Obviously, choosing c sufficiently small (but independent of v), we can guarantee (3.89). So far we have verified (i), (ii), and (iii). Finally, by the Regularity Lifting Theorem II (Theorem 3.3.5) and also by Remark 3.3.5, L°°(Rn) and C°11(Rn)

form an "XY-pair;" we conclude that the positive solution u is Lipschitz continuous. This completes the proof of the Theorem.

3.3.7 Applications to Fully Nonlinear Systems of Wolff Type The Regularity Lifting Theorem II can also be applied to the following fully nonlinear integral systems involving Wolff potentials to prove that every positive solution pair is Lipschitz continuous (See [CLM]). Consider J u(x) = W#,. (vq)(x), x E RTM,

v(x) =WW,y(uP)(x), x E Rn; where

W6,7(Mx) =

r J

LfB`(z) f(y)dy j to-Q7

"-' dt t

(3.90)

3 Regularity of Solutions

120

This system includes many known systems as special cases, in particular, when 0 = z and 7 = 2, system (3.90) reduces to u(x) = fa^ T=-

v(x) = f,,.

1

v(y)°dy, x E R",

r

u(y)ydye x E R".

(3 . 91 )

The solutions (u, v) of (3.91) are critical points of the functional associated with the well-known Hardy-Littlewood-Sobolev inequality. We can show that (3.91) is equivalent to a system of semi-linear elliptic PDEs (-4),/2u Jl (-Q)a/2'U = v4, in R", = u', in R".

When a = 2, it becomes the well-known Lane-Emden system, and in the special case when

p=q= n+2 n-2

it reduces to the Yamabe equation. System (3.90) is said to be in critical case if

7-1 T+_7_-1

+

7-1 q+7-1

n -Q7 n

(3.92)

In [CLM, the authors and Chao Ma proved the following:

Proposition 3.3.1 Let (u,v) E Lp}y-1(R") x Lq}ry 1(R") be a pair of positive solutions for system. (3.90) in the critical case (3.92). Further assume p > 1, q > 1, and 1 < 7 < 2. Then (u, v) is uniformly bounded. Based on this proposition, [CLM] used Regularity Lifting Theorem II to lift the positive solution pair from L°°(R") to C°-1(R").

Theorem 3.3.8 Under the same conditions as in Proposition 3.3.1, (u, v) is Lipschitz continuous.

Due to the complexity of the estimates, we only present the idea and outline of the proof here. Interested readers can find the details in [CLM].

Outline of the Proof of Theorem 3.3.8. Fix the positive solution pair (u, v). Let 11 = {g E L°°(R") : II91ILw < 2IIuhILoc +2IIvIILm} and

Q) = {9 E C°''(R") : II9IIL- < 2IIuJIL= + 2IIvIIL°°},

3.3 Regularity Lifting

121

is L°°(R") norm of g. One can see that X and 2J are closed subsets of Banach spaces equipped with L°°(R'") norm and Lipschitz norm where 1191IL°°

IgllC°,,(R..) respectively.

Define on X and r

fa<(s)g4(z)dz\ 11 to-7a

dt

Tig(x)

J

T2f(x)=

If (fB(r)fP(z)dz\

dt

JO

t

1\

t

and

Write F(x>

tn-7A

)

r (fB±()v(z)dz\1 t^'-7T, dt

and

(fB, (y) v,"(z) dz "-' dt

G(x) c

to-7Q

)

t

Equip the product X x X with norm II (f, 9) II Lm = IIf IIL°° + 11911L00

and 2,J x 2 with norm 11Y, 011c--= = IIfllco,l +IIgIIco,=.

Consider the equation (f,g) = S(f,g) ° T(f,g) + (F, G)

(3.93)

where T(f, g) = (Tig, T2f). Obviously, (u, v) satisfies equation (3.93). We can prove that (See [CLM]), for sufficiently small e,

(i) T is a contracting operator from X x X to X x X, (ii) T is a shrinking operator from 23 x 2J to Y x Y, and (iii)

F,Gc(XxX)n(2)x23), Then by Regularity Lifting Theorem II, we can conclude that (u, v) is Lipschitz continuous.

preliminary Analysis on Riemannian Manifolds

4.1

Differentiable Manifolds

4.2

Tangent Spaces

4.3

Riemannian Metrics

4.4

Curvature

4.4.1 4.4.2

4.4.3 4.5

Curvature of Plane Curves Curvature of Surfaces in Rs Curvature on Riemannian Manifolds

Calculus on Manifolds 4.5.1

Higher Order Covariant Derivatives and the Laplace-Beltrami Op-

crater

4.6

4.5.2

Integrals

4.5.3

Equations on Prescribing Gaussian and Scalar Curvature

Sobolev Embeddings

According to Albert Einstein, the Universe we live in is a curved space. We call this curved space a manifold, which is a generalization of the flat Euclidean space. Roughly speaking, each neighborhood of a point on a manifold is homeomorphic to an open set in the Euclidean space, and the manifold as

a whole is obtained by pasting together pieces of the Euclidean spaces. In this section, we will introduce the most basic concepts of differentiable manifolds, tangent space, Riemannian metrics, and curvatures. We will try to be as intuitive and brief as possible. For more rigorous arguments and detailed discussions, please see, for example, the book Riemannian Geometry by M. do Carmo [Ca].

124

4 Preliminary Analysis on Riemannian Manifolds

4.1 Differentiable Manifolds A simple example of a two-dimensional differentiable manifold is a regular surface in R3. Intuitively, it is a union of open sets of R2, organized in such a way that at the intersection of any two open sets, the change from one to the other can be made in a differentiable manner. More precisely, we have

Definition 4.1.1 (Differentiable Manifolds) A differentiable n-dimensional manifold M is (i) a union of open sets, M = Ua Va and a family of homeomorphisms ¢a from Vn to an open set Oa(Va) in Rn, such that

(ii) For any pair a and Q with V0 fl V1 = U $ 0, the images of the intersection 0, (U) and Op (U) are open sets in R' and the mappings 013 o ¢Q 1 are differentiable.

(iii) The family { V0, 0,} are maximal relative to the conditions (i) and (ii).

A manifold M is a union of open sets. In each open set U C M, one can define a coordinate chart. In fact, let q5U be the homeomorphism from U to Ou(U) in Rn; for each p E U, if ¢u(p) = (x1, x2, , xn) in Rn, then we can define the coordinates of p to be (x1,x2, .xn). This is called the local coordinates of p, and (U, ¢u) is called a coordinate chart. If a family of coordinates charts that covers M are well made so that the transition from one to the other is in a differentiable way as in condition (ii), then it is called a differentiable structure of M. Given a differentiable structure on M, one can easily complete it into a maximal one by taking the union of all the coordinate charts of M that are compatible with (in the sense of condition (ii)) the ones in the given structure. A trivial example of n-dimensional manifolds is the Euclidean space Rn, with the differentiable structure given by the identity. The following are two non-trivial examples.

Example 4.1. The n-dimensional unit sphere

Sn = {x E Rn+1 I xl +... + xn+1 =

1}.

Take the unit circle S1 for instance, we can use the following four coordinate charts: V1 = {x E S1 I x2 > 0), 01(x) = x1, V2 = {x E S1 I x2 < 0}, ,02(x) = x1,

V3 = {x E S1 I x1 > 0}, 03(x) = x2, V4 = {x E S1 I x1 < 0}, 04(x) = x2.

Obviously, S1 is the union of the four open sets V1, V2, V3, and V4. On the intersection of any two open sets, say on V1 n V3, we have

4.2 Tangent Spaces ' J

}

x2 =

1

x1=

1-x2, x2>0.

125

- x1, x1 > 0;

They are both differentiable functions. The same is true on the other intersections. Therefore, with this differentiable structure, S1 is a 1-dimensional differentiable manifold.

Example 4.2. The n-dimensional real projective space Pn(R). It is the set of straight lines of Rn+1 which pass through the origin (0, . , 0) E Rn+1 or the quotient space of Rn+1 \ {0} by the equivalence relation

(xl...xn+1)-(Axl,...,Axn+1), AER, A$0. We denote the points in Pn(R) by [x]. Observe that if xi # 0, then

[xi...,xn+1] _

[xi/xi,...xi-i/xi,1,xi+1/xi,...xn+1/xi]-

Fori=1,2,- ,n+1, let Vi={[x]Ixi#0}. Apparently,

n+1

Pn(R) = U Vi. i=1

Geometrically, Vi is the set of straight lines of Rn+1 which pass through the origin and which do not belong to the hyperplane xi = 0. Define

¶i([x])=(S1, ..,Si-i' i+1' ,Sn+1),

where k=xk/xi(i#k)and1
k

2,7

are obviously smooth, thus {Vi, ¢i} is a differentiable structure of P'(R) and hence P"(R) is a differentiable manifold.

4.2 Tangent Spaces We know that at every point of a regular curve or of a regular surface, there exists a tangent line or a tangent plane. Similarly, given a differentiable structure on a manifold, we can define a tangent space at each point. For a regular

126

4 Preliminary Analysis on Riemannian Manifolds

surfaces S in R3, at each given point p, the tangent plane is the space of all tangent vectors, and each tangent vector is defined as the "velocity" in the ambient space R3. Since on a differential manifold, there is no such ambient space, we have to find a characteristic property of the tangent vector that will not involve the concept of velocity. To this end, let's consider a smooth curve in R": 7(t) = (xi (t), ... , x,(t)), t e (-E, e), with 7(0) = p. The tangent vector of 7 at point p is

v = (x(0),.. . , x'n(0)). Let f (x) be a smooth function near p. Then the directional derivative of f along vector v E R" can be expressed as

(Lxo) f

Wt

=r

Hence the directional derivative is an operator on differentiable functions that depends uniquely on the vector v. We can identify v with this operator and thus generalize the concept of tangent vectors to differentiable manifolds M.

Definition 4.2.1 A differentiable function y : (-e, e)-*M is called a (differentiable) curve in M. Let D be the set of all functions that are differentiable at point p = 7(0) E M. The tangent vector to the curve y at t = 0 is an operator (or function) 7'(0) : DAR given by -Y (0)f = dt (f o 7) It=0

A tangent vector at p is the tangent vector of some curve y at t = 0 with 7(0) = p. The set of all tangent vectors at p is called the tangent space at p and denoted by TpM. Let (xj, , be the coordinates in a local chart (U, 0) that covers p with 0(p) = 0 E ¢(U). Let 7(t) = (xi(t), ,x"(t)) be a curve in M with -

7(0) = p. Then

7 (o)f = dtf(7(t)) It=e= atf(x

_ Exi(0) 8f

ax,

=> (xao)

(t),... ,x"(t)) It=o

(8)P) f. ax=

This shows that the vector 7'(0) can be expressed as the linear combination of (ae_) , i.e., P

4.2 Tangent Spaces

xe(0) (axi)P.

ry (0) _ Here

127

(88.) is the tangent vector at p of the "coordinate curve": xi-+0-1(0, ... , 0, xi 0 ... 0); ,

and

f=

ax.f(0-1(0

t

t

.. ,0,x1,0,... ,0)) Ix:=0

(ax_)P

One can see that the tangent space TPM is an n-dimensional linear space with the basis

The dual space of the tangent space TPM is called the cotangent space and is denoted by Tp M. We can realize it in the following. We first introduce the differential of a differentiable mapping.

Definition 4.2.2 Let M and N be two differential manifolds and let f : M-+N be a differentiable mapping. For every p E M and for each v E TPM, choose a differentiable curve y : (-e, c)-->M, with y(0) = p and y'(0) = v. Let 9 = f o y. Define the mapping dfP : TpM->Tf(P)N by dff(v) = 0'(0).

We call dfP the differential of f. Obviously, it is a linear mapping from one tangent space TPM to the other Tf(P)N; and one can show that it does not

depend on the choice of y (See [Ca]). When N = R1, the collection of all such differentials form the cotangent space. From the definition, one can derive

that

dfP((axi)P) = (axj)Pf' In particular, for the differential (dxi)P of the coordinates function xi, we have

(dxi)P((aax;)) = 6i;. P

And consequently, dfP

of

=

ax a

t

(dxi)P. P

128

4 Preliminary Analysis on Riemannian Manifolds

From here we can see that (dxl)p, (d' 2)p, ... , (dx-)P form a basis of the cotangent space TT M at point p.

Definition 4.2.3 The tangent space of M is

TM = U TM. pEM

And the cotangent space of M is

UTpM.

T'M

pEM

4.3 Riemannian Metrics To measure the arc length, area, or volume on a manifold, we need to introduce some kind of measurements or `metrics'. For a surface S in the three dimensional Euclidean space R3, there is a natural way of measuring the length of vectors tangent to S, which are simply the length of the vectors in the ambient

space R3. These can be expressed in terms of the inner product <, > in R3. Given a curve 7(t) on S for t E [a, b], its length is the integral b

ly (t) ldt,

where the length of the velocity vector y'(t) is given by the inner product in R3:

1ti (t) l =

< y' (t), y'(t) >.

The definition of the inner product <, > enables us to measure not only the lengths of curves on S, but also the area of domains on S, as well as other quantities in geometry. Now we generalize this concept to differentiable manifolds without using the ambient space. More precisely, we have

Definition 4.3.1 A Riemannian metric on a differentiable manifold M is a correspondence which associates to each point p of M an inner product < >p on the tangent space TTM, which varies differentiably, that is, 95.7 (q) =< (C72;)q' (a!j)q >q is a differentiable function of q for all q near p. A differentiable manifold with a given Riemannian metric will be called a Riemannian manifold.

4.3 Riemannian Metrics

129

It is clear that this definition does not depend on the choice of coordinate system. Also one can prove the following (see [Ca]):

Proposition 4.3.1 A differentiable manifold M has a Riemannian metric. Now we show how a Riemannian metric can be used to measure the length of a curve on a manifold M.

Let I be an open interval in R1, and let y : I-+M be a differentiable mapping ( curve) on M. For each t E I, dry is a linear mapping from TtI to dy(dt) is a tangent vector of Try(t)M. The restriction Try(t)M, and y'(t) dt of the curve ry to a closed interval [a, b] C I is called a segment. We define the

length of the segment by b

1/2 dt, Here the are length element is

ds =< y (t),y (t) >1/2 dt.

As we mentioned in the previous section, let (xi, nates in a local chart (U, 0) that covers p = 7(t) (xl (t),

be the coordi,

x,(t)). Then

y(t)=Exe(t)(&) Consequently,

ds2

= < y'(t), ry'(t) > <

axq,

dt2

x

>p xi(t)dtxj(t)dt

W=1

_

gij(p)dxidxj. i,7=1

This expresses the length element ds in terms of the metric gij. To derive the formula for the volume of a region (an open connected subset) Din M in terms of metric gij, let's begin with the volume of the parallelepiped formed by the tangent vectors. Let {el, en} be an orthonormal basis of TPM. Let

(')

aij ej,

Then

gik(p) _<

a

axi

a ,

a xk

>p= E aijaki < ej, el >= jd

aijakj. j

(4.2)

4 Preliminary Analysis on Riemannian Manifolds

130

We know the volume of the parallelepiped formed by as , as is the determinant det(aij), and by virtue of (4.2), it is the same as det(gij). Assume that the region D is covered by a coordinate chart (U,0) with coordinates (xi, - xn). Ftom the above arguments, it is natural to define the volume of D as vol(D) = det(gj)dx1 dx,. J ¢(U)

A trivial example is M = R", the n-dimensional Euclidean space with

a axi

The metric is given by

gij =< ei, ej >= bij. A less trivial example is Sz, the 2-dimensional unit sphere with standard metric, i.e., with the metric inherited from the ambient space R3 = {(x, y, z)}. We will use the usual spherical coordinates (B, 0) with

x=sinBcoso y = sin B sin 3/i

z=cos0 to illustrate how we use the measurement (inner product) in R3 to derive the expression of the standard metric gij in terms of the local coordinate (0,i) on Sz.

Let p be a point on Sz covered by a local coordinate chart (U,0) with coordinates (G4). We first derive the expression for ( ae)P andalp (a) , the tangent vectors at point p = 0-1(0, 0) of the coordinate curves and B = constant respectively.

P

= constant

In the coordinates of R3, we have,

a aB

and

7a

ax ay Oz (aB, aB, aB) =

(cos0cos0,cos0sinsin B),

ax, ay, az

a _ (av, a, a) _ (

sin B sin 3/i,sinecos /i,0).

It follows that gii (P) _< aB aB >P- cosz B cosz

t cost B sinz V) + sinz B = 1,

a

a

80 W

4.4 Curvature and

a

922(p) =<

a

o, ao >p=

131

sing 0.

Consequently, the length element is

ds =

d02 + sine d ?p2,

g11d02 + 2g12dOdili + g22d1G2 =

and the area element is dA =

det(gjj)dOd

=sinOdOdi/'.

These conform with our knowledge of the length and area in the spherical coordinates.

4.4 Curvature 4.4.1 Curvature of Plane Curves Consider curves on a plane. Intuitively, we feel that a straight line does not bend at all, and a circle of radius one bends more than a circle of radius 2. To measure the extent of bending of a curve, or the amount it deviates from straightness, we introduce curvature. Let p be a point on a curve, and T(p) be the unit tangent vector at p. Let q be a nearby point. Let da be the amount of change of the angle from T(p) to T(q), and As the arc length between p and q on the curve. We define the curvature at p as

Aa k(p) = q-+p lim As

_

ds

(P).

If a plane curve is given parametrically as c(t) _ (x (t), y(t)), then the unit tangent vector at c(t) is

T(t) _

(x'(t),y (ft)) [x'(t)]2 + [y'(t)]2 1

and

ds = [x'(t)]2 + [y'(t)]2dt. By a straightforward calculation, one can verify that y'(t)x"(t)

k(c(t)) = IdTI =

x'(t)y"(t) {[x'(t)]2 + [y'(t)72}3/2

If a plane curve is given explicitly as y = f (x), then in the above formula, replacing t by x and y(t) by f (x), we find that the curvature at point (x, f(x)) is f"(x)

k(x,f(x)) =

{1 + [f'(x)]2}3/2

.

(4.3)

4 Preliminary Analysis on Riemannian Manifolds

132

4.4.2 Curvature of Surfaces in R3 1. Principal Curvature

Let S be a surface in R3, p be a point on S, and N(p) a normal vector of S at p. Given a tangent vector T(p) at p, consider the intersection of the surface with the plane containing T(p) and N(p). This intersection is a plane curve, and its curvature is taken as the absolute value of the normal curvature, which is positive if the curve turns in the same direction as the surface's chosen normal, and negative otherwise. The normal curvature varies as the tangent vector T(p) changes. The maximum and minimum values of the normal curvature at point p are called principal curvatures and are usually denoted by kl(p) and k2(p).

2. Gaussian Curvature. One way to measure the extent of bending of a surface is by Gaussian curvature. It is defined as the product of the two principal curvatures

K(p) = ki(p) . k2(P)

From this, one can see that no matter on which side normal is chosen (say, for a closed surface, one may chose outer normals or inner normals), the Gaussian curvature of a sphere is positive, of a one sheet hyperboloid is negative, and of a torus or a plane is zero. A sphere of radius R has Gaussian curvature R . Let f (x, y) be a differentiable function. Consider a surface S in R3 defined

as the graph of x3 = f (x1, x2). Assume that the origin is on S, and the x-y plane is tangent to S, that is,

f(0,0) = 0 and fl(0, 0) =0 = f2(0,0), where, f o r simplicity, we write f; = a and will use f 7 to denote a We calculate the Gaussian curvature of S at the origin 0 = (0, 0, 0). Let u = (u1, u2) be a unit vector on the tangent plane. Then by (4.3), the normal curvature k (u) in u direction is

U [(a 2

) 2 + 1]3/2 (O)

where a and a are first and second directional derivative of f in u direction. Under the assumption that f1(0) = 0 = f2(0), we arrive at kn(n) = (fiin + 2f12u1u2 + f22n2)(0) = uTF u,

where

4.4 Curvature

133

F = (fir f12 l (0) f21 f22 J

,

and uT denotes the transpose of u.

Let Al and A2 be two eigenvalues of the matrix F with corresponding unit eigenvectors el and e2, i.e.,

Fei=Aiei, i=1,2.

(4.4)

Since the matrix F is symmetric, the two eigenvectors el and e2 are orthogonal, and it is obvious that

eTFei=Ai, i=1,2. Let 0 be the angle between u and el. Then we can express u = cos 0 el + sin 0 e2. It follows that

Assume that Al < A2. Then Al < Al + sine 0 (A2 - A1) =

cost 0 (A1 - A2) + A2 < A2.

Hence Al and A2 are minimum and maximumvalues of normal curvature, and therefore they are principal curvatures k1 and k2.

From (4.4), we see that Al and A2 are solutions of the quadratic equation

fu-Af12 f2l

f22-A

_ '\2 - (fll + f22)A+ (fllf22 - f12f21) = 0.

Therefore, by definition, the Gaussian curvature at 0 is

K(0) = klk2 = A1A2 = (fllf22 - f12f21)(0)

4.4.3 Curvature on Riemannian Manifolds On a general n-dimensional Riemannian manifold M, there are several kinds of curvatures. Before discussing them, we need to introduce a few preparatory concepts.

1. Vector Fields and Brackets

A vector field X on M is a correspondence that assigns each point p E M a vector X (p) in the tangent space TpM. It is a mapping from M to T M. If

134

4 Preliminary Analysis on Riemannian Manifolds

this mapping is differentiable, then we say the vector field X is differentiable. In local coordinates, we can express X (p)

a ai (p)

.

ag2y

When applied to a smooth function f on M, it is a directional derivative operator in the direction X (p): (X f) (p) _

ai (P)

Definition 4.4.1 For two differentiable vector fields X and Y, define the Lie Bracket as

[X, Y] = XY - YX. If X (p) _

° ai (p) aa; and Y(p) _ E' b; (p) as , then

\ of bia' [X,Y]f=XYf-YXf= I (aiab' ax; axi / I

ax;.

i,;=1

2. Covariant Derivatives and Riemannian Connections

Intuitively, we begin with a surface S in R3. Let c : I-+S be a curve on S, and let V(t) be a vector field along c(t), t E I. One can see that Ts is a vector in the ambient space R3, but it may not belong to the tangent plane of S. To consider the rate of change of V(t) restricted to the surface S, we make an orthogonal projection of dL onto the tangent plane, and denote it DV by We call this the covariant derivative of V(t) on S - the derivative viewed by the two-dimensional creatures on S. On an n-dimensional differentiable manifold M, the covariant derivative is defined by affine connections. Let D(M) be the set of all smooth vector fields.

Definition 4.4.2 Let X, Y, Z E D(M), and let f and g be smooth functions on M. An affine connection D on a differentiable manifold M is a mapping from D(M) x D(M) to D(M) which maps (X, Y) to DxY and satisfies

1. Dfx+gyZ= fDxZ+gDYZ, 2. Dx(Y + Z) = DxY + DxZ, 3. Dx(fY) = fDxY+X(f)Y.

4.4 Curvature

135

Given an affine connection D on M, for a vector field V(t) = Y(c(t)) along a differentiable curve c : I-4M, define the covariant derivative of V along c(t) as

DV dt

dt

In local coordinates, let c(t)=(xi(t),...x.(t))

and V(t)

vi(t) -.

Then by the properties of the affine connection, one can express

DV dt

dvi 8 dt 8xi

+ ij

dt vD ax'8 . dxi

(

4.5)

,i

Recall that in Euclidean spaces with inner product <,>, a vector field V(t) along a curve c(t) is parallel if only if dl = 0; and for a pair of vector fields X and Y along c(t), we have < X, Y >= constant. Similarly, we have the following:

Definition 4.4.3 Let M be a differentiable manifold with an affine connecDV = 0. tion D. A vector field V (t) along a curve c(t) E M is parallel if Definition 4.4.4 Let M be a differentiable manifold with an affine connection D and a Riemannian metric <, >. If for any pair of parallel vector fields X and Y along any smooth curve c(t), we have < X, Y >c(t)= constant

then we say that D is compatible with the metric <, >.

The following fundamental theorem of Levi and Civita guarantees the existence of such an affine connection on a Riemannian manifold.

Theorem 4.4.1 Given a Riemannian manifold M with metric <, >, there exists a unique connection D on M that is compatible with <, >. Moreover this connection is symmetric, i.e.,

DXY - DyX = [X, Y1. We skip the proof of the theorem. Interested readers may see the book of do Carmo [Cal (page 55). In local coordinates (xr, , x.), this unique Riemannian connection can be expressed as

4 Preliminary Analysis on Riemannian Manifolds

136

ka

Daxj

k'riiaxk'

where

I,k

1

ax,

z

09mi

a9ii

ax;

axm

g ink

is called the Christoffel symbols, gij =< as., as. >, and (gig) is the inverse matrix of (g,i). 3. Curvature

Definition 4.4.5 Let M be a Riemannian manifold with the Riemannian connection D. The curvature R is a correspondence that assigns every pair of smooth vector field X, Y C D(M) a mapping R(X, Y) D(M) -*D(M) defined by

R(X,Y)Z=DyDXZ-DXDYZ+D(X,Y1Z, VZED(M). Definition 4.4.6 The sectional curvature of a given two-dimensional subspace E C TyM at point p is

K(E)

< R(X,Y)X,Y > IXI2IYI2-<X,Y>2'

where X and Y are any two linearly independent vectors in E, and XI2IYI2- < X,Y >2 is the area of the parallelogram spanned by X and Y. One can show that K(E) so defined is independent of the choice of X and Y. It is actually the Gaussian curvature of the section. Example. Let M be the unit sphere with standard metric, then one can verify that its sectional curvature at every point is K(E) = 1.

Definition 4.4.7 Let X = Y. be a unit vector in TyM, and let {Y1,.. ,Yn_1} be an orthonormal basis of the hyperplane in TyM orthogonal to X. Then the Ricci curvature in the direction X is the average of the sectional curvature of the n -1 sections spanned by X and Yl, X and Y2, and X and Yn-1: n-1

Ricy(X) = n ll E < R(X,Y,)X,Y; > =r

The scalar curvature at p is

R(p) _ -1 j: Ricy(Y:) :=r

.

4.5 Calculus on Manifolds

137

One can show that the above definition does not depend on the choice of the orthonormal basis. On two-dimensional manifolds, Ricci curvature is the same as sectional curvature, and they both depend only on the point p.

4.5 Calculus on Manifolds In the previous chapter, we introduced the concept of affine connection D on a differentiable manifold M. It is a mapping from 1'(M) x r(M) to T(M), where T(M) is the set of all smooth vector fields on M. Here we will extend this definition to the space of differentiable tensor fields and thus introduce higher order covariant derivatives. For convenience, we adapt the summation convention and write, for instance, a xi a'.= j ikdxk

rk dxk k

and so on. and

4.5.1 Higher Order Covariant Derivatives and the Laplace-Beltrami Operator Let p and q be two non-negative integers. At a point x on M, as usual, we

denote the tangent space by TxM. Define Tp(TxM) as the space of (p,q)-

tensors on TM, that is, the space of (p + q)-linear forms

rl:TMx .xTMxTMx...xT.M-rR1. P

q

In local coordinates, we can express

ajxj

T=V1.'°dx"®...®dxiP®a ®...0

49

Recall that, for the affine connection D, in local coordinates, if we set

then

Vi

axe

(x) = rij aak x,

4 Preliminary Analysis on Riemannian Manifolds

138

where I ..' are the Christoffel symbols. For smooth vector fields X,Y E P(M) with

y = (y1'. ..

X = (Xl, ... , Xn) and

, y"),

by the bilinear properties of the connection, we have

DxY =

X'DiY

X'

\88Yj

Xi (

09

+Yj1Oak/ll

xi axi

xi + q,akYk )

/x

(4.6)

Now we naturally extend this affine connection D to differentiable (p, q)-

tensor fields T on M. For a point x on M, and a vector X E TxM, DxT is defined to be a (p, q)-tensor on TM by

DxT(x) = X'(viT)(x), where '7i...Jq

P

bL,

(17J)(x)41...2P

axi

X

))I'7r-7y rri.(x)T l (x

k=1

q

Jk-1a7k+'j'.

+

ri'.k (x)T(x)it

-i'

(4.7)

k=1

Notice that (4.6) is a particular case when (4.7) is applied to the (0, 1)tensor Y. If we apply it to a (1,0)-tensor, say dx1, then

,7i(dx1) = _rikdxk.

(4.8)

Given two differentiable tensor fields T and T, we have

Dx(T®T)=DxT®T+T®DxT. For a differentiable (p, q)-tensor field T, we define VT to be the (p + 1, q)tensor field whose components are given by

Similarly, one can define 72T, 73T, - - -.

For example, a smooth function f (x) on M is a (0, 0)-tensor. Hence 7f is a (1, 0)-tensor defined by

dxi. of = Difdxi = of axi

4.5 Calculus on Manifolds

139

And v2f is a (2, 0)-tensor:

2f

of dx') ® dxj

\ ax-i of (7j raxi a

=

I

_ \aaaj

I dx' + of 7j (dx )) dxj allai

I'ak/ dx'0dxj.

Here we have used (4.7) and (4.8). In the Riemannian context, v2 f is called the Hessian of f and denoted by Hess(f). Its ij'h component is a2

gf

axiaxj

The trace of the Hessian matrix ((v2f)ij) is defined to be the LaplaceBeltrami operator A acting on f,

of=ga'( axi asx

-r'af1 ark

where (g'j) is the inverse matrix of (gij). Through a straightforward calculation, one can verify that

I=1

191 i,j=1

8

axi (VI-Igi gii axj

where gl is the determinant of (gij).

4.5.2 Integrals On a smooth n-dimensional Riemannian manifold (M, g), one can define a natural positive Radon measure on M, and the theory of Lebesgue integral applies.

Let (Ui, ¢i)i be a family of coordinate charts that covers M. We say that a family (Ui, Oi, 77i)i is a partition of unity associated with (Ui, Oi)i if

(i) (77j)i is a smooth partition of unity associated with the covering (U1)j, and

(ii) for any i, supp 77i C U. One can show that, for each family of coordinate chart (U1,01)1 of M, there exists a corresponding family of partition of unity (Us' Oi, 770i. Let f be a continuous function on M with compact support, and given a family of coordinate charts (Ui, 0i)i of M, we define its integral on M as the sum of the integrals on open sets in Rn:

140

4 Preliminary Analysis on Riemannian Manifolds

f mfdVg= I 1

(r1if)°0irdx,

0:(Ut)

where (Ui, Oj, r)i)i is a family of partition of unity associated with (Ui, bi)i, gj is the determinant of (gij) in the charts (U1,Oi)i, and dx stands for the volume element of R. One can prove that such a definition does not depend on the choice of the charts (Ui, ¢i)i and the partition of unity (Ui, ti, rli)i. For smooth functions it and v on a compact manifold M, one can verify the following integration by parts formula

-f LguvdVg= f dV

(4.9)

M

where

is the Laplace-Beltrami operator associated with the metric g, and

< 7u' pv >= g11 V/ up v =

9ji

an On axi axe

is the scalar product associated with g for 1-forms

4.5.3 Equations on Prescribing Gaussian and Scalar Curvature A Riemannian metric g on M is said to be pointwise conformal to another metric go if there exists a positive function p, such that 9(x) = P(x)9o(x), V X E M.

Suppose M is a two-dimensional Riemannian manifold with a metric go and the corresponding Gaussian curvature K0(x). Let g(x) = e2"(x)go(x) for some smooth function u(x) on M. Let K(x) be the Gaussian curvature associated with the metric g. Then by a straightforward calculation, one can verify that

-Aou+K0(x) = K(x)e2u(x) V X E M.

(4.10)

Here oa is the Laplace-Beltrami operator associated with go. Similarly, for conformal relation between scalar curvatures on dimensional manifolds (n > 3), say on standard sphere S", we have

+n(n4

2) u

4(n-1))R(x)u

,

VxES",

n-

(4.11)

where go is the standard metric on S" with the corresponding LaplaceBeltrami operator gyp, R(x) is the scalar curvature associated with the pointwise conformal metric g = u^4=go. In Chapter 5 and 6, we will study the existence and qualitative properties of the solutions it for the above two equations, respectively.

4.6 Sobolev Embeddings

141

4.6 Sobolev Embeddings Let (M, g) be a smooth Riemannian manifold. Let it be a smooth function k on M and k be a non-negative integer. We denote 7 u the kth covariant derivative of it as defined in the previous section, and its norm Ipkul as defined in a local coordinate chart by ik(,7kn)ji...jk.

I okul2 = Oil ... 9ikik (V ku)

In particular, we have Ip°ul = Jul and 49E 49u

Io1u12 = I VuI2

= 9"(7u)i(VU)j = s Ot nofu = 9ij 49xi 49xj

For an integer k and a real number p > 1, let Ck(M) be the space of C°° functions on M such that

f

M

Definition 4.6.1 The Sobolev space H1,P(M) is the completion of CE(M) with respect to the norm IIUIIHk.P(M) _

j=o

(f M IYulPdVg )

'iP

Many results concerning the Sobolev spaces on R" introduced in Chapter 1 can be generalized to Sobolev spaces on Riemannian manifolds. For application purposes, we list some of them in the following. Interested readers may find the proofs in [Hel] or [Au]. Theorem 4.6.1 For any integer k, Hk,2(M) is a Hilbert space when equipped with the equivalent norm. 112

k

Null =

f Io'ul2dVg)

s-1 M

The corresponding scalar product is defined by k

=Ef

dV9,

i=1 M

where < . > is the scalar product on covariant tensor fields associated with the metric g.

4 Preliminary Analysis on Riemannian Manifolds

142

Theorem 4.6.2 If M is compact, then Hk,p(M) does not depend on the metric.

Theorem 4.6.3 (Sobolev Embeddings I) Let (M, g) be a smooth, compact n-dimensional Riemannian manifold. As-

sume 1 < q < n and p =

Q . Then

H''q(M) C LP(M).

More generally, if m, k are two integers with 0 < m < k, and if p n-q(k-m

then Hl,,q(M) C Hm,p(M).

Theorem 4.6.4 (Sobolev Embeddings II) Let (M, g) be a smooth, compact n-dimensional Riemannian manifold. As-

sume that q > 1 and m, k are two integers with 0 < m < k. If q > then

^ kk-m,

Hks(M) C C-(M). Theorem 4.6.5 (Compact Embeddings) Let (M, g) be a smooth, compact n-dimensional Riemannian manifold.

(i) Assume that q > 1 and m, k are two integers with 0 < m < At. Then for any real number p such that 1 < p < the embedding Hk,q(M) C H-,n(M)

is compact. In particular, the embedding of Hhq(M) into LP(M) is compact

ifl
(ii) For q > n, the embedding of H''q(M) into CA(M) is compact, if (k-A) > Q. In particular, the embedding of Hl,q(M) into C°(M) is compact.

Theorem 4.6.6 (Poincare Inequality) Let (M, g) be a smooth, compact n-dimensional Riemannian manifold and let p E [1, n) be real. Then there exists a positive constant C such that for any

u E H'&(M), (rM

\ lip In - ujPdV9 I

/

1/p

< C (IM jVurpdVy)

4.6 Sobolev Embeddings Where

u

is the average of u on M.

1

Vot(,M,y)

J udV9 M

143

5

prescribing Gaussian Curvature on Compact 2-Manifolds

5.1.

Variational Methods in General

5.2.

Introduction on Prescribing Gaussian Curvature

5.3

The Negative Case 5.3.1 5.3.2

5.4

Kazdan and Warner's Results-Lower and Upper Solutions The Limiting Situation

The Non-negative Case-Variational Approaches 5.4.1

5.4.2

5.4.3

5.4.4 5.4.5

Obstructions Variational Approaches and Key Inequalities Existence of Weak Solutions in Subcritical Case A Remedy for Critical Case-Recovering Coerciveness Existence of Weak Solutions in Critical Case

5.1 Variational Methods in General In addition to Chapter 2, in this and the upcoming chapters, we will present the reader with real research examples-semi-linear elliptic equations arising from prescribing Gaussian and scalar curvatures. We will illustrate, among other methods, how the calculus of variations can be applied to seek weak solutions of these equations. To show the existence of weak solutions for a certain equation, we consider the corresponding functional J(.) in an appropriate Hilbert space. According

146

5 Prescribing Gaussian Curvature on Compact 2-Manifolds

to the compactness of the variational scheme, people usually divide all situa, tions into one of three cases: a) The subcritical case, where the level set of J is compact, say, J satisfies the (PS) condition (readers may recall such examples from Chapter 2): Every sequence {uk} that satisfies J(uk)->c and J'(uk)-+O possesses a convergent subsequence,

b) The critical case, which are the limiting situations of subcritical cases where the functionals lose compactness, or c) The super critical case, namely, the remaining cases. Roughly, to see when a functional may lose compactness, let us consider a very simple example: J(x) = (x2 - 1)ez

defined on a one-dimensional space, R'. One can easily see that there exists a sequence {xk}, for instance, xk = -k, such that J(xk)-*O and J'(xk)-4o as k->oo; however, {xk} possesses no convergent subsequence because this sequence is not bounded. The functional has no compactness at level J = 0. We know that in finite-dimensional space, a bounded sequence has a convergent subsequence, whereas in infinite dimensional space, even a bounded sequence may not possess a convergent subsequence. For instance, {sin kx} is a bounded sequence in L2([0,1]) but does not have a convergent subsequence.

In the situation where the lack of compactness occurs, to find a critical point of the functional, one would try to recover the compactness through various means. Take again the one dimensional example J(x) = (x2 - 1)e'', which has no compactness at level J = 0. However, if we restrict it to levels less than 0. we can recover the compactness. That is, if we pick a sequence {xk} satisfying J(xk) < 0 and J'(xk)--*O, then one can verify that it possesses a subsequence which converges to a critical point x0 of J. Actually, this xo is a minimum of J. To further illustrate the concept of compactness, we consider some examples in infinite-dimensional space. Let 11 be an open bounded subset in R", and let Ho (1?) be the Hilbert space introduced in previous chapters. Consider the Dirichlet boundary value problem ZSu. = up(x), x E !2

U(X)=0,

xEBfl.

To seek weak solutions, we study the corresponding functional

JP(u) = )o lulp+rdx on the constraint

H = In EHa(Q) I

LIvul2dx=1}.

(5.1)

5.1 Variational Methods in General

147

When p < +2, it is in the subcritical case. As we have seen in Chapter 2, the functional satisfies the (PS) condition. Let MP = sup JP(u).

UEH

Consider a maximizing sequence {uk} with

JP(ak)-4MP and Jp'(7lk)iO. Then one can show that {uk} possesses a convergent subsequence in Ho(.(?), and the limiting function uo would be the maximum of J. in H, hence it is a desired weak solution. Here an alternative and equivalent way is to minimize the functional

IP(u) =

Jn

Ipul2dx

under the constraint

IuIP+'dx = 1,

u E Ho (fl) and Jn

as we have seen in Chapter 2. When p = T := n±2, it is in the critical case. One can find a maximizing

sequence of Jr that does not converge. For example, when I? = B1(0) is a ball, consider [n(n - 2)k]"<

uk(x) = u(x)

(1+kIxI2)2

7(x)

where ri E Co (dl) is a cut-off function satisfying

x E B, (0)

1,

between 0 and 1

,

elsewhere

Let Uk(X)

vk(x) _

fn IVUkI2dx

Then one can verify that {vk} is a maximizing sequence of Jr in H with J.(vk)-1O. However, it does not have a convergent subsequence. Actually, as

kioo, the energy of {vk} is concentrating at one point, 0. We say that the sequence "blows up" at point 0. In essence, the compactness of the functional JP here relies on the Sobolev embedding:

H'(0) y IF+'(Q).

5 Prescribing Gaussian Curvature on Compact 2-Manifolds

148

As we learned in Chapter 1, this embedding is compact when p < n±2 and is not when p = nn±2 n-2'

In the critical case, the compactness may be recovered by considering other level sets or by changing the topology of the domain. For example, if ft is an annulus, then the above Jr has a maximizer in H. In this and the next chapters, the reader will see examples from prescribing Gaussian and scalar curvature, where most of the corresponding- equations are in the critical case and where various means have been exploited to recover the compactness.

5.2 Introduction to Prescribing Gaussian Curvature Let M be a smooth and compact two-dimensional Riemannian manifold. Let go(x) be a metric on M with the corresponding Laplace-Beltrami operator A and Gaussian curvature K0(x). Given a function K(x) on M, can it be realized as the Gaussian curvature associated to the pointwise conformal metric g(x) = e2u(.)go(x)? As we mentioned in Chapter 4, to answer this question, it is equivalent to solve the following semi-linear elliptic equation:

-Du+K0(x) = K(x)e2u,

X E M.

(5.2)

To simplify this equation, we introduce the following simple existence result.

Lemma 5.2.1 Let f (x) E L2(M). Then the equation

-Au = f (x),

XEM

possesses a solution if and only if

f(x)dA=0.

(5.4)

IM

The proof of this Lemma is standard and will be presented in the Appendices. We now simplify equation (5.2). First let ii = 2u, then

-Aii + 2Ko(x) = 2K(x)e".

(5.5)

Let v(x) be a solution of

-Zv = 2Ko(x) - 2K0, where

(5.6)

5.2 Introduction to Prescribing Gaussian Curvature Ko

149

JMK0(x)dA

IMI

is the average of K0(x) on M. Because the integral of the right hand side of (5.6) on M is 0, the solution of (5.6) exists due to Lemma 5.2.1.

Let w = it + v. Then it is easy to verify that -,nw + 2Ko = 2K(x)e-9e°1.

(5.7)

Finally, letting

a = 2Ko and R(x) = 2K(x)e "(a) and rewriting in as it, we reduce equation (5.2) to

-An+a=R(x)e"ls), xEM.

(5.8)

By the well-known Gauss-Bonnet Theorem, if g is a Riemannian metric on M with corresponding Gaussian curvature K9(x), then

f K9(x)dA9 = 2rX(M), M

where X(M) is the Euler Characteristic of M, and dA9 the area element associated to metric g. Obviously,

a= MXM, where IMI is the volume (area) of M. According to the difficulty and the methods used in the process of prescribing Gaussian curvature, one may consider the following three cases

(i) The negative case: X(M) < 0 or a < 0; (ii) The subcritical case: 0 < X(M) < 2; and (iii) The critical case: X(M) = 2. In the negative case, people usually use the method of upper and lower (or super and sub) solutions, because in this situation, some kinds of maximum principles holds. In the other cases, the upper and lower solutions method does not work, and people usually resort to variational approaches. In the subcritical case, the corresponding variational scheme is compact, and one can rather easily take limit of a minimizing sequence to arrive at a weak solution. The critical case is the situation when M is conformal to the standard twodimensional sphere S2. We will consider each of these cases, in turn, in the next two sections.

For more results concerning prescribing Gaussian curvature, please see [BE], [CD1], [CD2], [ChL], [ChS], [CL3], [CL4], [CL7], [CL8], [CL12], [CL19], [CS], [CGY], [CY2], [CY3], [CY4], [DL], [Ha], [Ho], [Ka], [Ji], [J2], [KW1], [KW2], [Mo], [XY], and the references therein.

5 Prescribing Gaussian Curvature on Compact 2-Manifolds

150

5.3 The Negative Case 5.3.1 Kazdan and Warner's Results-Method of Lower and Upper Solutions Let M be a smooth and compact two-dimensional manifold. Let a < O. In [KW], Kazdan and Warner considered the equation

-'.u+a=R(x)e"h>, x E M

(5.9)

and proved the following theorem:

Theorem 5.3.1 (Kazdan-Warner). Let R be the average of R(x) on M. Then

i) A necessary condition for (5.8) to have a solution is R < 0.

ii) If R < 0, then there is a constant a,, with -oo < ao < 0, such that one can solve (5.8) if a > ao, but cannot solve (5.8) if a < ao. iii) as = -oo if and only if R(x) < 0. The existence of a solution was established by using the method of upper and lower solutions, and our following proof is adapted from [KW] with minor modifications. We say that u+ is an upper solution of equation (5.9) if

-Au++a>R(x)e"+, xEM. Similarly, we say that u_ is a lower solution of (5.9) if

-Au_ + a < R(x)e'"

,

x E M.

The main scheme in the proof of existence is as follows.

(a) If R < 0, then there is an ao > -oo, such that for

ao
L\¢+a¢=R(x), xEM must be positive.

(5.10)

5.3 The Negative Case

151

proof. We first prove the second part. Using the substitution v = e--, one sees that (5.9) has a solution u if and only if there is a positive solution v of z

v+av-R(x) -0- =0.

(5.11)

V

Assume that v is such a positive solution. Let 0 be the unique solution of (5.10) and let w = 0 - v. Then 'Lw+awIOvz

l

<0.

(5.12)

V

It follows that w > 0. Otherwise, there is a point xo E M, such that w(xo) < 0 and xo is a minimum of w, hence Lw(x,) > 0. Consequently, we must have

Aw(xo) +aw(xa) > 0, a contradiction with (5.12). Therefore w > 0 and 45 > v > 0. This shows that a necessary condition for (5.9) to have a solution is that the unique solution qs of (5.10) must be positive. Integrating both sides of (5.10), we see immediately that R < 0. This completes the proof of the Lemma.

Lemma 5.3.2 (The Existence of Upper Solutions). i) If R(x) < 0 but not identically 0, then there exist an upper solution u+ of (5.9).

ii) If R < 0, then there exists a small b > 0, such that for -b < a < 0, there exists an upper solution u+ of (5.9). Proof. Let v be a solution of

R(x) - R. We try to choose constants a and b such that u+ = av+b is an upper solution. We have

-Au+ + a - R(x)eu+ = aR(x) - aR + a - R(x)eav+b.

(5.13)

We want to choose constants a and b so that the right hand side of (5.13) is non-negative.

In Case i), when R(x) < 0, we regroup the right hand side of (5.13) as f (a, b, x)

(-aR + a) - R(x)(e69+b - a).

(5.14)

For any given a < 0, we first choose a sufficiently large, so that -of? +a > 0. Then we choose b large, so that eav(a)+6

- a> 0, V x E M.

152

5 Prescribing Gaussian Curvature on Compact 2-Manifolds

This is possible because v(x) is continuous and M is compact. It follows from (5.13) and (5.14) that

-Au+ + a - R(x)e"+ > 0. In Case ii), we choose a > °'2 and e° = a, where a is a positive number to be determined later. Then (5.14) becomes f (a, b, x) > -

2 - aR(x) (eau - 1)

>(-R a = aIJR11Lm

-IIRIIL;Ieav-1I)

R

(2

- leav - )

IIRIILQQ

Again by the continuity of v and the compactness of M, we see that uniformly on M as a-*0. Therefore, we can make eav(x)

-

11

<

ZRR

,

m

for all x E M

by choosing a sufficiently small. Thus u+ so constructed is an upper solution of (5.9) for a > a2 This completes the proof of the Lemma. .

Lemma 5.3.3 (Existence of Lower Solutions). Given any R E LP(M) and a function u+ E W2'P(M), there is a lower solution u_ E W2"P(M) of (5.9) such that u_ < u+.

Proof. If R(x) is bounded from below, then one can clearly use any sufficiently large negative constant as u_. For general R(x) E LP(M), let

K(x) = max 1, -R(x)}. Choose a positive constant a so that aK = -a. Then aK(x) + a = 0 and (aK(x) + a) E LP(M). Thus there is a solution to of

,Lw = aK(x) + a.

By the L. regularity theory, to E W2^P(M) and hence is continuous. Let

u- =w-A. Then -Au_ + a - R(x)ev- = -Aw + a = -aK(x) - R(x)e"'-.\

R(x)e'°_a

< -K(x)(a - eW_A).

(5.15)

5.3 The Negative Case

153

Choosing A sufficiently large so that a - ew

a

> 0,

and noticing that K(x) > 0, by (5.15), we arrive at

-,Lu_+a-R(x)e"- <0. Therefore u_ is a lower solution. Moreover, one can make A large enough such

that u_ = to - A < u+. This completes the proof of the Lemma.

Lemma 5.3.4 Let p > 2. If there exist upper and lower solutions, a+, u_ E w2,P(M) of (5.9), and if u_ < u+, then there is a solution u E W2,p(M). Moreover, u_ < u < u+, and u is C°° in any open set where R(x) is C°°. proof. We will rewrite the equation (5.9) as

Lu = f(x,u), then start from the upper solution u+ and apply iterations:

Lu1=f(x,u+) ,Lui+r=f(x,u:), i=1,2," In order that for each i, u_ < ui < u+, we need operator L to obey some Maximum Principle and f (x, ) to possess some monotonicity. The Laplace operator in the original equation does not obey Maximum Principles; however, if we let

Lu = -Du + k(x)u, where k(x) is any positive function on M, then the new operator L obeys the Maximum Principle:

IfLu>0, then u(x)>0, forallxEM. To see this, we suppose on the contrary that there is some point on M where it < 0. Let x° be a negative minimum of u; then, -Au(x°) < 0, and it follows that -Au(x°) + k(x°)u(x°) < k(x°)u(x°) < 0,

a contradiction. Hence the Maximum Principle holds for L. We now write equation (5.9) as Lu = -emu + k(x)u = R(x)eu - a + k(x)u := f(x,u).

Based on the Maximum Principle on L, keeping u_ < ui < u+ requires that f (x,) be monotone increasing, that is au = R(x)eu + k(x) > 0.

154

5 Prescribing Gaussian Curvature on Compact 2-Manifolds

To this end, we choose k(x) = kl(x)e"+lsl,

where

ki(x) = max{1, -R(x)}.

Based on this choice of L and f, we show by mathematical induction that the iteration sequence so constructed is monotone decreasing:

ui+1
(5.16)

In fact, from Lu+ ? f(x,u+)

and

Lul = f(x,u+)

we derive immediately

L(u+ - u1) ? 0.

Then the Maximum Principle for L implies u+ > ul. This completes the first step of our induction. Assume that for any integer in, u,a+1 < u,r, < u+; we will show that um+2 < u,,,+1. Since f (x, -) is monotone increasing in a for it < u+, we have f (x, Urn+1) <- f (x, u.)

0. Therefore u,,,,+1 > u,.,+2. This verifies (5.16) and hence L(um+r through mathematical induction. Similarly, one can show the other side of the inequality and arrive at

u_ < ui+1 <- ui < ... < u+,

i = 1, 2, ....

(5.17)

Since u+, u_, and ui are continuous, inequality (5.17) shows that {lull} is uniformly bounded. Consequently, in the LP norm, IILui+1IILP = IIR(x)e'

-a+k(x)uillLP

<-C.

Hence {ui} is bounded in W2'P(M). By the compact embedding of W2'p into C1, there is a subsequence that converges uniformly to some function u in C1(M). In view of the monotonicity (5.17), we conclude that the entire sequence {ui} itself converges uniformly to u. Moreover, Ilui+1- uj+111 w2,P <_ CIIL(ui+1 - uj+1)IILP


By Schauder theory, one proves inductively that it E C°° in any open set where R(x) is C°°. More precisely, if R E C'"+y, then it E Cm+2+7 This completes the proof of the Lemma.

5.3 The Negative Case

155

Based on these Lemmas, we are now able to complete the proof of Theorem 5.3.1.

proof of Theorem 5.3.1. part i) can obviously be derived from Lemma 5.3.1.

From Lemma 5.3.3 and 5.3.4, if there is an upper solution, then there is a solution. Hence by Lemma 5.3.2, there exist a b > 0, such that (5.9) is solvable

for -b< a < 0. Let ao = inf{a (5.9) is solvable }.

Then for any 0 > a > a0, one can solve (5.9). In fact, if for al, (5.9) possesses a solution ul, then for any a > al, we have

-,Lui + a > R(x)eu" that is, ul is an upper solution for equation (5.9) at a. Hence equation

-Du + a = R(x)e" also has a solution.

Lemma 5.3.2 implies that ao = -oo if R(x) < 0 but R(x) is not identically 0. Now to complete the proof of the Theorem, it suffices to prove that if R(x)

is positive somewhere, then ao > -oo. By virtue of Lemma 5.3.1, we only need to show

Claim: There exist a < 0, such that the unique solution of

Lcb + a¢ = R(x)

(5.18)

is negative somewhere.

Actually, we can prove a stronger result: aO(x) -*R(x) almost everywhere as a-> - oo.

(5.19)

We argue in the following 2 steps.

i) For a given a < 0, let u = u(x, a) be the solution of (5.18). Multiply both sides of equation tion (5.18) by u and integrate over M to obtain

-

J1pu12dA+aJ u2dA= r R(x)u(x)dA. M

M

Consequently,

1r M

Then by the Holder inequality,

RudA.

156

5 Prescribing Gaussian Curvature on Compact 2-Manifolds IIt1IL2 <- aIIRIIL2.

This implies that

as a-+ - oo, u(x, a) -+0, almost everywhere .

(5.20)

Since for any a < 0, the operator A + a is invertible, we can express

u = (A + a)-1R(x). From the above reasoning, one can see that, as a-* - m,

if R E L2, then (0 + a)-1R(x)--+0 almost everywhere

.

(5.21)

ii) To see (5.19), we write

-a¢+R(x) = (A+a)-10R.

(5.22)

One can see the validity of (5.22) by simply applying the operator A + a to both sides. Now by the assumption, we have LR E L2, and hence (5.21) implies that

-a4'+R(x)-O, a.e. Or

a.e.

Since R(x) is continuous and positive somewhere, for sufficiently negative a, O(x) is negative somewhere.

This completes the proof of the Theorem.

5.3.2 The Limiting Situation From the previous section, one can see from Kazdan and Warner's result that in the case when R(x) is non-positive, equation (5.8) has been solved completely for any a < 0. However, in the case when R(x) changes signs, we

have ao > -oo, and hence some questions remain. In particular, one may naturally ask: Does equation (5.8) have a solution for the critical value a = a0? In [CL1], Chen and Li answered this question affirmatively.

Theorem 5.3.2 (Chen-Li) Assume that R(x) is a continuous function on M. Let ao be defined in the previous section. Then equation (5.8) has at least one solution when a = ao.

5.3 The Negative Case

157

proof. The Outline. Basically, we approximate the limiting situation when a = ao by a sequence of solvable cases when a = ak > ao with corresponding solution uk. We then try to derive certain necessary estimates on the sequence and thus to obtain its convergence in certain spaces to a function, which we prove to be a solution. More precisely, let {ak} be a sequence of numbers such that ak > a0, and ak-->a0, as kioo.

We will prove the Theorem in 3 steps. In step 1, we minimize the functional, for each fixed ak,

Jk(u) =

JM 2

1vu12dA+ak

JM

udA-

JM

R(x)e°dA

in a class of functions that are between a lower and an upper solution. Let the minimizer be uk. Then let ak-4aa. In step 2, we show that the sequence of minimizers {uk} is bounded in the region where R(x) is positively bounded away from 0. In step 3, we prove that {uk} is bounded in H1(M) and hence converges to a desired solution.

Step 1. For each ak > a0, by Kazdan and Warner, there exists a solution ¢k of

-/4'k + ak = R(x)e'k.

(5.23)

We first show that there exists a constant co > 0, such that

¢k(x)?-co,VxEM andVk=1,2,-

.

(5.24)

Otherwise, for each k, let xk be a minimum of Ok (x) on M. Then obviously,

Ok(xk)-Y - oo, as On the other hand, since xk is a minimum, we have

-pOk(xk) <0,Vk=1,2,.. These contradict equation (5.23) because ak-Ya0 < 0, as k->oo. This verifies (5.24).

Choose a sufficiently negative constant A < -c0 to be the lower solution of equation (5.8) for all ak, that is,

158

5 Prescribing Gaussian Curvature on Compact 2-Manifolds

-,LA +ak
For each fixed ak, choose ao < dk < ak. By the result of Kazdan and Warner, there exists a solution of equation (5.8) for a = &k. Call it I k. Then

-0'V)k + ak >

&k = R(x)e''Pk(-)

i.e., bk is an upper solution of the equation

-emn + ak = R(x)e°(z).

(5.25)

Inspired by an idea of Ding and Liu [DL], we minimize the functional Jk (u) in a class of functions

H = {uECI(M) IA
Jk(u) >

2 JM

IVU12dA+ak fm udA - C, M

UIuII _ C2)2 - Ca.

e0kdA M (5.26)

Here we have used the Holder and Poincare inequalities. From (5.26), one can easily see that Jk is bounded from below. Also by (5.26) and a similar argument as in Chapter 2, we can show that there is a subsequence of a minimizing sequence that converges weakly to a minimum Uk of 4(u) in the set H. In order to show that Uk is a solution of equation (5.25), we need it to be in the interior of H, i.e., the strict inequality holds: A < uk(x) < Ok(x), V X E M.

(5.27)

We know that if the minimum is on the boundary of H, then it only satisfies a differential inequality, while what we need is a differential equation,

which requires it to be an interior point of H. To this end, we start from the fact that Uk satisfies a differential inequality, and then by a Maximum Principle, we deduce that Uk is in the interior of H. To verify (5.27), we first show that

uk(x) <'Ok(x), Vx E M.

(5.28)

5.3 The Negative Case

159

Suppose on the contrary that there exists xo E M, such that uk(xo) = bk(xo); we will derive a contradiction. Since uk(xo) > A, there is a b > 0, such that

A < uk(x), Vx E B5(xo). It follows that, for any positive function v(x) with its support in B5(xo),

there exists an e > 0, such that as -c < t < 0, we have uk + tv E H. Since Uk is a minimum of Jk in H, we have in particular that the function g(t) = Jk(uk+tv) achieves its minimum at t = 0 on the interval [-e, 0]. Hence 9r(0) < 0. This implies that

J [-Auk + ak - R(x)e°k]v(x)d A < 0.

(5.29)

M Consequently

-Auk(xo) +ak - R(xo)euk(=e) < 0.

(5.30)

Let w(x) = YPk(x) - uk(x). Then w(x) > 0, and xo is a minimum of in. It follows that -Aw(xo) < 0. While on the other hand, we have, by (5.30),

-Aw(xo) = -A''k(xo) - (-Auk(xo))

-ak +ak > 0,

again a contradiction. Therefore we must have

uk(x) < Ok(x), Vx E M. Similarly, one can show that

A
small t, and hence at Jk(u + tv) It=o = 0. Then a direct computation as we have done previously will lead to

-Auk + ak = R(x)e"k , V x E M. Step 2. It is obvious that the sequence {uk} so obtained in the previous step is uniformly bounded from below by the negative constant A. To show that it is also bounded from above, we first consider the region where R(x) is positively bounded away from 0. We will use a sup + inf inequality of Brezis, Li, and Shafrir [BLS] ( See also [CL15]): Lemma 5.3.5 (Brezis, Li, and Shafrir). Assume that V is a Lipschitz function satisfying

0
160

5 Prescribing Gaussian Curvature on Compact 2-Manifolds

-,Lu=V(x)e",xE$2, satisfies

su`pu+infu
(5.31)

Let x° be a point on 12 at which R(x°) > 0. Choose e so small such that

R(x) > a> 0 , V x E 0

B2E (x° ),

for some constant a. Let vk be a solution of

Ovk-ak=OxE,(2

vk(x)1

xEt3(2.

Let wk = uk + vk. Then wk satisfies

-pwk = R(x)e- °ke'°'. By the Maximum Principle, the sequence {vk} is bounded from above and from below in the small ball R. Since {uk} is uniformly bounded from below, {wk} is also bounded from below. Locally, the metric is pointwise conformal to the Euclidean metric, hence one can apply Lemma 5.3.5 to {wk} to conclude that the sequence {wk} is also uniformly bounded from above in the smaller ball BE(x°). Since x° is an arbitrary point where R is positive, we conclude that

{wk} is uniformly bounded in the region where R(x) is positively bounded away from 0. Now the uniform bound for {uk} in the same region follows accordingly.

Step 3. We show that the sequence {uk} is bounded in H'(M). Choose a small 6 > 0, such that the set

D = {xEMIR(x)>6} is non-empty. It is open since R(x) is continuous. From the previous step, we know that {uk} is uniformly bounded in D. Let K(x) be a smooth function such that

K(x) < 0, for x E D; and K(x) = 0 , elsewhere. For each ak, let vk be the unique solution of

-tvk + ak = K(x)e°k , x E M. Then since

5.3 The Negative Case

161

-Ovk + ak < 0 , and ak-+a0 The sequence {vk} is uniformly bounded. Let wk = Uk - Vk, then

-AWk = R(x)e"k - K(x)e'4.

(5.32)

We show that {wk} is bounded in H'(M). On the one hand, multiplying both sides of (5.32) by ewk and integrating on M, we have

R(x)e°k+wkdA-J K(x)e°kdA.

JIowk12ewkdA=J

M

M

(5.33)

D

On the other hand, since each uk is a minimizer of the functional Jk, the second derivative Jk (uk) is positively definite. It follows that, for any function

¢ E H'(M), we have 2

11

rrI

LWt-2

J(uk + t¢)J

> 0, L=0

or equivalently,

- R(x)e"k¢2]dA > 0. f[Io0I2 M Choosing 0 = eWk, we derive 4

Ipwkl2ewkdA > f R(x)e"k+wkdA.

JM

(5.34)

M

Combining (5.33) and (5.34), we arrive at

-J K(x)e"kdA> 3J Ipwkl2ewkdA. 4M

D

(5.35)

Noticing that {uk} is uniformly bounded in region D and {wk} is bounded

from below due to the fact that {uk} is bounded from below and {vk} is bounded, (5.35) implies that fM 1ewkl2dA is bounded, and so is fm JpukI2dA.

Consider uk = Uk - Uk, where ilk is the average of uk. Obviously ukdA = 0, IM

hence we can apply the Poincare inequality to ilk and conclude that {ilk} is bounded in H1(M). Apparently, we have

JM

u2dA<21 J ukdA+ukIM1l \M

162

5 Prescribing Gaussian Curvature on Compact 2-Manifolds

If fm uk dA is unbounded, then {v,k} is unbounded. Taking into account that {uk} is bounded in the region D where R(x) is positively bounded away from 0, we would have

r

JD

r ukdA=J Iuk-v.k12dA->oo, D

for some subsequence, a contradiction with the fact that {v,k} is bounded in H1(M). Therefore, {uk} must also be bounded in H1(M). Consequently, there exists a subsequence of {uk} that converges to a function uo in H' (M), and uo is the desired weak solution for

-Auo + ao = R(x)eu' This completes the proof of the Theorem.

5.4 The Non-negative Case-Variational Approaches Now we assume a > 0 and consider the equation

-Au + a = R(x)e°l'>,

x E M,

(5.36)

where A is the Laplace-Beltrami operator associated with the base metric go. 5.4.1 Obstructions

Integrating both sides of the equation (5.36), and using the fact that

-JM AudA=JM 7u.V1dA=0, we have

R(x)e°dA = aIMI. IM

It follows that, in the case a = 0, R(x) must change signs (We assume R(x) 0 0); and in the case a > 0, R(x) must be positive somewhere. In any case, we have derived that a necessary condition for equation (5.36) to have a solution is as follows:

R(x) > 0 somewhere.

(5.37)

In the critical case when X(M) = 2, besides this obvious necessary condition, there are other necessary conditions. For simplicity, we may just consider M = S2, the two-dimensional sphere with standard metric. In this case equation (5.36) becomes

5.4 The Non-negative Case-Variational Approaches

-,Lu + 2 = R(x)e'('),

X E S2.

163

(5.38)

It is known as the Nirenberg problem.

Kazdan and Warner [KW1] proved that, if u is a solution of equation (5.38), then

VoiVR(x)e" dx = 0, i = 1, 2, 3.

(5.39)

Js

Here Oi are the first spherical harmonic functions, i.e.,

20i (x), i=1,2,3; and for simplicity, we write S = S2 and dA = dx.

Actually, 4i (x) = xi in the coordinates x = (X1, X2, X3) of R. Later, Bouguignon and Ezin generalized this condition to X (R)e" dx = 0.

(5.40)

is

where X is any conformal Killing vector field on standard S2. Condition (5.39) gives rise to many examples of R(x) for which the equation (5.38) has no solution. In particular, a monotone rotationally symmetric function admits no solution. Then for which kinds of functions R(x) can one solve (5.38)? This has been an interesting problem in geometry.

5.4.2 The Variational Approach and Key Inequalities To obtain the existence of a solution for equation (5.36), people usually use a variational approach. First prove the existence of a weak solution; then, by a regularity argument, one can show that the weak solution is smooth and hence is the classical solution. To obtain a weak solution, we let

H1(M) := H1,2(M) be the Hilbert space with norm 11/2

muli1= 11M (IVuI2+u2)dAJ Let

H(M) = {u E H1(M) I fm udA = 0,

f

JM

R(x)e"dA> 0}.

164

5 Prescribing Gaussian Curvature on Compact 2-Manifolds

Then by the Poincare inequality, we can use /2

dull = [ f 1,7U12 dAl J

as an equivalent norm in H(M). Consider the functional

r

J(u) = 2 J IpuI2 dA - 4irX In / R(x)eu dA M

in H(M), where X = x(M). To estimate the value of the functional J, we introduce some useful inequalities.

Lemma 5.4.1 (Moser-Trudinger Inequality) Let M be a compact, two-dimensional Riemannian manifold. Then there exists constant C, such that the inequality

M

e4,ru2dA


(5.41)

holds for all u E H1(M), satisfying Ipuj2dA < 1

IM

and f udA = 0.

(5.42)

M

We skip the proof of this inequality. Interested readers many see the article of Moser [Moll or the article of Chen [C1]. From this inequality, one can derive immediately that

Lemma 5.4.2 There exists constant C, such that for all u E H' (M) holds

fm

e"dA < Cexp { 167 J Ipul2dA+ IMI f udA1 M

Proof. Let u

MII u(x)dA

be the average of u on M. Still denote 1/2

Hull = [IM Ioul2 dAl J

.

(5.43)

5.4 The Non-negative Case-Variational Approaches

165

i)ForanyVEH1(tLl)with V=0,letw=4.Then llwll=landw=0. Applying Lemma 5.4.1 to such w, we have 4xv2

ei

IM

dA
(5-44)

From the obvious inequality

'IV" -

2fv12 > 0, IIvil I

we we deduce immediately 2

2

v
e°dA < C exp { 16Olvul21

,

fm

Vv E Hl (M), with v = 0.

(5.45)

From the above proof, one can actually replace v by IvI on the left hand

side and obtain a stronger inequality: fm

e1°dA
Vv E H1(M), with v = 0.

ii) For any it E H1(M), let v = u-5. Then v = 0, and we can apply (5.45) to v and obtain

f

e'

dA < C exp

M

5ll 16_' l u l l 21

Consequently

J e"dA
ll

11

0

This completes the proof of the Lemma.

From Lemma 5.4.2, we can conclude that the functional J(-) is bounded from below in H(M) (Note X(M) < 2). In fact, by the Lemma, IM

e°dA < maxR(x) . C exp t

R(x)e" dA < rnax R I

16zr'lull2l

.

J

l

M

It follows that

J(u) ? 2 IluII2 - 47rx l ln(CmaxR) +

6,11U112

> -Sirln(CmaxR)+ (2 - 4 I Mull > -87rln(C max R).

\

/

]

(5.46) (5.47)

166

5 Prescribing Gaussian Curvature on Compact 2-Manifolds

Therefore, the functional J is bounded from below in H(M). Now we can take a minimizing sequence {uk} C H(M): inf J(u). J(uk)-> .EH(M)

-

5.4.3 Existence of Weak Solutions in Subcritical Case Theorem 5.4.1 Assume that R(x) is positive somewhere and 0 G X < 2. Then equation (5.36) possesses at least one weak solution. Proof. Since R(x) is positive somewhere, the set H(M) is non-empty. Let {uk} be a minimizing sequence of J(u) in H(M). By (5.46), there is co, cl > 0, such

that J(uk) 1 CoIIUkII2 - Cl.

It follows that {uk} is bounded in H1(M) and hence possesses a subsequence (still denoted by {uk}) which converges weakly to some element uo C- H1(M), As shown in Chapter 2, the integral is weakly lower semi-continuous, i.e.,

limki00 f IVUkl2dA>1 1pu0I2dA.

M

M

To show that the functional J is weakly lower semi-continuous,

J(ua) < imJ(uk) =

inf

uEH(M)

J(u),

we need the following:

Lemma 5.4.3 If {uk} converges weakly to uo in Hl (M), then there exists a subsequence (still denoted by {uk}), such that euo(z)dA, as k->oo.

M

M

We postpone the proof of this lemma for a moment. Now from this lemma, we obviously have f R(x)eukdAi J R(x)e"dA,

M

m

and it follows that JO is weakly lower semi-continuous. On the other hand, by the strong convergence uk-+uo

in L1(M)

5.4 The Non-negative Case-Variational Approaches

167

and the observation that

JM

R(x)e' dA> 0

(otherwise, we would have J(uk)->co), we have uo E H(M), and hence

J(uo) >

inf

.EH(M)

J(u).

Therefore, uo is a minimum of J in H(M) and is hence a weak solution of (5.36).

The Proof of Lemma 5.4.3. From Lemma 5.4.2, for any real number p > 0, we have fm

eP"dA < Cexp{

167r

JM

1pul2dA+ 4

And by the Holder inequality, for any 0 < p < 2, rM

Iv(eu)IPdA= J ep"IVUIPdA < ( L JM

\

M

e

JM

udA}.

\.

dA I ///

(5.48)

\E

IpuI2dA 12

(IM

.

(5.49)

Inequalities (5.48) and (5.49) imply that {e"k} is a bounded sequence in H1"P(M) for any 1 < p < 2. By the compact Sobolev embedding of Hz,P into L', there exists a subsequence of {e"k} which converges strongly to euo in L'(M). This completes the proof of the Lemma.

5.4.4 A Remedy for Critical Case-Recovering Coerciveness In the critical case when x = 2, from (5.46), we only know that the functional

J(u) is bounded below in H(M). Although we can still take a minimizing sequence, as we mentioned earlier, a minimizing sequence may not be bounded,

that is, it may `leak' to infinity. Take M = S2 for example. In the spherical polar coordinates (B, q5), 0 < B < ar, 0 < 0 < 2ir,

where 0 is the geodesic distance from the point (0, 0) to the north pole. Consider 1 - 12 ?PA (0) = In (1+Acos0)2 This is a family of well-known standard functions that satisfy -4V'A + 2 = 2e"A, x = (0, 0) E S2,

(5.50)

168

5 Prescribing Gaussian Curvature on Compact 2-Manifolds

where 4 is the Laplace-Beltrami operator associated with the standard metric on S2. One can easily see that, as A -* 1, tJA blows up at the south pole where 0 = 7r and goes to -oo elsewhere. One can verify that u,. = t/iA -T\ is a minimizing sequence of the functional

J(u) =

1

2

JS=

Ipu 2dA-82rlnJ 2e"dA Sz

corresponding to equation (5.50). Actually, they are all minima: J(uA) = -81r In 2.

However, one can verify that, as

IHHAll =

To prevent this from happening, we need to recover some coerciveness for the functional J. Although it is not true in general, we can still do this in some special subset of H(M), in particular, in the set of even functions-antipodal symmetric functions, where we can make the constant rsn in the key inequality (5.43) smaller and hence retain the coerciveness of J here. More precisely and generally, we have the following "Distribution of Mass Principle" from Chen and Li [CL7] [CL8]. For clarity, we state it only for S = S2.

Lemma 5.4.4 Let fll and fl2 be two subsets of S such that dist(f21,ff22) > b" > 0.

Let 0 < ao < 2. Then for any e > 0, there exists a constant C = C(a", bo, e), such that the inequality

J e"dA
(32-+e

Ill

l ]Ju]12+u

//

(5.51) ))))

holds for all u E Hl (S) satisfying fS2 e"dA

fs e"dA

- a°

fS2a e"dA and

fs e dA -

CIO.

Intuitively, if we think of e"lxl as the density of a solid sphere S at point x, then fS2 e"dA is the mass of the region S ?j, and fs e"dA the total mass of the sphere. The Lemma says that if the mass of the sphere is somewhat distributed in two parts of the sphere, more precisely, if the total mass does not concentrate at one point, then we can have a stronger inequality than (5.43). In particular, for even functions, the stronger inequality (5.51) holds.

5.4 The Non-negative Case-Variational Approaches

169

If we apply this stronger inequality to the family of even functions in H(S), we have

J(u) >

2IIull2

- 8ir fln(C maax R) + 132_ + el 1Iuii21 J

-8irln(CmaLxL

R) + \4 -87re)

>

IIull2.

/

Choosing c small enough, we arrive at

J(u) > g lltII2 - Ci, for all even functions it in H(S).

(5.53)

This stronger inequality guarantees the boundedness of a minimizing sequence, and hence it converges to a weak limit u, in H'(S). As we will show in the next section, ua so obtained is a weak solution of equation (5.38). In many articles, authors use the "center of mass" analysis, i.e., consider the center of mass of the sphere with density e"lsl at point x:

P(u) - fs x e"dA fs e"dA

and use this to study the behavior of a minimizing sequence (See [CD], [CD1], [CY], and [CYI]). Both analyses are equivalent on the sphere S2. However,

since "the center of mass" P(u) lies in the ambient space R3, on general two dimensional compact Riemannian surfaces, this kind of analysis can not be applied, while the "Distribution of Mass" analysis can be applied to any compact surfaces-even surfaces with conical singularities. Interested readers may see the author's paper [CL7] [CL8] for a more general version of inequality (5.51).

Proof of Lemma 5.4.4. Let gi, g2 be two smooth functions, such that

1>9i>-0, 9i(x)=1, forxEfli, i=1,2 and supp gin supp 92 = 0. It suffices to show that for it E H1(S) with fs u dA = 0, (5.52) implies e"dA < C ex p p

I (327r+E)ul2

Case i) If I19iuJI < II92uIl, then by (5.43)

(5.54)

170

5 Prescribing Gaussian Curvature on Compact 2-Manifolds

f eudA < s

1 f e" dA < 1 f ao n, ao Js C

ao

C

< ao <

1

e91 LdA

2

exp{16,"gluII +9iu} 1

exp{32irI91u+92uII2+9iu}

C exp{321 (1+Ei)IIuII2+c1(E1)IIuIIi2}. ao

(5.55)

for some small El > 0. Here we have used the fact that

II91u+92uII2 = f I'7(91u+92u)I2dA s

=JI('791+V92)u+(91+82),7uI2dA

S

< IIuII2 + C1IIuII IIfhIL= +

C21JU112

In order to eliminate the term u ft2 on the right hand side of the above inequality, we employ the condition fs u dA = 0. Given any ?7 > 0, choose a, such that rheas{x E SI u(x) > a} =,q. Applying

(5.55) to the function (u - a)+ - max{0, (u - a)), we have

f

e'dA < ea

s

f

s

e"-adA < eo . f

e(u-a)+dA

+a,} (5.56) where C = C(6, a,). By the Holder and Sobolev inequality (See Theorem 4.6.3)

IKu -a)+Ilia <_ 9II(u-a)+IIi4
(5.57)

Choose q so small that C171/2 < 1; then the above inequality implies II(u

a)+II ,2 < 2072' IInII2.

(5.58)

Now by the Poincare inequality (See Theorem 4.6.6)

f>oudA< f

IuIdA
hence for any 6 > 0,

a<511u112 +4b2.

(5.59)

Now (5.54) follows from (5.56), (5.58), and (5.59).

Case ii) If I92uII < II91uII, by a similar argument as in case i), we obtain (5.55) and then (5.54). This completes the proof.

5.4 The Non-negative Case-Variational Approaches

171

5.4.5 Existence of Weak Solutions in the Critical Case Even Functions

Theorem 5.4.2 (Moser (Mo]) If the function R is even, that is, if R(x) = R(-x) V x E S2, where x and -x are two antipodal points, then the necessary and sufficient condition for equation (5.38) to have a solution is that R(x) be positive somewhere. To prove the existence of a weak solution, as in the previous section, we

consider the functional

J(u) = 2

J

1puI2 dx - Brr 1,

J is

R(x)e" dx

in He(S) = {u E H(S) I u is even almost everywhere}. Let {uk} be a minimizing sequence of J in He(S). Then by (5.53), {uk} is bounded in H'(S) and hence possesses a subsequence (still denoted by {uk}) that converges weakly to some element up in Ht(S). Then by the compact Sobolev embedding of Hr into L2 uk-m, in La(S).

(5.60)

Consequently

u0(-x) = uo(x) almost everywhere

,

and fs uo(x)dA= 0,

that is, uo E He(S), and hence J(uo) > inf

J.

He(S)

As we have seen in the previous section, the functional J is weakly lower semi-continuous:

J(ua) < hmJ(uk) = inf J. H.(S)

Therefore, uo is a minimum of J in He(S). It follows that, for any v E He(S), we have

0 =< J'(uo), v >=

Js

7uovyv dA - fs R(8x )eu-dA is R(x)e"°vdA. s

For any w E H(S), let

v(x) = 2(w(x)+w(-x))

2(w(x)+"w-(x))-

172

5 Prescribing Gaussian Curvature on Compact 2-Manifolds

Then obviously, v E H,(S), and it follows that

0 =< J'(uo), v >= j(< J'(uo), w > + < J'(uo), w- >) _< X(uo), w >

.

Here we have used the fact that uo(-x) = uo(x). This implies that uo is a critical point of J in H1(S) and hence a constant multiple of uo is a weak solution of equation (5.38). p

General Symmetric Functions Chen and Ding [CD] generalized Moser's result to a broader class of symmetric functions, as we will describe next.

Let 0(3) be the group of orthogonal transformations of R. Let G be

a

closed finite subgroup of 0(3). Let

FG={xES2Igx=x,VgEG} be the set of fixed points under the action of G. The action of G on S2 induces an action of G on the Hilbert space HI(S): 9 [u] (x) " u(9x)

under which the fixed point subspace of H'(S) is given by

X = {u E H'(S) I u(gx) = u(x), Vg E G}. Assume that R(x) is G-symmetric, or G-invariant, i.e.,

R(gx) = R(x), Vg E G.

(5.61)

Let x. = X f l H(S). Recall that

H(S) = {u e HI (S) f udA =0, J R(x)e'dA > 01. S

Under the assumption (5.61), the functional

J(u) = l

Js

1,7UI2

dx - 8rr 1,

JS R(x)e,- dx

is G-invariant in X i.e.,

J(g[u])=J(u), VgEG, VuEX,. We seek a minimum of J in X.. The following lemma guarantees that such a minimum is the critical point we desired.

5.4 The Non-negative Case-Variational Approaches

173

Lemma 5.4.5 Assume that uo is a critical point of J in X.; then it is also a critical point of J in H(S). proof. First, for any g E G and any u, v E H(S), we have

< X(U), g['v] >=< J'(9-1[u]), v >,

(5.62)

where g-' is the inverse of g. This can easily be seen from

< J'(u),v >=

s

7u,7vdA-

8ir

fs ReudA s

Re' vdA.

Assume

G = {gl, ... gm}.

For any w E H(S), let

v=

1

(gi[w]+.+gm[w]).

Then for any g E G g [VI = I (ggi [w] + .

+ ggm [w]) = v.

Hence v E X.. It follows that

Y<J'(uo),gi[w]> 0=<J'(uo),v>= 1 m m

=

m

<J'(ga1[u,]),w>=-<J'(uo),w>

=<X(vo),w>. Therefore, uo is a critical point of J in H(S).

Theorem 5.4.3 (Chen and Ding [CDJ). Suppose that R E C°(S2) and satisfies (5.61) with maxs R > 0. If FG # 0, let in = maxFF R. Then equation (5.38) has a solution provided one of the following conditions holds:

(i) Fa = 0, (ii) in < 0, or (iii) in > 0 and c. = infx. J < -87r In 47rm.

Remark 5.4.1 In the case when the function R(x) is even, the group G = {id, -id}, where id is the identity transform, i.e.,

id(x) = x and - id(x) _ -x V x E S2. Obviously, in this case FG is empty. Hence the above theorem contains Moser's existence result as a special case.

5 Prescribing Gaussian Curvature on Compact 2-Manifolds

174

To prove this theorem, we need another two lemmas.

Lemma 5.4.6 (Onofri [Onj) For any it E H1(S) with fs u dA = 0, holds e"dA < 47r exp { is

ll

167r

f IpuI2dA } . s

(5.63)

11

This lemma provides another best constant 47r besides 167r, and it is needed in the finer analysis in the critical case. Interested readers may find its proof in [On] or [Ho].

Lemma 5.4.7 Suppose that {ui} is a sequence in H(S) with J(ui) < Q. if lluijj-*oo, then there exists a subsequence (still denoted by {u;}) and a point E S2 with R(e) > 0, such that

iR(x)e"idA=(R(e)+o(1))ieuidA,

(5.64)

lim J(ui) > -87rln47rR(e).

(5.65)

is

is

and

Proof. We first show that there exists a subsequence of {ui} and a point L; E S2, such that for any c > 0, we have

e"idA-40.

(5.66)

Otherwise, there exists two subsets of S2, 01 and 122, with dist(S7i, .22) > eo > 0 such that {u;} satisfies all the conditions in Lemma 5.4.4. Hence the inequality

f e"idA 0. Then by the previous argument, we conclude that IIuiII is bounded. This contradicts our assumption and hence verifies (5.66). Now (5.66) and the continuity assumption on R(x) implies (5.64). Roughly speaking, the above argument shows that if a minimizing sequence

of J is unbounded, then passing to a subsequence, the mass of the surface S with density e"i(x) at point x will be concentrating at one point e on S. To verify (5.65), we use (5.64) and Onofri's Lemma:

5.4 The Non-negative Case-Variational Approaches

1vnI2dA - 87rln(R(4) +o(1))

J(ui) = 2

>

r

2

r e',dAr

Ipu12dA - 8aln(R(£) +o(1)) J-

=2

f

175

8rlnJ e"'dA rs

Ipu12dA - 8irln(R(4) +o(1)) - 87r(ln47r+ 16 J Ipu12dA)

_ -87r ln(47R(4) + o(1)).

Letting i->oo, we arrive at (5.65). Since the right hand side of (5.65) is bounded above by 0, we must have R(4) > 0. This completes the proof of the Lemma.

The Proof of Theorem 5.4.3. Define

c,=infJ. X. As we argued in the previous section (see (5.47)), we have c, > -oc. Let {ui} C X. be a minimizing sequence, i.e., J(ui)ic., as i-+eo. Similar to the reasoning at the beginning of this section, one can see that we only need to show that {ui} is bounded in H'(S). Assume on the contrary that IIuiII-*oo. Then from the proof of Lemma 5.4.7, there is a point £ E S2, such that for any e > 0, we have e"'dA->0.

(5.67)

s\BE(C)

Since ui is invariant under the action of G, we must also have e'dA->0.

(5.68)

S'\BE(gf)

Because a is arbitrary, we deduce that gf = f, that is 4 E FG. Now under condition (i), Fc is empty. Therefore, {ui} must be bounded. Moreover, by Lemma 5.4.7,

R(4)>O and c , > -87rln47rR(4). Noticing that m > R(4) by definition, this again contradicts with conditions (ii) and (iii). Therefore, under any one of the conditions (i), (ii), or (iii), the minimizing 'sequence {ui} must be bounded and hence possesses a subsequence converging

weakly to some u" E X. Thus a constant multiple of uo is the desired weak solution of equation (5.38). Here conditions (i) and (ii) in Theorem 5.4.3 can be easily verified through the group G or the given function R. However, one may wonder under what

5 Prescribing Gaussian Curvature on Compact 2-Manifolds

176

conditions on R do we have condition (iii). The following theorem provides a sufficient condition.

Theorem 5.4.4 (Chen and Ding ]CDI]). Suppose that R E C2(S2) soinvariant, and in = maxF0 R > 0. Let D = {x E S2 I R(x) = m}.

If AR(x") > 0 for some x0 E D, then C. < -8ir In 4irm,

(5-69)

and therefore (5.38) possesses a weak solution.

Proof. Choose a spherical polar coordinate system on S := S2:

(9, 0), - 2 < 9 < 2 , 0 < ¢ < 21r, with xo = (2,

as the north pole.

Consider the following family of functions:

_ U), (0, ¢) = In

(1

2

1 a sin9)2' VA = ua - 4n J uadA, 0 < A < 1.

One can verify that this family of functions satisfy the equation -Aua + 2 = 2e"a.

(5.70)

As a-+1, the family {u,\} `blows up' (tends to oo) at the one point x,, while approaching -oo everywhere else on S2. Therefore, the mass of the sphere with density e°,(y) is concentrating at point xo as A-41. It is easy to see that vA E H(S) for A close to 1. Moreover, we claim that va E X, i.e., v,\(gx) = va(x), Vg E G.

In fact, since g is an isometry of S2, and gx, = x0, we have d(9x, xo) = d(9x, gxo) = d(x, xo),

where

is the geodesic distance on S2. But vA depends only on 0 =

d(x, x.), so V,\ (9x) = V,\ (X), V9 E G.

It follows that va E X. for A sufficiently close to 1. By a direct computation, one can verify that S

Ipua12dA+87ru,\

=0.

5.4 The Non-negative Case-Variational Approaches

177

Consequently,

J(va)=-8trlnJ R(x)e'AdA.

is

Notice that c, < J(va) by the definition of c,. Thus, in order to show (5.69), we only need to verify that, for A sufficiently close to 1 R(x)e"adA> 47rm.

(5.71)

Js

Let b > 0 be small, we have r2,r

r Re"'dA = (1 - A2) If

s

m]

a-s

o

+J

f,-a

cos O

(1-ain0)2

Cos 9

[R(O,¢) - m]

J a

(1 - A sin B)2

d&dO

dodo + 47m

= (1 - A2){I + 77} + 4am. Here we have used the fact

I J

e"a dA = 4tr,

which can be derived directly from equation (5.70). For each fixed 5, it is easy to check that the integral II remains bounded as A tends to 1. Thus (5.71) will hold if we can show that the integral I -4 +oo as A-r1. Notice that xo is a maximum point of the restriction of R on Fc. Hence by the principle of symmetric criticality, xo is a critical point of R, that is, VR(xo) = 0. Using this fact and the second order Taylor expansion of R(x) near x0, we can derive

[AR(xa) Cos20+0 (10 -

7=T r

(1-Asin0)2 z

COS3 B

c°j _a (1-Asin0)2

z 12) ] cos0do

dB.

z

Here co is a positive constant. We have chosen b to be sufficiently small and have used the fact that AR(xo) is positive. From the above inequality, one

-do

can easily verify that as A--+ 1, I-+ + oo, since the integral Cos3 0

_a (1-sin0)2 diverges to +oo.

This completes the proof of the Theorem.

The Yamabe Problem and Prescribing Scalar Curvature on S'E, for n > 3

6.1

Introduction 6.1.1

6.1.2 6.2 6.3

The Variational Approach for the Yamabe Problem The Variational Approach for Prescribing Scalar Curvature 6.3.1

6.3.2 6.4

The Yamabe Problem Prescribing Scalar Curvature on Sn

Estimate the Values of the Functional The Variational Scheme

The a priori Estimates for Prescribing Scalar Curvature 6.4.1

6.4.2 6.4.3

In the Region Where R < 0 In the Region Where R is small In the Region Where R > 0

6.1 Introduction 6.1.1 The Yamabe Problem Given a compact, smooth, and orientable manifold M of dimension n > 3 with a Riemannian metric go, does there exist a metric g conformal to go for which the scalar curvature of g is constant? In other words, does there exist a smooth positive function p(x) on M for which the metric g = P(x)g, has constant scalar curvature? This is the well-known Yamabe problem. It was named after the mathematician Hidehiko Yamabe, who claimed to have a solution in 1960 [Y], while a critical error in his proof was discovered in 1968

6 The Yamabe Problem and Prescribing Scalar Curvature on S^, for n > a

180

by Neil Trudinger [T]. The combined work of Neil Trudinger [T], Thierry Aubin [Aul], and Richard Schoen [Sc] provided a complete solution to the problem as of 1984.

As we introduced in Chapter 4, if we let g = u^^==go, then

4n - 1) R0(x)u = 4(n - 1) Rx)u

x E M,

,

(6.1)

where D is the Laplace-Beltrami operator associated with the metric go, R0(x) is the scalar curvature of go, and R(x) the scalar curvature of g.

Hence to solve the Yamabe problem, it is equivalent to find a positive function u on M such that

n-2

-emu +

"+2 R,(x)u = A u^-_

x E M,

(6.2)

4(n - 1)

for some constant A.

To show the existence of a solution for (6.2), a variational approach is usually employed. Let H'(M) be the Hilbert space of functions on M with norm IIkIIH = U

f

M

(IDuI2+u2)dVo.

Let dVo be the volume element of (M, go), and for simplicity, write 2

K0(x) = 4(n

)Ro(x)

and T=

1

Traditionally, people minimized the quotient Q(u)

_

JM

[I,7ul2 + Ko(x)u2] dVo Ilullr+1

in H1(M) \ {0}, where

Ilullp = (IM

IuIPdVo)

is the LP norm on (M, go). Or equivalently, one can minimize the functional [1,7UI2

J(u) = J

+ Ko(x)u2] dVo

M under the constraint

Gr = {u E H'(M) I IIullz+r =1}.

n±2.

6.1 Introduction

181

We will adapt the latter scheme here. It is easy to verify that, the minima, or more generally, the critical points of J() in G. are weak solutions of (6.2) with some constant A. Unfortunately, the Sobolev embedding H'(M) --+ LT+1(M) is not compact, and this causes the non-compactness of the corresponding variational scheme. This value r is called the critical exponent. Let

mo =nEG, inf J(u). It is well-known that the key ingredient in solving the Yamabe problem is to verify that (6.3) mo < So,

if M is not conformal to the unit sphere S' with standard metric, where S. is the best constant in the Sobolev embedding from Hr (Rn) to LT+1(R"), or from H1(Sn) to L7+1(Sn) as indicated by the following:

Theorem 6.1.1 For all u E Hl(Rn), we have Sju11z+r

- IIvuU

(6.4)

,

where

n S°-n(n-2) r(2)r(n+1)Wn_1 r(n+1)

-

n(n-2) a 4

Wn

(6.5)

is the largest possible constant. Here IIuIIy is the L" norm of u in Rn, w,i_1 and wn are the areas of (n - 1) and n-dimensional unit spheres, respectively, with

n7r 2

+

wn.-1 T-(22

where r(-) is the Gamma function. Inequality (6.3) is verified by Aubin [Aul] for manifolds of dimensions n > 6 that are not locally conformally flat, and by Schoen [Sc] for the remaining cases. The proofs in [Aul] and [Sc] were unified by Lee and Parker [LP] via introducing the conformal normal coordinates. Later, Wang [W] provided a simpler proof by using the positive mass theorem of Schoen and Yau [SY], but not the conformal normal coordinates. The proof of this inequality is too advanced to include in this book, and the interested reader may refer to the above mentioned articles. However, we will demonstrate, via a standard variational approach, the proof of the following theorem:

Theorem 6.1.2 Assume that (6.3) holds. Then the Yamabe problem (6.2) Possesses a positive solution.

We will prove this Theorem in Section 6.2.

6 The Yamabe Problem and Prescribing Scalar Curvature on S", for n > 3

182

6.1.2 Prescribing Scalar Curvature on S'° A natural and obvious generalization of the Yamabe problem is as follows: Given a compact, smooth, and orientable manifold M of dimension n > 3 with a Riemannian metric go, which functions R(x) can be realized as the scalar curvature of some metric g that is pointwise conformal to go? This is known as the "prescribing scalar curvature" problem.

As we mentioned in the previous subsection, if we let g = u=go, then it is equivalent to consider the existence of a positive solution to equation (6.1). When M = S^ with standard metric, this problem becomes the most difficult. In this case, equation (6.1) becomes

-Au + n(n 4

2) u

= 4(n - 1) R(x)u-2 , u> 0, x E S",

(6.6)

where A is the Laplace-Beltrami operator associated with the standard metric

on S'. The equation is "critical" in the sense that lack of compactness occurs. Besides the obvious necessary condition that R(x) be positive somewhere, there are well-known obstructions found by Kazdan and Warner [KW2] and later generalized by Bourguignon and Ezin [BE]. The conditions are these:

i X (R)dVg = 0

(6.7)

where dVg is the volume element of the conformal metric 9, and X is any conformal vector field associated with the standard metric go. We call these Kazdan-Warner type conditions. These conditions give rise to many examples of R(x) for which equation (6.6) has no solution. In particular, a monotone rotationally symmetric function R admits no solution. In the last two decades, numerous studies were dedicated to these problems and various sufficient conditions were found (please see the articles [Ba], [Ball, [BC], [BCCH], [C], [CC], [ChL], (CL3], [CL4], [CL6], [CL10], [CS], [CY1], [CGY], [ES] [Ka], [J3], [J4], [LiYl], [LiY2], [LiY3], [Linl], [Sc], (Sell, [SZ], and the references therein). However, among others, one problem of common

concern left open was this: Were the necessary conditions of the KazdanWarner type also sufficient conditions? In the case where R is rotationally symmetric, the conditions become:

R > 0 somewhere and R' changes signs.

Then (a) Is (6.8) a sufficient condition? and (b) if not, what are the necessary and sufficient conditions?

(6.8)

6.1 Introduction

183

Kazdan listed these as open problems in his CBMS Lecture Notes [Ka]. Recently, the authors answered question (a) negatively [CL3] [CL4]. We found some stronger obstructions. Our results imply that for a rotationally symmetric function R, if it is monotone in the region where it is positive, then problem (6.6) admits no solution unless R is a constant. In other words, a necessary condition to solve (6.6) is that

R'(r) changes signs in the region where R is positive.

(6.9)

However, is this a sufficient condition? For prescribing Gaussian curvature equation (5.38) on S2, Xu and Yang [XY] showed that if R is 'non-degenerate,' then (6.9) is a sufficient condition. For equation (6.6) on higher-dimensional spheres, a major difficulty is that

multiple blow-ups may occur when approaching a solution by a mini-max sequence of the functional or by solutions of subcritical equations. In dimensions higher than 3, the `non-degeneracy' condition is no longer sufficient to guarantee the existence of a solution. It was illustrated by a counter example in [Bi] constructed by Bianchi. He found some positive rotationally symmetric function R on S°, which is non-degenerate and nonmonotone, and for which the problem admits no solution. In this situation, a more proper condition is called the `flatness condition'. Roughly speaking, it requires that at every critical point of R, the derivatives of R up to order (n - 2) vanish, while some higher, but less than nth, order derivative is distinct from 0. For n = 3, the 'non-degeneracy' condition is a special case of the `flatness condition'. Although people are wondering if the `flatness condition' is necessary, it is still used widely today (see [CY2], [LiY2], and [SZ]) as a standard assumption to guarantee the existence of a solution. The above mentioned Bianchi counterexample seems to suggest that the `flatness condition' is somewhat sharp.

Now, a natural question to ask is this: Under the `flatness condition,' is (6.9) a sufficient condition? We answer the question affirmatively. In the next two sections, we will prove that, under the `flatness condition,' (6.9) is a necessary and sufficient condition for (6.6) to have a solution. This is true in all dimensions n > 3, and it applies to functions R with changing signs. Thus, we essentially answer the open question (b) posed by Kazdan. There are many versions of `flatness conditions,' and a general one was presented in [LiY2] by Y. Li. Here, to better illustrate the idea, in the statement of the following theorem, we only list a typical and easy-to-verify one.

Theorem 6.1.3 Let n > 3. Let R = R(r) be rotationally symmetric and satisfy the following flatness condition near every positive critical point ro:

6 The Yamabe Problem and Prescribing Scalar Curvature on S", for n > 3

184

R(r) = 11(r0) + alr - r" Ia + h (jr - r01), with a 00 and n-2< a< n (6.10) where h'(s) = o(sa-1). Then a necessary and sufficient condition for equation (6.6) to have a solution is that R' (r) changes signs in the regions where R > 0.

The theorem is proved by a variational approach. We blend in our new ideas with other ideas in [XY], [Ba], [LiY2], and [SZ]. We use the `center of mass' to define neighborhoods of `critical points at infinity,' obtain some quite sharp and clean estimates in those neighborhoods, and construct a max-mini variational scheme at sub-critical levels and then approach the critical level.

Outline of the proof of Theorem 6.1.3. Let 7" _ n(4 2) and r =

n±2. We

first find a positive solution up of the

subcritical equation

-Au + 7,u = R(r)u1',

(6.11)

for each p < r and close to T. Then let p-*r, and take the limit. To find the solution of equation (6.11), we construct a max-mini variational scheme. Let JJ(u):=

L Rup+rdV J

and

E(u)

f "(1vu12 +7"u2)dV.

We seek critical points of Jp(u) under the constraint

H = In G H1(S") I E(u) = 7 IS'"I, u> 0}, where IS"I is the volume of S. If R has only one positive local maximum, then by condition (6.9), on each

of the poles, R is either non-positive or has a local minimum. Then similar to the approach in [C], we seek a solution by minimizing the functional in a family of rotationally symmetric functions in H. In the following, we assume that R has at least two positive local maxima. In this case, the solutions we seek are not necessarily symmetric. Our scheme is based on the following key estimates on the values of the functional J. (u) in a neighborhood of the `critical points at infinity' associated with each local maximum of R. Let r1 be a positive local maximum of R. We prove that there is an open set G1 C H (independent of p), such that on the boundary 8G1 of G1, we have

6.1 Introduction

Jp(u) < R(rl)ISRI - a,

185

(6.12)

while there is a function 01 E G1, such that

Jp(01) > R(ri)IS"I -

(6.13)

Here 8 > 0 is independent of p. Roughly speaking, we have some kind of 'upside-down mountain pass' associated with each local maximum of R. The set Gl is defined by using the `center of mass'. Let r1 and r2 be two smallest positive local maxima of R. Let 1' and 02 be two functions defined by (6.13) associated with r1 and r2. Let ry be a path in H connecting 01 and 02, and let r be the family of all such paths. Define cp = sup min Jp(u).

(6.14)

lEr 7

For each p G r, by compactness, there is a critical point up of Jp(-), such that Jp(up) = Cp.

Intuitively, in the graph of functional Jp, the neighborhoods of (01, J(01)) and (02, J(02)) look like the two humps of a camel, and P = (u7,, J(up)) is the saddle point between the two humps, as shown in the following figure.

(''2, J('2))

(0i, J(t1))

H1(S")

Fig. 6.1. Saddle Point P

Obviously, a constant multiple of up is a solution of (6.11). Moreover, by (6.12),

Jp(up) < R(ri) I S"I - S,

for any positive local maximum ri of R.

(6.15)

6 The Yamabe Problem and Prescribing Scalar Curvature on S°, for n > 3

186

To find a solution of (6.6), we let p-+r, and take the limit. To show the convergence of a subsequence of {u,}, we established a priori bounds for the solutions in the following order. (i) In the region where R < 0: This is done by the `Kelvin Transform' and a Maximum Principle. (ii) In the region where R is small: This is mainly due to the boundedness of the energy E(up). (iii) In the region where R is positively bounded away from 0: First due to the energy bound,-{up} can only blow up at finitely many points. Using a Pohozaev type identity, we show that the sequence can only blow up at one point, and that point must be a local maximum of R. Finally, we use (6.15) to argue that even a one-point blow up is impossible, thus establishing an a priori bound for a subsequence of {up}. In Section 6.3, we carry on the max-mini variational scheme and obtain a solution up of (6.11) for each p. In Section 6.4, we establish a priori estimates on the solution sequence {up}.

The arguments in Sections 6.3 and 6.4 are rather complicated, and beginners in this area are welcome to skip them.

6.2 The Variational Approach for the Yamabe Problem In this section, we prove Theorem 6.1.2. The following special case of Aubin's theorem (See 2.21 Theorem in [Au]) plays an important role in the proof.

Theorem 6.2.1 Let M be a complete n-dimensional manifold with bounded curvature and injectivity radius b > 0. Then for any e > 0, there exists a constant A(e), such that every u e HI(M) satisfies: So11u117}I <_

(6.16)

where Ilullp is the LP norm of u on M and S. is the best constant defined by (6.5).

The proof of Theorem 6.1.2. Since at critical exponent r, the variational problem is not compact, we first consider the subcritical case. We minimize the same functional J in the set Gp for 1 < p < T. Let mp = inf J(u). nEGp

Lp+I(M), we will show in the Due to the compact embedding H1(M) following Step 1 that for each 1 < p < r, there exists a positive function up

6.2 The Variational Approach for the Yamabe Problem

187

which is a minimum of J in G, and hence is a weak solution of

-pup + K.(x)up = Apup,

(6.17)

for some constant Ap. It is easy to see that Ap = mp,

if we multiply both sides of (6.17) by up and then integrate over M. Then in Step 2, we let p-Yr and study the convergence of the sequence

{up}.

Step 1. The existence of a positive solution of equation (6.17) for each

1
J(uk)-imp = inf J. Gp

(6.18)

Since p > 1 and M is compact, by the Holder inequality, there is a constant C, such that IInkI12 < GIIukII2+1 = C.

(6.19)

IIuIIH = J(u) + fm (I - K0(x))u2dVo.

(6.20)

And by the definition of J

Here IIuIIH is the H1 of it defined in the previous section. Now (6.18), (6.19),

and (6.20) imply that the sequence {uk} is bounded in H1(M) and hence possesses a subsequence (still denoted by {uk}) that converges weakly in H1(M) to some element up E H'(M). By the compact Sobolev embedding

H1 "" LP}', {uk} converges strongly in Lp+l(M) as well as in L2(M). Hence

up E Gp.

It follows that J(up) > rnp.

(6.21)

On the other hand, similar to an argument made in Chapter 2, we can show that IM

Ipupl2dVo < liminf f Ipukj2dVo. k

M

Consequently,

J(up) <'mp. From this and (6.21), we conclude that up is a minimum of J in Gp and hence a weak solution of (6.17). Then by a similar bootstrap argument made in

6 The Yamabe Problem and Prescribing Scalar Curvature on S^, for n

188

3

Chapter 3, Section 3.3.1, one can lift the regularity of up up to C2 through the equation. We may assume that up is non-negative because we can replace up by jupl if necessary. Moreover, up is not identically zero, since f uPP+r dVo = 1. M

To see that up is strictly positive, we rewrite equation (6.17) as

-Aup + c(x)up = 0 with

c(x) = (Ko(x) - .pup-r). Since up > 0 and c(x) is bounded, we can apply the Strong Maximum Principle

(see Theorem 7.3.3 in Chapter 7) to conclude that up > 0. Step 2. Analyzing the limit as p->r. We consider three possible cases.

Case (i) The zero case: There is some 1 < P < r, such that rap = 0. Then by (6.17), we have

-Aup+K.(x)up=0-un=0-4P. It follows that the conformal metric g" - u;-'g, has constant scalar curvature 0, and we are done. Case (ii) The negative case: There is some 1 < p" < r, such that mp < 0. Then by (6.17), we have

-Ant + K0(x)up = mp

p

- mpup r u p

Compare this with equation (6.1), we see that the conformal metric uP '-'g, has a strictly negative scalar curvature

4n-1 Now we start from this conformal metric g. Denote the corresponding Laplace-Beltrami operator by A and let K(x) = 4 n ilR(x). Consider the new equation

-Au + K(x)u = A up, x E M.

(6.22)

Using a similar argument as in Step 1, we can show that, for each 1 < p < r, (6.22) possesses a positive solution. Let {pk} be a sequence that pk->T. Denote the corresponding solution by Uk and the constant by Ak:

6.2 The Variational Approach for the Yamabe Problem

-Auk + K(x)uk = Akukk, X E M,

189

(6.23)

where

Ak = inf J = J(uk) =

G,

JM

[17uk12 +K(x)uk] dV9.

(6.24)

We first show that all Ak are negative and bounded away from zero. That is, there is a constant co > 0, such that Ak < -CO,

for all k.

(6.25)

Notice that M is compact and k is negative, hence there exists a constant cl, c2 > 0, such that

-c2 < K(x) < -c1, V X E M.

(6.26)

Let I MI denote the volume of (M, g). Then it is easy to see that the constant function Ak = (poi) Pk+' is in Gpk. Consequently, from (6.24) and (6.26),

Ak <J(Ak)

<-CiIMIPPk+1-p-e1IMl+++ 7+-1,

as k->oo.

This verifies (6.25). Next we show that the solution sequence {uk} is uniformly bounded. Let xk be a maximum of Uk, then

-Auk(xk) J 0. It follows from this and (6.23), (6.24), and (6.25) that -C2Uk(xk)

K(xk)uk(xk)

AkUk (Xk) C -Coukk(xk),

i.e,

,apk-1(xk) < k

e2

co

Hence {uk} is uniformly bounded for all x E M and for all k. Then we deduce from equation (6.23) that {uk} is bounded in W2'q(M) for any 1 < q < oo, and by the compact embedding W 2,q (M) yy H1(M), there exists a subsequence which converges to some element uo E H1(M). The limit uo(x) is the desired solution of the Yamabe problem.

Case (iii) The positive case: Assume mk > 0 for all k.

Let {pk} be a sequence such that pk-.r. For each pk, let Uk > 0 be a corresponding minimum of J in Gpk:

J(uk) = Mk = inf J. GPk

6 The Yamabe Problem and Prescribing Scalar Curvature on Sn, for n > 3

190

Let

mo=infJ. GI

Claim (a):

EM Mk = mo. k +co

(6.27)

On the one hand, by the Holder inequality IIukIIPk+1

<_ IIukIIr+1 IMI '+1 ok+1

Obviously, the last term in the above approaches 1 as k-+oo, and hence

liminf

kim

Let ilk =

w IIr+1'

IIukIIr+1

>_ liminf IInkIIPk+1 = 1. k-Yoo

(6.28)

Then

Mk = J(uk) = IIukII+1J(uk) >_ IIukII+1 mo.

It follows from (6.28) that

liminfmk ? rno k-+w

(6.29)

On the other hand, let {vk} be a sequence that IvkIIr+1 = 1,

Let vk =

Ilvkllpi+l .

and J(vk)-*mo

Then J(yk)

mi < J(f)k) =

Iluk IIni+1

Fixing k at this moment, and letting i->oo, we have

limsupmi <J(vk), Vk.

i- o.

Then letting kioo, we arrive at

limsupmi <mo. i-+oo

This, together with (6.28), verifies the Claim. Claim (b):

mo < S.

(6.30)

6.2 The Variational Approach for the Yamabe Problem

191

In fact, by Theorem 6.2.1, the best constant in the Sobolev embedding is the same on any compact manifold. Thus there exist a sequence of C°°(M) functions {vi}, such that IIviII +i = 1,

Iv:II2i0,

and IIvvillz-->So.

Obviously,

This verifies the Claim.

By a similar argument as in Step 1, one can see that {uk} is bounded in H1(M) and hence possesses a subsequence (still denoted by {uk}) that converges weakly to some element u° in H1(M). Each uk is a weak solution of (6.17) with p = Pk, that is, for any 0 E C°° (M), r

1 {VUkVO+K°(x)ukO}dV°=mkJ ukk¢dV M

(6.31)

M

Due to the weak convergence Uk s u° in H'(M), we derive LM {puoVO+K°(x)uo0}dV° =m°fM ua¢dV°. Hence u° is a weak solution of

-/ uo +K°(x)u° = m°uo.

(6.32)

It follows from the Maximum Principle that u° is either positive everywhere or identically zero.

Claim (c): If m° < S°, then u, is positive and hence is the desired solution. To see this, we apply Theorem 6.2.1 to uk: S°IIukIIr+1 <(1+6)IIvuk112+Ap(E)IIukII2

_ (I+E) { J(U.k) - J

Ko(x)ukdVo

M

11

} +A(E)IIukII2,

(6.33)

where c is any positive number.

By the Holder inequality

IItkIIT+1 ?IMI +: Pk+ IIUkIIPk+1= IMI

.4

(6.34)

If u° __ 0, then by the weak convergence Uk 1 u° in H1 (M) and hence the strong convergence uk->u° in L2(M), we derive from (6.33) and (6.34) that

S. < (1+E)mo for any positive number c, a contradiction with our assumption that m° < Sp. This verifies the Claim and therefore completes the proof of Theorem 6.1.2.

6 The Yamabe Problem and Prescribing Scalar Curvature on S°, for n > 3

192

6.3 The Variational Approach for Prescribing Scalar Curvature In this section, we construct a max-mini variational scheme to find a solution of

-Lu+7u = R(r)up

(6.35)

for each p < T:= nn+2 Let

JP(u) :=

Jsn

RuP+rdV

and

E(u)

Jps

Let

(u,v)= J (vupv+7nuv)dV s^

be the inner product in the Hilbert space H1(Sn), and

lull:=

E(u)

be the corresponding norm.

We seek critical points of JP(u) under the constraint

H := In E H1(S1) I E(u) = E(1) = 7n 1Sn1, u > 0}.

where IS'l is the volume of S. One can easily see that a critical point of JP in H multiplied by a constant is a solution of (6.35).

We divide the rest of the section into two parts. First, we establish the key estimates (6.12) and (6.13). Then, we carry on the max-mini variational scheme.

6.3.1 Estimate the Values of the Functional To construct a max-mini variational scheme, we first show that there is some kind of `upside-down mountain pass' associated with each positive local maximum of R. Unlike the classical ones, these `mountain passes' are in some neighborhoods of the `critical points at infinity' (see Proposition 6.3.1 and 6.3.2 below).

Choose a coordinate system in Rn+r, so that the south pole of S^ is at the origin 0 and the center of the ball Bn}1 is at (0, - - - , 0, 1). Here our Sn

6.3 The Variational Approach for Prescribing Scalar Curvature

is the boundary of this B"+1 As usual, we use Rn}1.

193

to denote the distance in

Define the center of mass of u as 4(u)

f xut+i(x)dV

(6.36)

fsn uT+1(x)dV

It is actually the center of mass of the sphere S" with density uT+l (x) at the point X.

We recall some well-known facts in conformal geometry.

Let qq be the standard solution with its `center of mass' at q E B"+1 that is, 0q E H and satisfies (6.37) -Au + 7nu = 7nUT. We may also regard Oq as depending on two parameters A and 4, where 4 is the intersection of Sn with the ray passing through the center and through the point q. When 4 = 0 (the south pole of Sn), we can express, in the spherical polar coordinates x = (r, 9) of Sn (0 < r < 7r, 0 E Ss-1) centered at the south pole where r = 0, n-z 2 Oq = Oa,4

cos2 2 + sine Z

(6.38)

with 0 < A < 1. As A- O, this family of functions blows up at the south pole, while tending to zero elsewhere. Correspondingly, there is a family of conformal transforms

T g : H i- 4H;

Tgu := u(h),(x))[det(dha)] a_2

(6.39)

with

ha(r,0) = (2tan 1(A tan 2),0). Here det(dha) is the determinant of dha and can be expressed explicitly as

det(dha) =

A

cos2 2 + A2 Sin2 2

It is well known that this family of conformal transforms leave the equation (6.37), the energy EO, and the integral fs ur}1dV invariant. More generally and more precisely, we have Lemma 6.3.1 For any u,v E H1(Sn), and for any non-negative integer k,

we have

(i) (Tqu,Tgv) = (u, v) (ii)

r (Tqu)k(Tgv)T+1-kdV s.

(6.40)

= f ukvr+1-kdv Jsn

(6.41)

6 The Yamabe Problem and Prescribing Scalar Curvature on S°, for n > 3

194

In particular,

E(Tqu) = E(u) and

J is.

(T9u)T+1dV =

Jsn

ur}ldV.

Proof. (i) We will use a property of the conformal Laplacian to prove the invariance of the energy

E(Tqu) = E(u). The proof for (i) is similar. On a n-dimensional Riemannian manifold (M, g),

L9 :_ %9 - c(n)R9 is called a "conformal Laplacian," where c(n) = 4(n-1) , O9 and R. are the Laplace-Beltrami operator and the scalar curvature associated with the metric g, respectively.

The conformal Laplacian has the following well-known invariance property w4/(n-2)g, w > 0, we have under the conformal change of metrics. For

L9u = w--2 L9(uw) ,

for all u E C°°(M).

(6.42)

Interested readers may find its proof in [Bes]. If M is a compact manifold without boundary, then multiplying both sides of (6.42) by u and integrating by parts, one would arrive at fM{Iobu12 +c(n)RyIuI2}dVg

=

fM{I09(uw)12 +c(n)RgIuw12}dV9. (6.43)

In our situation, we choose g to be the Euclidean metric g in R" with gig = bij and

Co, the standard metric on Sn. Then it is well-known that g° _ W "

(x)g

with

satisfying the equation We also have n(n - 2)

4 and c(n)Ry = c(n)0 = 0.

6.3 The Variational Approach for Prescribing Scalar Curvature

195

Now formula (6.43) becomes

f {IV.u12+7n.iu12}dVo= f nIo(uw)l2dx,

(6.44)

R

S

where vo and p are gradient operators associated with the metrics go and respectively.

Again, let q" be the intersection of S" with the ray passing through the center of the sphere and the point q. Make a stereo-graphic projection from Sn to Rn, where q" is placed at the origin of Rn. Under this projection, the point (r, 0) E Sn becomes x = (xi, , xn) E Rn, with 1x1

2

r

=tan g

and

(Tqu)(x) = u(Ax) I A(4+Ix12) 4 + Aixi2

I

2

It follows from this and (6.44) that E(Tqu)

=

fs {loo

z'

U('\x) 12+7n1u(Ax)

(A4+ajxj2)\

1 12J

4-2

f

R

=

f

117

Lu (Ax)

t

H

(

M(4+ x12)l I- w(x)] 4+A1x12

v-

(4Ix12)n

12dx

] 12x

l

= fRT 10 [u(y) (4+41y12 n22] I2dy

=

f

R^

117 (u(y)w(y))12dy

= E(u).

(ii) This can be derived immediately by making the change of variable y = ha(x). This completes the proof of the lemma. One can also verify that Tq¢q = 1.

196

6 The Yamabe Problem and Prescribing Scalar Curvature on S", for n > 3

The relations between q and A, 4 and Tq for 4 # 0 can be expressed in a similar way.

We now carry on the estimates near the south pole (0, 0) which we assume to be a positive local maximum. The estimates near other positive local maxima are similar. Our conditions on R imply that

R(r) = R(0) - ara, for some a > 0, n - 2 < a < n

(6.45)

in a small neighborhood of O. Define

E = {u E H I

q(u)I <_ P0, IIvII

min In - tQ1 <- Po}.

(6.46)

t,q

Notice that the `centers of mass' of functions in E are near the south pole O. This is a neighborhood of the `critical points at infinity' corresponding to O. We will estimate the functional Jp in this neighborhood. We first notice that the supremum of J, in Z approaches R(0)IS"l as P-+T. More precisely, we have

Proposition 6.3.1 For any Si > 0, there is a p1 < r, such that for all

r>P>pi,

sup Jp(u) > R(0)jS"I - 61. E

(6.47)

Then we show that on the boundary of E, J, is strictly less and bounded away from R(0)IS"I.

Proposition 6.3.2 There exist positive constants p0, p0, and 60, such that for all p > po and for all u E 8E, holds Jp(u) < R(0)IS"I - 60.

(6.48)

Proof of Proposition 6.3.1 Through a straightforward calculation, one can show that

Jr(0a,o)-4R(0)IS°I, as A-r0.

(6.49)

For a given 61 > 0, by (6.49), one can choose A0, such that ¢ao,o E E, and

JJ(Oao,o) > R(0)IS"I - 21.

(6.50)

It is easy to see that for a fixed function 4ao,p, Jp is continuous with respect to p. Hence, by (6.50), there exists a p1, such that for all p > pl,

6.3 The Variational Approach for Prescribing Scalar Curvature

197

Jp('ao,o) > R(0)IS" I - 6r. This completes the proof of Proposition 6.3.1.

The proof of Proposition 6.3.2 is rather complex. We first estimate Jp for a family of standard functions in E.

Lemma 6.3.2 For p sufficiently close to r, and for h and Iq"I sufficiently small, we have

Jp(Oa,4) <_ (R(0) - Cii4I")IS"I(l +op(1)) - Cra"+5,,

where 6p :=T - p and

(6.51)

as p--+T.

Proof of Lemma 6.3.2. Let e > 0 be small such that (6.45) holds in B5(O) C Sn. We express Jp(oa,4) =

R(x)Oa±idV = f

... +

f

..

.

(6.52)

9

Js°

B,(O)

From the definition (6.38) of 1'A,4 and the boundedness of R, we obtain

the smallness of the second integral:

fs^\&(a)

R(x)Aa 91dV < C2a

,z for 4 E Bz (O).

(6.53)

To estimate the first integral, we work in a local coordinate centered at 0.

f

BE(O)

QrdV =

R(x)

5

f

BE(o)

< B ,(o)

JIB BE(0)

R(x+4)

o dV

[R(0)-aIx+41"]odV

[R(0)-C3Ixi"-C3j4I") pdV

[R(0) - C3I4I°jIS°j[1 +op(1)) - C4Aa+a,(7-2)/2.

(6.54)

Here we have used the fact that ix+q"I" > c(IxI" + 141°) for some c > 0 in one half of the ball B,(0) and the symmetry of 0a,o. Noticing that a < n and 6p->0, we conclude that (6.52), (6.53), and (6.54) imply (6.51). This completes the proof of the Lemma. To estimate Jp for all u E 8E, we also need the following two lemmas that describe some useful properties of the set E.

6 The Yamabe Problem and Prescribing Scalar Curvature on S", for n > 3

198

Lemma 6.3.3 (On the `center of mass') (i) Let q, A, and q be defined by (6.38). Then for sufficiently small q, Ig12 < C(IaJ2 + ),4).

(6.55)

(ii) Let po, q, and v be defined by (6.46). Then for po sufficiently small, (6.56)

Po <_ I4I + CIIvil.

Lemma 6.3.4 (Orthogonality) Let u E E and v = u - toOgo be defined by (6.46). LetTgo be the conformal transform such that Tq,Ogo = 1.

(6.57)

Then

J

TgovdV = 0

(6.58)

n

and

(6.59) J sn

where ?pi are first spherical harmonic functions on Sn: -0'V)i = n'Oi (x) , i = 1, 2, -

, n.

If we place the center of the sphere S" at the origin of R"+' (not in our case), then for x = (xi, ... x"), ,

'+Pi(x)=xi,i=1,2,...,n.

Proof of Lemma 6.3.3. (i) From the definition that calculation, one arrives at Iq - 41 - A2,

'q

and (6.38), and by an elementary

for A sufficiently small

.

(6.60)

Drawing a perpendicular line segment from 0 to qq, one can see that (6.55) is a direct consequence of the triangle inequality and (6.60). (ii) For any it = t¢q + v E BE, the boundary of E, by the definition, we have either IIvII = Po or q(u)I = po. If IIvii = po, then we are done. Hence we assume that q(u)I = po. It follows from the definition of the `center of mass'

that

6.3 The Variational Approach for Prescribing Scalar Curvature Po = I

fsn (to,

199

(6.61)

I.

Noticing that jlvii < pn is very small, we can expand the integrals in (6.61) in terms of IIvII:

Po=

g+1 fxOq+1+(r+1)tT fxO9v+0( vI) tT+1 f0q+1+(T+1)tT f oqv+O(IIvII) f XoT+l f0q+1 I+CIIvII <- qI+CIIvII I

Here we have used the fact that v is small and that t is close to 1. This completes the proof of Lemma 6.3.3.

Proof of Lemma 6.3.4. Write L = 0 + 7n. We use the fact that v = u - tocbgo is the minimum of E(u - t¢q) among all possible values of t and q. (i) By a variation with respect to t, we have 0 = (v, ¢qo) =

Js^

vLggodV

(6.62)

It follows from (6.37) that

Jsnvq5godV=0.

Now, apply the conformal transform Tqo to both v and 0q0 in the above integral. By the invariant property of the transform, we arrive at (6.58). (ii) To prove (6.59), we ma/k/e a variation with respect to q.

0= Vq j vLWq Iq=qo7nVq Vq J Tg0v(Tg0Oq)T Iq=go= Oq

f

f

vOq Iq=qo

Tg0v(Ikq-qo)T

Iq=qo

T-1

T J TgovO/pq-go 9,, rO9-go Iq,,/=qo

-TJ Tgov(Wlr' This completes the proof of the Lemma.

Proof of Proposition 6.3.2 Make a perturbation of R(x):

' , On).

200

6 The Yamabe Problem and Prescribing Scalar Curvature on S', for n >3

J R(x) x E Ben (C) x E S' \ B2 0 (C), M

R(x)

(6.63)

where m = R IaB2po(o) Define

JP(u) = f R(x)uP+1dV. sn

The estimate is divided into two steps. In step 1, we show that the difference between JP(u) and J,(u) is very

small. In step 2, we estimate J,(u). Step 1.

First we show (6.64)

JP(U) <_ Jr(u)(1 + Op (1)),

where op(1)--10 as p-*T.

In fact, by the Holder inequal y P+1

ur+1dV)'+'

1 f?up+1 < (f

IS"I=-P

R P+1

\JRvr+ldV),+1

IT-A

I

R(0)IS^I

This implies (6.64). Now estimate the difference between JP(u) and JP(u).

JP(u) - JP(u) I < Ct

.aP+1 Sf"\B2Po(0) (tOrz)P+1 + C2

< C2 J Sn\B2Pa (O)

< C3A"-6P

na

22 + C3IIvIIP+1.

f

vn+1(6.65)

Sn\B2no(O)

(6.66)

Step 2.

By virtue of (6.64) and (6.66), we now only need to estimate JJ(u). For any it E 8E', write it = v + tOg. (6.62) implies that v and q5g are orthogonal with respect to the inner product associated with that is

Jsn Notice that IIuII2

(VVV

E(u), we have IIuII2 = IIvII2+t2II0g112.

6.3 The Variational Approach for Prescribing Scalar Curvature

201

Using the fact that both u and cbq belong to H with IIulI = 'rnlS" I = II0gII,

we derive

t2=1-

IIvII2

(6.67)

'yniSn

It follows that R(x)uT+1 Sn

tT+1

I

(x)OQ+1+(r+1) f TR(x)¢gv+T(r2 1) f T'Q-1v2+o(IIv1I2)

= Ir + (r + 1)12 +

'(I3 1)

(6.68)

I3 +o(IIvII2).

To estimate 11, we use (6.67) and (6.51),

11 <(1_T21ry

clams)+0(An)+o(IIv 2) (6.69)

for some positive constant c1. To estimate 12, we use the H1 orthogonality between v and 0q (see (6.62)),

12 =

f (R(x) -

m)oQvdV =

fgaoo(o)(R(x) -m)¢QvdV

1

(R(x) - m)LOvdV 7n aavo(0)4 < CPo I f

B2po (0)

C CPo

IIgq II IIvII

LOq - vdV I = CPo (0q, v)

<_ C4Po IIvII

(6.70)

To estimate I3, we use (6.58) and (6.59). It is well known that the first and second eigenspaces of -A on Sn are constants and 0i corresponding to eigenvalues 0 and n respectively. Now (6.58) and (6.59) imply that Tqv is orthogonal to these two eigenspaces and hence S.

IV(Tgv)I2dV > A2

f TgvI2dV, Sn

where A2 = n + c2 for some positive number c2 is the second non-zero eigenvalue of -0. Consequently, IIv1I2 = IITgvll2 ? (7n+n + c2)

(Tgv)2. is,

202

6 The Yamabe Problem and Prescribing Scalar Curvature on S°, for n > 3

It follows that I3

R(0)

f

J

¢q 1v2dV = R(0)

r Jsn

(TgOq)r-1(Tgv)2dV

R(0)Jsn(T°v)2 <

(6.71)

yn R(n0) c2IlvII2

Now (6.69), (6.70), and (6.71) imply that there is a Q > 0 such that

JJ(u)
(6.72)

Here we have used the fact that IIv11 is very small and po livi1 can be controlled by 141a+Ack (see Lemma 6.3.2). Notice that the positive constant c2 in (6.71) has played a key role, and without it, the coefficient of IIvU12 in (6.72) would be 0 due to the fact that y, = n(4 1

For any u E 8E, we have either IIv11= po or q(u)1= PO

By (6.55) and (6.72), in either case there exist a J. > 0, such that Jr(u) < R(0)iSnl - 25o.

(6.73)

Now (6.48) is an immediate consequence of (6.64), (6.66), and (6.73) . This completes the proof of the Proposition. O

6.3.2 The Variational Scheme In this part, we construct variational schemes to show the existence of a solution to equation (6.35) for each p < r. Case (i): R has only one positive local maximum.

In this case, condition (6.9) implies that R must have local minima at both poles. Similar to the ideas in [Cl, we seek a solution by maximizing the functional Jp in a class of rotationally symmetric functions:

H,={uEHI u=u(r)}.

(6.74)

Obviously, Jp is bounded from above in Hr and it is well known that the variational scheme is compact for each p < r (see the argument in Chapter 2). Therefore any maximizing sequence possesses a converging subsequence in

Hr, and the limiting function multiplied by a suitable constant is a solution of (6.35).

Case (ii): R has at least two positive local maxima.

6.4 The a priori Estimates for Prescribing Scalar Curvature

203

Let rr and r2 be the two smallest positive local maxima of R. By Proposition 6.3.1 and 6.3.2, there exist two disjoint open sets E1, E2 C H, V)i E E;, po < r and b > 0, such that for all p > po, 6

JP(V)i) > R(ra)SRJ

i=

1, 2;

(6.75)

and

Jp(u) < R(r:)IS^I - 6, Vu E OEa, i = 1, 2.

(6.76)

Let -y be a path in H joining fir and 02. Let P be the family of all such paths. Define cp = sup min JP (u). 7Er uEy

(6.77)

For each fixed p < r, by the well-known compactness, there exists a critical (saddle) point up of Jp with JP(uP) = ep-

Moreover, due to (6.76) and the definition of cp, we have J. (up) < min R(ri) I STI - 6. .-1,2

(6.78)

One can easily see that a critical point of Jp in H multiplied by a constant

is a solution of (6.35) and for all p close to r, the constant multiples are bounded from above and bounded away from 0.

6.4 The a Priori Estimates for Prescribing Scalar Curvature In the previous section, we showed the existence of a positive solution up to the subcritical equation (6.35) for each p < r. In this section, we prove that as p->r, there is a subsequence of {up}, which converges to a solution uo of (6.6). The convergence is based on the following a priori estimate.

Theorem 6.4.1 Assume that R satisfies condition (6.10) in Theorem 6.1.3, then there exists po < r, such that for all po < p < r, the solution of (6.35) obtained by the variational scheme are uniformly bounded.

To prove the Theorem, we estimate the solutions in three regions where R < 0, R close to 0, and R > 0 respectively.

204

6 The Yamabe Problem and Prescribing Scalar Curvature on S", for n > 3

6.4.1 In the Region Where R < 0 The a priori bound of the solutions is stated in the following proposition, which is a direct consequence of Proposition 6.3.2 in our paper [CLG].

Proposition 6.4.1 For all 1 < p < r, the solutions of (6.35) are uniformly bounded in the regions where R < -6, for every 6 > 0. The bound depends only on 6, dist({x I R(x) < -6}, {x I R(x) = 0}), and the lower bound of R.

6.4.2 In the Region Where R is Small In [CL6], we also obtained estimates in this region by the method of moving planes. However, in that paper, we assume that pR be bounded away from zero. In [CL7], for rotationally symmetric functions R on S3, we removed this condition by using a blowing up analysis near R(x) = 0. That method may be applied to higher dimensions with a few modifications. Now within our variational frame in this paper, the a priori estimate is simpler. It is mainly due to the energy bound.

Proposition 6.4.2 Let {up} be solutions of equation (6.35) obtained by the variational approach in Section 6.3. Then there exists a po < r and a 6 > 0, such that for all po < p < r, {up} are uniformly bounded in the regions where IR(x)I <-6.

Proof. It is easy to see that the energy E(up) are bounded for all p. Now suppose that the conclusion of the Proposition is violated. Then there exists a subsequence {ui} with ui = up;, pi+r, and a sequence of points {xi}, with xi->x° and R(x°) = 0, such that ui(x`)->oo.

We will use a re-scaling argument to derive a contradiction. Since xi may not be a local maximum of ui, we need to choose a point near xi, which is almost a local maximum. To this end, let K be any large number and let

ri = 2K[ui(xi)]In a small neighborhood of x°, choose a local coordinate and let

si(x`)=ui(x)(ri-Ix-x'D° ' Let ai be the maximum of si (x) in B, (x'). Let Ai = [ui(ai)]-

2

6.4 The a priori Estimates for Prescribing Scalar Curvature

205

Then from the definition of ai, one can easily verify that the ball BA,K(ai)

is in the interior of Bri(x) and the value of ui(x) in BA,K(ai) is bounded above by a constant multiple of ui(ai). To see the first part, we use the fact si(ai) > si(xi). From this, we derive 2

2

ui(ai)(ri - jai - x''I)Pi > ui(xi)ri=' = (2K)P-i It follows that

ri-jai-x`j>AiK,

(6.79)

2 hence

BAiK(ai) C Bri(x`).

To see the second part, we use si(ai) > si(x) and derive from (6.79), 2

ui(x) < ui(ai)

1

r1 - jai - xil \ Pi-1

ri - Ix - x`lJ a

G ui(ai) 2Pi=1

Vx E BA:K(ai).

Now we can rescale: ui(a1

Vi(x)

=

i)ui(Aix + ai).

Then it satisfies R(Aix + ai)vri(x)

with R(Aix + ai)--30.

Obviously vi is bounded in Bx(0), and it follows by a standard argument that {vi} converges to a harmonic function vo in BK(0) C R", with vo(0) = 1. Consequently for i sufficiently large, vi(x)r+1dx > cK",

(6.80)

for some c > 0.

On the other hand, the boundedness of the energy E(ui) implies that uir}1dV
(6.81)

Js^

for some constant C. By a straightforward calculation, one can verify that, for any K > 0, uq+1dV > f va+ldx. (6.82)

J S.

JBx(0)

Obviously, (6.81) and (6.82) contradict with (6.80). This completes the proof of the Proposition.

206

6 The Yamabe Problem and Prescribing Scalar Curvature on S", for n > 3

6.4.3 In the Regions Where R > 0. Proposition 6.4.3 Let {up} be solutions of equation (6.35) obtained by the variational approach in Section 6.3. Then there exists a po < r, such that for all po < p < r and for any e > 0, {up} are uniformly bounded in the regions where R(x) > e. Proof. The proof is divided into 3 steps. In Step 1, we argue that the solutions

can only blow up at finitely many points because of the energy bound. In Step 2, we show that there is no more than a one-point blow up and the point must be a local maximum of R. This is done mainly by using a Pohozaev type identity. In Step 3, we use (6.78) to conclude that even a one-point blow up is impossible.

Step 1. The argument is standard. Let {x'} be a sequence of points such that and xi-*xo with R(xo) > 0. Let ri, ai, si(x), and vi(x) be defined as in the previous subsection. Then

-Ovi(x) = R(Aix + ai)vP;(x)

(6.83)

The difference between this and the previous case is that now

R(Aix + ai)-.R(xo) > 0. Similar to the previous case, one can derive that vi(x) is uniformly bounded in BK(0) for any fixed K > 0. It follows from (6.83) that wiIIw=.9(BK(o)) < C

for any q > 1. Hence by Sobolev embedding, wiIIC ,a(BK(o)) < C.

Applying the Schauder estimate to equation (6.83), we have IViIIC3.°(BK(0)) < C.

Therefore, {vi(x)} converges in C2,°(BK(0)) to a function vo(x), and

R(xo)vo(x), Vx E BK(0).

(6.84)

Since K > 0 is arbitrary, equation (6.84) actually holds for all x in R". Then by the classification result, vo(x) must be radially symmetric about some point in R" and assume a certain form (see for example [CL1), or Chapter 8 for more details). Hence va+i(x)dx > co > 0.

R^

It follows that, for large i,

6.4 The a priori Estimates for Prescribing Scalar Curvature

f

ur+l(x)dV >

2> Co

207

0.

Because the total energy of ui is bounded, we can have only finitely many

such x0. Therefore, {ui} can only blow up at finitely many points. Step 2. We have shown that {ui} has only finitely many blow-up points. As a consequence of a result in [LiY2] (also see [CL6] or [SZ]), we have

Lemma 6.4.1 Let R satisfy the `flatness condition' in Theorem 1, Then {ui} can have at most one simple blow-up point, and this point must be a local maximum of R. Moreover, {ui} behaves almost like a family of the standard with Ai = functions and with 4 at the maximum of R. (maxui)_n?z

Step 3. Finally, we show that even a one-point blow up is impossible. For convenience, let {ui} be the sequence of critical points of Jp, we obtained in Section 6.3. From the proof of Proposition 6.3.2, one can obtain

JJ(ui) <minR(rk)IS"I -b,

(6.85)

for all positive local maxima rk of R because each ui is the minimum of Jr. on a path. Now if {ui} blows up at x0, then by Lemma 6.4.1, we have JT(u )-YR(x0)IS^I.

This is a contradiction with (6.85). Therefore, the sequence {ui} is uniformly bounded and possesses a subsequence converging to a solution of (6.6). This completes the proof of Theorem 6.1.3.

7

Maximum Principles

7.1

Introduction

7.2

Weak Maximum Principle

7.3

The Hopf Lemma and Strong Maximum Principle

7.4

Maximum Principle Based on Comparison

7.5

A Maximum Principle for Integral Equations

7.1 Introduction As a simple example of the Maximum Principle, let's consider a C2 function u(x) of one independent variable x. It is well known in calculus that at a local maximum point x° of u, we must have

-(x°) < 0. Based on this observation, then we have the following simplest version of the Maximum Principle: Assume that u"(x) > 0 in an open interval (a, b); then it cannot have any interior maximum in the interval. One can also see this geometrically. Since under the condition u"(x) > 0, the graph is concave up, it can not have any local maximum. More generally, for a C2 function u(x) of n independent variables x =

(xi,

,

at a local maximum x°, we have D2u(x°) := (us1 ; (x')) < 0,

7 Maximum Principles

210

that is, the symmetric matrix is non-positive definite at point x°. Correspondingly, the simplest version of the Maximum Principle reads as follows:

If

(aij (x)) ?0 and Eaij(x)uz;y;(x)>0

(7.1)

ij

in an open bounded domain fl, then u cannot achieve its maximum in the interior of the domain. An interesting special case is when (aij (x)) is an identity matrix, in which case condition (7.1) becomes

Au>0. Unlike its one-dimensional counterpart, condition (7.1) no longer implies that the graph of u(x) is concave up. A simple counterexample is this: z u(xl,xz) = x l -

2 x2'

One can easily see from the graph of this function that (0, 0) is a saddle point.

In this case, the validity of the Maximum Principle derives from the following simple algebraic fact:

For any two n x n matrices A and B, if A > 0 and B < 0, then AB < 0. In this chapter, we will introduce various Maximum Principles, and most of them will be used in developing the method of moving planes in the next chapter. Of course, there are numerous other applications of maximum principles, and we list some of these below. i) Providing Estimates

Consider the boundary value problem

Au = f (x) , x E Br(0) C Rn U(X) = 0,

(7.2)

x E OB1(0).

If a < f(x) < b in B1(0), then we can compare the solution u with the two functions

2n(1-x12) and (1-Ix12) which satisfy the equation with f (x) replaced by a and b, respectively, and which vanish on the boundary as it does. Now, applying the Maximum Principle for the A operator (see Theorem 7.1.1 in the following), we obtain

2n(1 -

Ix12)

< u(x) < 2n (l

ii) Proving Uniqueness of Solutions

-

Ix12)_

7.1 Introduction

211

In the above example, if f (x) _ 0, then we can choose a = b = 0, and this implies that it - 0. In other words, the solution of the boundary value problem (7.2) is unique. iii) Establishing the Existence of Solutions (a) For a linear equation such as (7.2) in any bounded open domain S2, let

u(x) = sup O(x), where the sup is taken over all functions that satisfy the corresponding differential inequality

5 p¢C f(x),xES2 1O-(x)=0,

xEOQ.

Then u is a solution of (7.2). (b) Now consider the nonlinear problem

Lu=f(u),xES2 {u(x)=0,xEOQ. Assume that f() is a smooth function with f'( ) > 0. Suppose that there exist two functions u(x) < u(x), such that -ou < f(u) < f(u) < -Au.

These two functions are called sub (or lower) and super (or upper) solutions, respectively, as we have seen in Chapter 5. To seek a solution of problem (7.3), we use successive approximations. Let

-,Lul = f(jk) and - Dui+r = f(ui). Then by the Maximum Principle, we have

v_, < ul < 1t2 < ... < ni < ... < 9r. Let u be the limit of the sequence {ui}:

u(x) = limui(x). Then u is a solution of the problem (7.3). This chapter is organized as follows. In the remainder of this section, we introduce both strong and weak maximum principles; in Section 7.2 we prove weak maximum principles, and in Section 7.3 we prove strong maximum principles using the Hopf Lemma. In Section 7.4, as a consequence of Theorem

7.1.5, we derive the `Narrow Region Principle' and the `Decay at Infinity Principle'. These principles can be applied very conveniently in the `Method of. Moving Planes' to establish the symmetry of solutions for semi-linear elliptic equations, as we will see in later sections. Finally, in Section 7.5, we establish a Maximum Principle for integral inequalities.

7 Maximum Principles

212

Theorem 7.1.1 (Weak Maximum Principle for -A.)

i)If -Au(x) > 0, x E .fl, then

min u > min u.

- as?

f2

ii) If -Au(x) < 0,

x E £7,

then

max u < max u.

an

n

This result can be extended to general uniformly elliptic operators. Let x

09

Di = a

i,

Di

Define

L = - E aii (x)Dii + E bi (x)Di + c(x).

ii

i

In the following we always assume that affi(x), bi(x), and c(x) are bounded continuous functions in £7, and we also assume that L is uniformly elliptic,

that is, aii(xgi6i ? bI6I2 for any x E .fl, any e E R" and for some 5 > 0.

Theorem 7.1.2 (Weak Maximum Principle for L) Assume that c(x) i) If Lu > 0 in .2, then

0.

min u > min u. f2

- 8111

ii) If Lu < 0 in .2, then max u < max u. Q

an

These Weak Maximum Principles infer that the minima or maxima of u are attained at some points on the boundary a.fl. However, they do not exclude the possibility that the minima or maxima may also occur in the interior of Q. Actually this cannot happen unless it is constant, as we will see in the following.

7.1 Introduction

213

Theorem 7.1.3 (Strong Maximum Principle for L with c(x) _ 0) Assume that 12 is an open, bounded, and connected domain in Rn with smooth bound-

ary 812. Let it be a function in C2(Q) n C(Q). Assume that c(x) = 0 in 12.

i) If Lu(x) > 0, x E .(2, then it attains its minimum value only on 80 unless it is constant.

ii) If

Lu(x)<0, xE12, then it attains its maximum value only on 812 unless u is constant. This Maximum Principle (as well as the weak one) can also be applied to the case when c(x) > 0 with slight modifications.

Theorem 7.1.4 (Strong Maximum Principle for L with c(x) > 0) Assume that S? is an open, bounded, and connected domain in Rn with smooth bound-

ary 8I1. Let it be a function in C2(0) n C(0). Assume that c(x) > 0 in R.

i) If Lu(x) > 0, x E fl, then it can not attain its non-positive minimum in the interior of 0 unless it is constant.

ii) If Lu(x) < 0, x E ,fl, then u can not attain its non-negative maximum in the interior of Q unless u is constant.

Notice that all the previous theorems require that c(x) > 0. Roughly speaking, Maximum Principles hold for `positive' operators. -A is `positive',

and obviously so is -A + c(x) if c(x) > 0. However, as we will see in the next chapter, in practice the condition c(x) > 0 frequently cannot be met. But do we really need c(x) > 0? The answer is `no.' Actually, if c(x) is not "too negative", then the operator '-A+ c(x)' can still remain "positive" to ensure the Maximum Principle. As mentioned earlier, these will be studied in Section 7.4 where we prove the `Maximum Principles Based on Comparisons'. Let ¢ be a positive function on R satisfying

-AO + A(x)O > 0.

(7.4)

Let it be a function such that

Du + c(x)u ? O x E 1? u on 490. I >0

(7.5)

7 Maximum Principles

214

Theorem 7.1.5 (Maximum Principle Based on Comparison) Assume that 12 is a bounded domain. If c(x) > A(x) ,Vx E 12,

then it > 0 in ,2.

7.2 Weak Maximum Principles In this section, we prove the Weak Maximum Principles.

Theorem 7.2.1 (Weak Maximum Principle for -z.) i) If -Du(x) > 0, x E I2,

(7.6)

then

minu > minu. an B

ii) If -Au(x) < 0,

x E 12,

then

max it < max it. an Li Proof. Here we only present the proof of part i). An entirely similar proof also works for part ii). To better illustrate the idea, we treat the one-dimensional and higher-

dimensional cases separately.

First, let 12 be the interval (a, b). Then condition (7.6) becomes u"(x) < 0. This implies that the graph of u(x) on (a, b) is concave downward, and therefore one can roughly see that the values of u(x) in (a, b) are large or equal to the minimum value of it at the end points (See figure).

a

b

To prove the above observation rigorously, we first carry it out under the stronger assumption that

7.2 Weak Maximum Principles

215

-u°(x) > 0. (7.10) Let in = minan it. Suppose, contrary to (7.7), that there is a minimum x° E (a, b) of it, such that u(x°) < rn. Then by the second order Taylor expansion of it around x°, we must have -u" (x°) < 0. This contradicts with the assumption (7.10).

Now for it satisfying only the weaker condition (7.6), we consider a perturbation of u: U,(x) = u(x) - Ext.

Obviously, for each E > 0, u,(x) satisfies the stronger condition (7.10), and hence

min u, > min u,. 00

T2

Now letting E-40, we arrive at (7.7). To prove the theorem in dimensions higher than one, we need the following lemma:

Lemma 7.2.1 (Mean Value Inequality) Let x° be a point in Si. Let Br(x°) C 11 be the ball of radius r, center x°, and boundary DBr(x°)

i) If -Au(x) > ( or =) 0 for x E Bro(x°) with some r° > 0, then for

anyr°>r>0,

u(x°)

1

> ( or =) BBr(x°)I

u(x)dS.

(7.11)

JOB,(.-)

It follows that, if x° is a minimum of it in 12, then

-Lu(x°) < 0.

(7.12)

ii) If -Au(x) < 0 for x E Bro(x°) with some r° > 0, then for any r° > r > 0, 1 (7.13) u(x°) < (x°)I JaBr(so) u(x)dS. 1'9B,

It follows that, if x° is a maximum of it in fl, then

-Au(x°) > 0.

(7.14)

We postpone the proof of the lemma for a moment. This lemma tells us

that, if -Au(x) > 0, then the value of u at the center of the small ball Br(x°) is larger than its average value on the boundary 8Br(x°). Now based on the lemma, to prove the Theorem, we first consider u(x) EIxI2. Obviously,

216

7 Maximum Principles

-AuE = -Au + 2en > 0.

(7.15)

Hence we must have

nnu, > min us.

(7.16)

a

Otherwise, if there exists a minimum x° of u in !2, then by Lemma 7.2.1, we have -Auj(x°) < 0. This contradicts with (7.15). Now in (7.16), letting e-*0, we arrive at the desired conclusion (7.7). This completes the proof of the Theorem.

The Proof of Lemma 7.2.1. By the Divergence Theorem,

JB,(.,)

Au(x)dx =

au dS = r'-'

J&B,.(x") 8v

r

au (x° + rw)dS° ur

(7.17)

where dSw is the area element of the (n - 1)-dimensional unit sphere Sn-1 =

{w w1 =1}. If An < 0, then by (7.17), { Jga

1

u(x +rw)dS,} < 0.

(7.18)

Integrating both sides of (7.18) from 0 to r yields

u(x°+rw)dS - u(x)IS"-1I <0, S^_1

where IS" I is the area of Sn-1. It follows that 1

u(x°) >

u(x)dS.

This verifies (7.11).

To see (7.12), we suppose on the contrary that -.u(x°) > 0. Then by the continuity of Au, there exists a d > 0, such that

-Au(x) > 0, Vx E BS (X').

Consequently, (7.11) holds for any 0 < r < J. This contradicts with the assumption that x° is a minimum of u. This completes the proof of the Lemma.

From the proof of Theorem 7.2.1, one can see that if we replace the -A operator by -A+c(x), with c(x) > 0, then the conclusion of Theorem 7.2.1 is still true (with slight modifications). Furthermore, we can replace the Laplace operator -,L with general uniformly elliptic operators. Let

7.2 Weak Maximum Principles

D, Define

-,a 0xi

217

92

Di; = axax;'

L=->a,j(x)Di;+>bi(x)Di+c(x).

(7.19)

Here we always assume that ai;(x), bi(x), and c(x) are bounded continuous functions in R. We say that L is uniformly elliptic if ai;

for any x E 12, any e E R" and for some b > 0.

Theorem 7.2.2 (Weak Maximum Principle for L with c(x) - 0). Let L be the uniformly elliptic operator defined above. Assume that c(x)

0.

i) If Lu > 0 in Q, then minu > min u. an n

(7.20)

ii) If Lu < 0 in 12, then max u < max u. an Q

For c(x) > 0, the principle still applies with slight modifications.

Theorem 7.2.3 (Weak Maximum Principle for L with c(x) > 0). Let L be the uniformly elliptic operator defined above. Assume that c(x) > 0. Let u (x) = min{u(x),0}

and u+(x) = max{u(x),0}.

i) If Lu > 0 in ,fl, then

minu>minu . D

- an

ii) If Lu < 0 in 0, then maxu < maxu+. Q

- an

Interested readers may find its proof in many standard books (see [Ev], p. 327).

7 Maximum Principles

218

7.3 The Hopf Lemma and Strong Maximum Principles In the previous section, we proved a weak form of the Maximum Principle. In

the case Lu > 0, it concludes that the minimum of u attains at some point on the boundary 812. However it does not exclude the possibility that the minimum may also attain at some point in the interior of fl. In this section, we will show that this cannot actually happen; that is, the minimum value of u can only be achieved on the boundary unless u is constant. This is called the "Strong Maximum Principle," which we will prove using the following:

Lemma 7.3.1 (Hopf Lemma). Assume that d2 is an open, bounded, and connected domain in Rn with smooth boundary 811. Let u be a function in C2 (0) fl C(12) . Let

L = - L aij (x)Dij + E bi(x)Di + c(x) z7

a

be uniformly elliptic in 12 with c(x) __ 0. Assume that

Lu> 0

in Q.

(7.21)

Suppose there is a ball B contained in 12 with a point x° E 812 fl 8B and suppose

u(x) > u(x°), Vx ED.

(7.22)

Then for any outward directional derivative at x°,

8u(x°) 8v

<0

(7 . 23)

.

In the case c(x) > 0, if we require additionally that u(x°) < 0, then the

same conclusion of the Hopf Lemma holds.

Proof. Without loss of generality, we may assume that B is centered at the origin with radius r. Define w(x) = e

e-a

n12.

Consider v(x) = u(x) + ew(x) on the set D = Br (x°) fl B (See figure).

7.3 The Hopf Lemma and Strong Maximum Principles

219

We will choose a and e appropriately so that we can apply the Weak

Maximum Principle to v(x) and arrive at

v(x) > v(x°) Vx E D.

(7.24)

We postpone the proof of (7.24) for a moment. Now from (7.24), we have (y) < 0.

(7.25)

Noticing that C9W

8v (x°) > we arrive at the desired inequality

0,

-(x) < 0. Now to complete the proof of the lemma, we only need verify (7.24). We will carry this out in two steps. First we show that

Lv > 0.

(7.26)

Hence we can apply the Weak Maximum Principle to conclude that minv > min v. D

(7.27)

OD

We then show that the minimum of v on the boundary 8D is actually attained at x°: (7.28) v(x) > v(x°) Vx E 8D. Obviously, (7.27) and (7.28) imply (7.24). To see (7.26), we directly calculate

7 Maximum Principles

220

Lw =

(7.29) n

e °IsI2 {4a2 E aij(x)xixj - 2a E[affi(x) - bi(x)xi] - c(x)} +c(x)e'a'' i,j=1

>

i=1

e-aIx2{4a2

n

E aij(x)xixj-2aE[aii(x)-bi(x)x,j-c(x)} (7.30) i,j=1

i=1

By the ellipticity assumption, we have

1 aj(x)xixj > 61xI2

> b(2)2 > 0 in D.

(7.31)

i,j=1

Hence we can choose a sufficiently large, such that Lw > 0. This, together with the assumption Lu > 0 implies Lv > 0, and (7.27) follows from the Weak Maximum Principle. To verify (7.28), we consider two parts of the boundary 8D separately.

(i) On 8D n B, since u(x) > u(x°), there exists a c° > 0, such that u(x) > u(x°) + c°. Take e small enough such that eIwI < b on 8D n B. Hence v(x) > u(x°) = v(x°) Vx E 8D n B.

(ii) On 8D n 8B, w(x) = 0, and by the assumption u(x) > u(x°), we have v(x) > v(x°). This completes the proof of the Lemma. We are now ready to prove the following theorem:

Theorem 7.3.1 (Strong Maximum Principle for L with c(x) - 0.) Assume that 1? is an open, bounded, and connected domain in Rn with smooth bound-

ary 80. Let a be a function in C2(dl) n C(Q). Assume that c(x)

0 in

Q.

i) If Lu(x) > 0, x E ,fl, then it attains its minimum only on OR unless it is constant.

ii) If Lu(x) < 0, x E ,fl, then u attains its maximum only on 8.2 unless u is constant.

Proof. We prove part i) here. The proof of part ii) is similar. Let m be the minimum value of it in ,2. Set E _ {x E 9 1 u(x) = m}. It is relatively closed in ,2. We show that either E is empty or E = R.

7.3 The Hopf Lemma and Strong Maximum Principles

221

We argue by contradiction. Suppose E is a nonempty proper subset of Q. Then we can find an open ball B C 1? \ E with a point on its boundary

belonging to E. Actually, we can first find a point p E .(2 \ E such that d(p, E) < d(p, 017), then increase the radius of a small ball center at p until it hits E (before hitting 852). Let x° be the point at 8B n E. Obviously we have in B Lu > 0 and u(x) > u(x°).

Now we can apply the Hopf Lemma to conclude that the normal outward derivative i9u

(x) < 0.

(7.32)

8v On the other hand, x° is an interior minimum of u in S2, and we must have Du(x°) = 0. This contradicts with (7.32) and hence completes the proof of the Theorem. In the case when c(x) > 0, the strong principle still applies with slight

modifications.

Theorem 7.3.2 (Strong Maximum Principle for L with c(x) > 0.) Assume that 52 is an open, bounded, and connected domain in R with smooth bound-

ary 852. Let it be a function in C2(S2) n C(.fl). Assume that c(x) > 0 in Q.

i) If

Lu(x) ?0, x(= Sl, then u can not attain its non-positive minimum in the interior of 5l unless it is constant.

ii) If Lu(x) < 0, x E Sl, then it can not attain its non-negative maximum in the interior of S2 unless u is constant.

Remark 7.3.1 In order that the Maximum Principle hold, we assume that the domain 52 be bounded. This is essential, since it guarantees the existence of maximum and minimum of it in Q. A simple counter example is when 52 is

the half space {x E R' I x > 0}, and u(x) = xn. Obviously, Lu = 0, but u does not obey the Maximum Principle: max u < max u. an 92

222

7 Maximum Principles

Equally important is the non-negativeness of the coefficient c(x). For exam-

ple, set Q = { (x, y) E R2 I - 2 < x < 2 , - 2 < y < 2 }. Then u = cos x cos y satisfies

Du-2u=0, in S? on 80.

U = 0,

But obviously u attains its maximum at the origin: u(0, 0) = 1. However, if we impose some sign restriction on u, say u > 0, then both conditions can be relaxed. A simple version of such a result will be present in the next theorem. Also, as will be shown in the next section, c(x) is actually allowed to be negative, but not "too negative."

Theorem 7.3.3 (Maximum Principle and Hopf Lemma for not necessarily bounded domain and not necessarily non-negative c(x). ) Let .fl be a domain in Rn with smooth boundary. Assume that u E C2 (.2) n C(Q) and satisfies

Lu+E bi(x)Diu+c(x)u>0, u(x)>0,xE.2 1

U(X) = 0

x E OQ

(7.33)

with bounded functions bi(x) and c(x). Then

0 in .2; and

i) if u vanishes at some point in 0, then u

ii) if u *0 in .2, then on 80, the exterior normal derivative av < 0. To prove the Theorem, we need the following Lemma concerning eigenvalues.

Lemma 7.3.2 Let .X be the first positive eigenvalue of enso = Ai (x), x E Bl(0) ¢(x) = 0, x e 8B1(0)

(7.34)

with the corresponding eigenfunction ¢(x) > 0. Then for any p > 0, the first positive eigenvalue of the problem on Bp(0) is

a2

. More precisely, if we let

O(x) _ ¢(P), then ,n,V) =

O(x) = 0,

Pl0(x), x E

Bp(0)

x E 8Bp(0).

The proof is straightforward and is left for the reader.

(7.35)

7.3 The Hopf Lemma and Strong Maximum Principles

223

The Proof of Theorem 7.3.3. i) Suppose that u = 0 at some point in 12, but u ,E 0 on Q. Similar to the proof of the Strong Maximum Principle, let E = {x

S? I u(x) = 0}.

It is relatively closed in d2. We show that either E is empty or E = 12. We argue by contradiction. Suppose E is a nonempty proper subset of 12. Then we can find an open ball B C 12 \ E with a point on its boundary

belonging to E. Actually, we can first find a point p E 12 \ E such that d(p, E) < d(p,812), then increase the radius of a small ball center at p until it hits E (before hitting 812). Let x° be the point at 8B n E. Obviously we have in B u(x) > u(x°).

For p > 0 sufficiently small, one can choose a ball Bp 2(x) C B with x° as its boundary point. Let 0 be the positive eigenfunction of the eigenvalue Al. problem (7.35) on Bp(x°) corresponding to the eigenvalue Obviously, P

Bp(x°) completely covers B,/200. Let v = U. Then from (7.33), it is easy to

deduce that

0<-pv-2pv-

+bi(x)Div+I

n

-AV + E bi(x)Div + c(x* i=1

Let 0 be the positive eigenfunction of the eigenvalue problem (7.34) on B1,

then

c(x) =

n

A

1

P

P

a+

be(x)D:0 + c(x). 0

This allows us to choose p sufficiently small so that c(x) > 0. Now we can apply the Hopf Lemma to conclude that the outward normal derivative at the boundary point x° of Bp12(a), (7.36) because v(x) > 0 Vx E Bp 2(x) and v(x°) _

0.

On the other hand, since x° is also a minimum of v in the interior of 12, we must have

7 Maximum Principles

224

'7v(x°) = 0.

This contradicts with (7.36) and hence proves part i) of the Theorem. ii) The proof is almost the same as in part i) except we consider the point

x° on ad2 and the ball B,,/2(±) is in fl with x° E 8B.12(x). Then for the outward normal derivative of u, we have an av (x)

=

av av

a v(im) av

49V

av

0.

Here we have used a well-known fact that the eigenfunction 7P on B,(x°) is ra-

dially symmetric about the center x°, and hence 7'(x°) = 0. This completes the proof of the Theorem.

7.4 Maximum Principles Based on Comparisons In the previous section, we showed that if (-L + c(x))u > 0, then the Maximum Principle, i.e., (7.20), applies. There, we required c(x) > 0. We can think

of -0 as a `positive' operator, and the Maximum Principle holds for 'positive' operators. For c(x) > 0, -L + c(x) is also `positive'. Do we really need c(x) > 0 here? To answer the question, let us consider the Dirichlet eigenvalue

problem of -&

¢ - ao(x) = 0, x E fl x E an.

{ ¢(x) = 0,

(7.37)

We notice that the eigenfunction 0 corresponding to the first positive eigenvalue al is either positive or negative in ,fl. That is, the solutions of (7.37) with A = ar obeys the Maximum Principle; that is, the maxima or minima of 0 are attained only on the boundary a0. This suggests that, to ensure the Maximum Principle, c(x) need not be nonnegative and is allowed to be as negative as -A1. More precisely, we can establish the following more general Maximum Principle based on comparison.

Theorem 7.4.1 Assume that I? is a bounded domain. Let ¢ be a positive function on fl satisfying

-AO +A(x)O > 0.

(7.38)

Assume that u is a solution of

Du+c(x)u>OxEQ

I u>O

on OR.

(7.39)

If c(x) > A(x)

then is > 0 in ,fl.

,

Vx E 0,

(7.40)

7.4 Maximum Principles Based on Comparisons

225

proof. We argue by contradiction. Suppose that u(x) < 0 somewhere in Q.

Let v(x) = -(-T) . Then since O(x) > 0, we must have v(x) < 0 somewhere in ,fl. Let x° e f2 be a minimum of v(x). By a direct calculation, it is easy to verify that -,Lv = 2pv 0 + 1(-Au + u). (7.41) On the one hand, since x° is a minimum, we have

0 and pv(x°) = 0,

(7.42)

while on the other hand, by (7.38), (7.39), and (7.40), and taking into account

that u(x°) < 0, we have at point x°,

-Au+

u(X') > -Du+A(x°)u(x°)

> -Lu+c(x°)u(X') > 0. This is an obvious contradiction with (7.41) and (7.42) and thus completes the proof of the Theorem.

Remark 7.4.1 From the proof, one can see that conditions (7.38) and (7.40) are required only at the points where v attains its minimum, or at points where u is negative. The Theorem is also valid on an unbounded domains if u is "nonnegative" at infinity:

Theorem 7.4.2 If 0 is an unbounded domain, besides condition (7.39), we assume further that liminfu(x)/fi(x) > 0, (7.43) I=I*o where 0 is the function mentioned in Theorem 7.4.1. Then u > 0 in P. Proof. Still consider the same v(x) as in the proof of Theorem 7.4.1. Now condition (7.43) guarantees that the minima of v(x) do not "leak" away to infinity. Then the rest of the arguments are exactly the same as in the proof of Theorem 7.4.1.

For convenience in applications, we provide two typical situations where there exist such functions 0 and c(x) satisfying condition (7.38) and (7.40), so that the Maximum Principle Based on Comparison applies:

226

7 Maximum Principles

i) Narrow regions, and

ii) c(x) decays fast enough at oo. i) Narrow Regions. When

f2={xl0<x1
We can choose O(x) = sinO. Then it is easy

to see that -0¢

I U

I

x1

where A(x) =

I

F2

can be very negative when 1 is sufficiently small.

Corollary 7.4.1 (Narrow Region Principle.) If u satisfies (7.39) with bounded function c(x). Then when the width l of the region ,fl is sufficiently small, c(x) satisfies (7.40), i.e., c(x) > A(x) = Hence we can directly apply Theorem 7.4.2 to conclude that u > 0 in fl, provided lim inf x _- u(x)/¢(x) > 0. l2.

ii) Decay at Infinity. In dimension n > 3, one can choose some positive number q < n - 2, and let ¢(x) = s , Then it is easy to verify that q(n Ixl2 2 - q)

In the case c(x) decays fast enough near infinity, we can derive from Theorem 7.4.2 that

Corollary 7.4.2 (Decay at Infinity) Assume there exists R > 0, such that C(X) > - q(n

Ixl2 -

q)

V ixi > R.

(7.44)

Suppose

lim u(x)

IX19 = 0.

Let .2 be a region containing in BR (0) = R" \ BR(0). If it satisfies (7.39) on .2, then

u(x) >0 for all xe!?.

7.5 A Maximum Principle for Integral Equations

227

Remark 7.4.2 F1om Remark 7.4.1, one can see that actually condition (7.44) is only required at points where u is negative.

Remark 7.4.3 Although Theorem 7.4.1 as well as its Corollaries are stated in linear forms, they can easily be applied to a nonlinear equation, for example,

-Au - IuIP-1u = 0 x c R".

(7.45)

Assume that the solution u decays near infinity at the rate of lIs with x s(p- 1) > 2. Let c(x) = -Iu(x)Ip-1. Then for R sufficiently large, and for the I

region Sl as stated in Corollary 7.4.2, c(x) satisfies (7.44) in S2. If we further assume that

UIon>0, then we can derive from Corollary 7.4.2 that u > 0 in the entire region fl.

7.5 A Maximum Principle for Integral Equations In this section, we introduce a Maximum Principle for integral equations. Let Sl be a region in Rn, which may or may not be bounded. Assume

K(x,y)?0, V(x,y)E.2xS2. Define the integral operator T by

(Tf)(x) = Let

.

Jn

K(x,y)f(y)dy

be a norm in a Banach space satisfying If II <-11911,

whenever 0 < f (x) < g(x) in Sl.

(7.46)

There are many norms that satisfy (7.46), such as LP (P) norms, the L' (S?) norm, and the C(12) norm.

Theorem 7.5.1 Suppose that IITgII s OIIgII

with some 0 < 0 < 1 Hg.

(7.47)

If f(x) < (Tf)(x),

(7.48)

f (x) < 0 for all x E 92.

(7.49)

then

7 Maximum Principles

228

Proof. Define

.i2+= {xE0I f(x)>0},

and

f +(x) = max{0, f(x)}, f-(x) = min{0, f (x)}. Then it is easy to see that for any x E .fl+,

0 5 f+(x) 5 (Tf)(x) = (Tf)+(x) + (Tf) (x) 5 (Tf)+(x),

(7.50)

while in the rest of the region .0, we have f+(x) = 0 and (Tf)+(x) > 0 by the definitions. Therefore, (7.50) holds for any x E Q. It follows from (7.50)

that IIf+II 5 II(Tf)+II <- e11f+11

Since 0 < 1, we must have IIf+II = 0,

which implies that f+(x) - 0, and hence f (x) < 0. This completes the proof of the Theorem.

Recall that in Section 7.4, after introducing the "Maximum Principle Based on Comparison" for Partial Differential Equations, we explained how it can be applied to the situations of "Narrow Regions" and "Decay at Infinity'. Parallel to this, we have similar applications for integral equations. Let a be a number between 0 and n. Define

(Tf)(x) = L

Ix - 11

°

c(y)f(y)dy

Assume that f satisfies the integral equation

f=Tf

in ,fl,

or more generally, the integral inequality

f
(7.51)

IITfIILP(.a) SCIIc(y)IIL-(n)IIfIILP(n),

(7.52)

Further assume that

for some p, ,T > 1.

If we have some right integrability condition on c(y), then we can derive, from Theorem 7.5.1, a Maximum Principle that will be applied to "Narrow Regions" and "Near Infinity'. More precisely, we have

7.5 A Maximum Principle for Integral Equations

229

Corollary 7.5.1 Assume that c(y) > 0 and c(y) E LT (R^). Let f E L1'(Rn) be a nonnegative function satisfying (7.51) and (7.52). Then there exist positive numbers Ra and co depending on c(y) only, such that if µ(,f2 fl BR. (0)) < eo, then f+ - 0 in I2,

where p(D) is the measure of the set D. Proof. Since c(y) E LT (R'), by Lebesgue integral theory, when the measure of the intersection of ,fl with BR, (0) is sufficiently small, we can make the integral fs2 I c(y) IT dy as small as we wish, and thus obtain CII c(y)II LT(A) < 1.

Now it follows from Theorem 7.5.1 that

f+(x)=_0,VxEQ. This completes the proof of the Corollary.

0

Remark 7.5.1 One can see that the condition µ(Q n BR, (0)) G c, in the Corollary is satisfied in the following two situations. i) Narrow Regions: The width of ,fl is very small. ii) Near Infinity: Say, D = BR (0), the complement of the ball BR(0), with sufficiently large R.

As an immediate application of this "Maximum Principle", we study an integral equation in the next chapter. We will use the "method of moving planes" to obtain the radial symmetry and monotonicity of the positive solutions.

8

Methods of Moving Planes and Moving Spheres

8.1 8.2

Outline of the Method of Moving Planes Applications of Maximum Principles Based on Comparison 8.2.1

8.2.2 8.2.3 8.3

Method of Moving Planes in a Local Way 8.3.1

8.3.2 8.4

The Background The A Priori Estimates

Method of Moving Spheres 8.4.1 8.4.2

8.5

Symmetry of Solutions on a Unit Ball Symmetry of Solutions of -Qu = uP in Rn. Symmetry of Solutions of -Qu = e° in R2

The Background Necessary Conditions

Method of Moving Planes in Integral Forms

The Method of Moving Planes (MMP) was invented by the Soviet mathematician Alexanderoff in the early 1950s. Decades later, it was further developed by Serrin [Se], Gidas, Ni, and Nirenberg [GNN], Caffarelli, Gidas, and Spruck [CGS), Chang and Yang [CY], and many others. For example, please also see [BLS], [CheLl], [CheL2], [CL1] [CL2], [DG], [C] [Gu], [GW], [HP], [Li), [Lil], [LiY4], [LiL], [LiL1], [Lin], [LiZ], [Po], [Pol], [SeZ], [Xu], [Zou], [Zoul],

and the references therein. This method has been applied to free boundary problems, semi-linear partial differential equations, and other problems. Particularly for semi-linear partial differential equations, there have been many significant contributions. We refer to the paper of Frenkel [F] for further description of the method.

8 Methods of Moving Planes and Moving Spheres

232

The Method of Moving Planes and its variant, the Method of Moving Spheres, have become powerful tools in establishing symmetries and monotonicities for solutions of partial differential equations. They can also be used to obtain a priori estimates, to derive useful inequalities, and to prove nonexistence of solutions. From the previous chapter, we have seen the beauty and power of Maximum Principles. The MMP greatly enhances the power of Maximum Principles. Roughly speaking, the MMP is a continuous way of repeated applications of Maximum Principles. During this process, the Maximum Principle has been used infinitely many times, and the advantage is that each time, we only need to use the Maximum Principle in a very narrow region. From the previous chapter, one can see that, in such a narrow region, even if the coefficients of the equation are not `good,' the Maximum Principle can still be applied. In the authors' research practice, we also introduced a form of Maximum Principle at infinity to the MMP and therefore simplified many proofs and extended the results in more natural ways. We recommend that readers study this part carefully so that they will be able to apply it in their own research. It is well known that by using Green's function, one can change a differential equation into an integral equation, and under certain conditions, they are equivalent. To investigate the symmetry and monotonicity of integral equations, the authors, together with On, created an integral form of MMP. Instead of using local properties, say, differentiability of a differential equation, they employed the global properties of the solutions of integral equations.

In this chapter, we will apply the Method of Moving Planes and the Method of Moving Spheres to study semi-linear elliptic equations and integral equations. We will establish symmetry, monotonicity, a priori estimates, and non-existence of solutions. During the process of Moving Planes, the Maximum Principles introduced in the previous chapter are applied in innovative ways.

In Section 8.2, we will establish radial symmetry and monotonicity for the solutions of the following three semi-linear elliptic problems

Au = f (u), x E B1(0) u = 0, on 8B1(0);

-Lsu = u

(x), x E R", n > 3;

and

-Du = e°iyl, x E R2. During the moving of planes, the Maximum Principles Based on Comparison

will play a major role. In particular, the Narrow Region Principle and the Decay at Infinity Principle will be used repeatedly in dealing with the three examples.

8.1 Outline of the Method of Moving Planes

233

In Section 8.3, we will apply the Method of Moving Planes in a `local way' to obtain a priori estimates on the solutions of the prescribing scalar curvature equation on a compact Riemannian manifold M 4(n

n

I)

L0u+

R0(x)u = R(x)u-2 x E M. ,

2

We allow the function R(x) to change sign. In this situation, the traditional blowing-up analysis fails near the set where R(x) = 0. We will use the Method of Moving Planes in an innovative way to obtain a priori estimates. Since the

Method of Moving Planes cannot be directly applied to the solution it,, we introduce an auxiliary function to circumvent this difficulty. In Section 8.4, we use the Method of Moving Spheres to prove a nonexistence of solutions for the prescribing Gaussian and scalar curvature equations

-Du + 2 = R(x)e" and

-Lu + n(n4 2)u

=

4(n

- 1)R(x)u

on S2 and on S' (n > 3), respectively. We prove that if the function R(x) is rotationally symmetric and monotone in the region where it is positive, then both equations admit no solution. This provides a stronger necessary condition than the well known Kazdan-Warner condition, and it also becomes a sufficient condition for the existence of solutions in most cases. In Section 8.5, as an application of the Maximum Principle for integral equations introduced in Section 7.5, we study the integral equation in R"

u(x)='

n

yl"a

U-(y)dy,

hx for any real number a between 0 and it. It arises as an Euler-Lagrange equation for a functional in the context of the Hardy-Littlewood-Sobolev inequalities. Due to the different nature of the integral equation, the traditional Method of Moving Planes does not work. Hence we exploit its global property and develop a new idea, the Integral Form of the Method of Moving Planes to obtain the

symmetry and monotonicity of the solutions. The Maximum Principle for Integral Equations established in Chapter 7 is combined with the estimates of various integral norms to carry out the moving of planes.

8.1 Outline of the Method of Moving Planes To outline how the Method of Moving Planes works, we take the Euclidean

space R" as an example. Let it be a positive solution of a certain partial

8 Methods of Moving Planes and Moving Spheres

234

differential equation. If we want to prove that it is symmetric and monotone in a given direction, we may assign that direction the xl axis. For any real number A, let TA = {x =

(xl,x2,...

xn) E R" I xl = A}.

This is a plane perpendicular to the xl-axis and is the plane with which we shall move. Let E'a denote the region to the left of the plane, i.e.,

La= {xER"Ix1
xa -(2A -xlf x2i... )xn) be the reflection of the point x = (xi,

,

about the plane TA (See Fig.

8.1).

. Ta

IN

Fig. 8.1. Moving Planes

We compare the values of the solution u at point x and xa, and we want to show that u is symmetric about some plane Tao. To this end, let wa(x) = u(x") - u(x).

In order to show that there exists some A such that WA0 (x) = 0 , Vx E Eao,

we generally go through the following two steps. Step 1. We first show that for A sufficiently negative, we have W,\(x) > 0 , Vx E Ea.

(8.1)

Then we are able to start off from this neighborhood of xl = -oo and move the plane TA along the xl direction to the right as long as the inequality (8.1) holds.

8.2 Applications of the Maximum Principles Based on Comparisons

235

Step 2. We continuously move the plane in this way up to its limiting position. More precisely, we define A0 = sup{A I wa(x) > 0, Vx E LA}.

We prove that u is symmetric about the plane Tao, that is, wao (x) ° 0 for all x E Lao. This is usually carried out by a contradiction argument. We show that if wae(x) 0, then there would exist A > A. such that (8.1) holds, and this contradicts with the definition of Ao.

From the above illustration, one can see that the key to the Method of Moving Planes is to establish inequality (8.1), and for partial differential equations, Maximum Principles are powerful tools for this task, while for integral equations, we use a different idea. We estimate a certain norm of wa on the set

La = {x E Ls I wa(x) < 0} where the inequality (8.1) is violated. We show that this norm must be zero, and hence La is empty.

8.2 Applications of the Maximum Principles Based on Comparisons In this section, we study some semi-linear elliptic equations. We will apply the Method of Moving Planes to establish the symmetry of the solutions. The essence of the Method of Moving Planes is the application of various Maximum Principles. In the proof of each theorem, readers will see vividly how the Maximum Principles Based on Comparisons are applied to narrow regions and to solutions with decay at infinity.

8.2.1 Symmetry of Solutions in a Unit Ball We first begin with an elegant result of Gidas, Ni, and Nirenberg [GNN1]:

Theorem 8.2.1 Assume that f (S) is a Lipschitz continuous function such that

- f (q) I

(8.2) - qI for some constant Co. Then every positive solution u (u E C2(Q) n C(f2)) of If (p)

Co jp

Au = f(u(x)), x E B1(0) U(X) = 0,

x E 8B1(0).

is radially symmetric and monotone decreasing about the origin.

(8.3)

8 Methods of Moving Planes and Moving Spheres

236

Proof. As shown in Figure 8.2 below, let Ta = {x I xl = A} be the plane perpendicular to the xt axis. Let Ea be the part of B1(0) on the left of the plane TA. For each x E EA,let xa be the reflection of the point x about the plane TA, more precisely, x' = (2A - xl, x2,

,

(See Fig. 8.2).

Fig. 8.2. Moving Planes on the Unit Ball

We compare the values of the solution u on Ea with those on its reflection. Let and wa(x) = ua(x) - u(x).

UA(x) = u(x'),

Then it is easy to see that ua satisfies the same equation as u does. Applying the Mean Value Theorem to f (u), one can verify that wa satisfies

Awa +C(x, A)wa(x) = 0, x E Lr, where

C(x A) =

f(UA(x)) - f('u(x)) U), (X) - u(x)

and by condition (8.2), IC(x,A)I 5 Co.

(8.4)

Step 1: Start Moving the Plane. We start from the near left end of the region. Obviously, for A sufficiently close to -1, EA is a narrow (in the xl direction) region, and on BEa, WA(x) > 0. (On TA, w,\ (x) = 0, while on the curve part of BEa, w,\ (x) > 0 since it > 0 in Bt (0).) We can now apply the "Narrow Region Principle" (Corollary 7.4.1) to conclude that for A close to -1, wa(x) > 0,

Vx E Ea.

(8.5)

8-2 Applications of the Maximum Principles Based on Comparisons

237

This provides a starting point for us to move the plane TA.

Step 2: Move the Plane to Its Right Limit. We now increase the value of A continuously; that is, we move the plane Ta to the right as long as the inequality (8.5) holds. We show that by moving in this way, the plane will not stop before hitting the origin. More precisely, let A = sup{A I wµ(x) > 0, Vx E Eµ,µ < a};

we first claim that

a>0. Otherwise, we will show that the plane can be further moved to the right by a small distance, and this would contradict with the definition of A. In fact, if A < 0, then the image of the curved surface part of 8Ea under the reflection about TX lies inside Bl (0), where u(x) > 0 by assumption. It follows that, on

this part of 8Ea, wx(x) > 0. By the Strong Maximum Principle, we deduce that wa(x) > 0

in the interior of Ea. Let do be the maximum width of narrow regions to which we can apply the "Narrow Region Principle." Choose a small positive number, 6, such that 6 < z , -A. We consider the function wa+a(x) on the narrow region:

a=Ea+an{xIxi>

2°°}.

(See Fig. 8.3).

The function wa+a(x) satisfies

Owa+6+C(x,.A+6)wa+6 = 0, x E Qa wx+s(x) > 0,

x E 0125.

The equation is obvious. To see the boundary condition, we first notice that it is satisfied on the two curved parts and one flat part where xl = A+6 of the boundary 8Q5 due to the definition of wa+a. To see that it is also

true on the rest of the boundary where xl = A - Z , we use a continuity argument. Notice that on this part, wa is positive and bounded away from 0. More precisely and more generally, there exists a constant co > 0, such that

wa(x)>co, VxEEa_dy. z Since wa is continuous in A, for 6 sufficiently small, we still have wa+6(x) > 0

,

Vx E Ex_s. 2

238

8 Methods of Moving Planes and Moving Spheres

Fig. 8.3. Narrow Region f26

Hence, in particular, the boundary condition in (8.7) holds for such small 6. Now we can apply the "Narrow Region Principle" to conclude that wa+a(x) > 0,

Vx E ,(la,

and therefore

wa+a(x) > 0,

Vx E E'a}6.

This contradicts with the definition of A and thus establishes (8.6). Now (8.6) implies that

u(-xi, x') < u(xr, XI) , Vxr > 0,

(8.8)

where x' = (x2, ... , x ). We then start from A close to 1 and move the plane Ta toward the left. Similarly, we obtain

u(-xr, x') > u(x1, x') ,Vx1 > 0.

(8.9)

Combining two opposite inequalities (8.8) and (8.9), we see that u(x) is symmetric about the plane To. Since we can orient the Ti axis in any direction, we conclude that u(x) must be radially symmetric about the origin. Also the monotonicity easily follows from the argument. This completes the proof.

8.2.2 Symmetry of Solutions of -Au = up in R^ In an elegant paper of Gidas, Ni, and Nirenberg [GNN], an interesting result is the symmetry of the positive solutions of the semi-linear elliptic equation:

8.2 Applications of the Maximum Principles Based on Comparisons

Au+up=0, xER',n>3.

239 (8.10)

They proved

Theorem 8.2.2 For p = n+2, all the positive solutions of (8.10) with reasonable behavior at infinity, namely

are radially symmetric and monotone decreasing about some point, and hence assume the form

u(x) -

2 [n(n - 2)A]' + n z for A > 0 and for some x° E R. (A2 + Ix - x°I2) 2

This uniqueness result, as was pointed out by R. Schoen, is in fact equivalent to the geometric result due to Obata [0]: A Riemannian metric on Sn which is conformal to the standard one and having the same constant scalar curvature is the pull back of the standard one under a conformal map of Sn to itself. Recently, Caffarelli, Gidas and Spruck [CG5] removed the decay assumption it = O(IxI2-n) and proved the same result. In the case that 1 < p < n+2, Gidas and Spruck [GS] showed that the only non-negative solution of (8.10) is identically zero. Then, in the authors' paper [CL1], a simpler and more elementary proof was given for almost the same result:

Theorem 8.2.3 i) For p = 2, every positive C2 solution of (8.10) must be radially symmetric and monotone decreasing about some point, and hence assumes the form n-2

u(x) _

[n(n - 2)A]" 2

(A2+Ix-x°I2)°2

, for some A > 0 and x° E Rn.

ii) For p < 2, the only nonnegative solution of (8.10) is identically zero. The proof of Theorem 8.2.2 is actually included in the more general proof of the first part of Theorem 8.2.3. However, to better illustrate the idea, we will first present the proof of Theorem 8.2.2 (mostly in our approach). And the readers will see vividly how the "Decay at Infinity" principle is applied here.

Proof of Theorem 8.2.2. Define

Ea={x=(xi,...,x,) ERnIxl
8 Methods of Moving Planes and Moving Spheres

240

and let xA be the reflection point of x about the plane TA, i.e.,

(See the previous Fig. 8.1.) Let UA(x) = u(xa),

and wa(x) = ua(x) - u(x).

The proof consists of three steps. In the first step, we start from the very left end of our region R", that is, near x1 = -oo. We will show that, for A sufficiently negative, WA (X) > 0, Vx E Ea.

(8.11)

Here, the "Decay at Infinity" is applied. Then in the second step, we will move our plane TA in the x1 direction toward the right as long as inequality (8.11) holds. The plane will stop at some limiting position, say at A = A0. We will show that wao(x)

0, Vx E Eao.

This implies that the solution it is symmetric and monotone decreasing about the plane Tao. Since the xl-axis can be chosen in any direction, we conclude that it must be radially symmetric and monotone about some point. Finally, in the third step, using the uniqueness theory in Ordinary Differential Equations, we will show that the solutions can only assume the given form.

Step 1. Prepare to Move the Plane from Near -oo. To verify (8.11) for A sufficiently negative, we apply the Maximum Principle to wa(x). Write T = n±2. By the definition of ua, it is easy to see that, u), satisfies the same equation as it does. Then by the Mean Value Theorem, it is easy to verify that

-OwA = ua(x) - uT(x) =

(8.12)

where 0,,(x) is some number between u), (x) and u(x). Recalling the Maximum

Principle Based on Comparison (Theorem 7.4.1), we see that here c(x) = -TlPT-1(x). By the "Decay at Infinity" argument (Corollary 7.4.2), it suffices to check the decay rate of 1/ia-1(x), and more precisely, only at the points x where wT is negative (see Remark 7.4.2 ). Apparently at these points, U),(z) < u(x), and hence

0 < ua(x) < 0a(X) < U(i)By the decay assumption of the solution

8.2 Applications of the Maximum Principles Based on Comparisons n(x)=p

241

P(/ \IzIn-2

we derive immediately that (x)

(xj1 2)=a/

=0\x41

Here the power of Ill is greater than two, which is what (actually more z than) we desire. Therefore, we can apply the Maximum Principle Based on Comparison to conclude that for A sufficiently negative (I±I sufficiently large), we must have (8.11). This completes the preparation for the moving of planes.

Step 2. Move the Plane to the Limiting Position to Derive Symmetry. Now we can move the plane TA toward the right, i.e., increase the value of A as long as the inequality (8.11) holds. Define As = sup{A I wl,,(x) > 0, Vx E Eµ,µ < A}.

Obviously, A0 < +oc due to the asymptotic behavior of it near xl = +oo. We claim that (8.13) W,\.(X) - 0, Vx E EAo.

Otherwise, by the Strong Maximum Principle on unbounded domains (see Theorem 7.3.3), we have

wA0(x) > 0 in the interior of EA0.

(8.14)

We show that the plane T 0 can still be moved a small distance to the right. More precisely, there exists a 60 > 0 such that, for all 0 < 6 < b0, we have WAo+s(x) > 0, Vx E EAo+a.

(8.15)

This would contradict with the definition of a0, and hence (8.13) must hold. Recall that in the last section, we used the "Narrow Region Principle" to derive (8.15). Unfortunately, it cannot be applied in this situation because the "narrow region" here is unbounded, and we are not able to guarantee that wA0 is bounded away from 0 on the left boundary of the "narrow region". To overcome this difficulty, we introduce a new function

w,\(x) =

A( ), )

where

fi(x)=

x1

with 0
8 Methods of Moving Planes and Moving Spheres

242

It is then a straightforward calculation to verify that

-LWA = 2pwA 0 +

A0

\

1

Owa + 0 W), 10.

(8.16)

We have

Lemma 8.2.1 There exists an R. > 0 (independent of A), such that if x° is a minimum point of wA and rx(x°) < 0, then 1x°1 < R°. We postpone the proof of the Lemma for a moment. Now suppose that (8.15) is violated for any S > 0. Then there exists a sequence of numbers f6 j) tending

to 0 and for each i, the corresponding negative minimum x' of 'WAo+si. By Lemma 8.2.1, we have

x'j
VzA0(x°) = limroOwAo+ai(x') = 0 and

wAo(x°) _ limwAo+ai(x') < 0. However, we already know wAo > 0, and therefore we must have 'wAo (x°) = 0.

It follows that

vwA0(x°)=vwA0(x°)0(x )+1VAo(x°)04(x)=0+0=0.

(8.18)

On the other hand, by (8.14), since wA0(x°) = 0, x° must be on the boundary of EAp. Then by the Hopf Lemma (see Theorem 7.3.3), we have the outward normal derivative OWA (x°) < 0.

This contradicts with (8.18). Now, to verify (8.15), we need only prove the Lemma.

Proof of Lemma 8.2.1. Assume that x° is a negative minimum of WA. Then -AwA(x°) < 0 and VWA(x°) = 0. (8.19) On the other hand, as we argued in Step 1, by the asymptotic behavior of is at infinity, if x°j is sufficiently large, c(x°)

> _ q(n - 2 - q) _ 0(x°) x012

-

O(x°)

8.2 Applications of the Maximum Principles Based on Comparisons

243

It follows from (8.12) that 'nSwa

+wa) (x°) > 0.

This, together with (8.19) contradicts with (8.16) and hence completes the proof of the Lemma. Step 3. In the previous two steps, we showed that the positive solutions of (8.10) must be radially symmetric and monotone decreasing about some point in R. Since the equation is invariant under translation, without loss of generality, we may assume that the solutions are symmetric about the origin. They then satisfy the following ordinary differential equation -u" (r) - n=1u'(r) = ,aT r

u'(0)=0 u(0)

n(n 2) (n-2)/4 Z =2

2)a ] 2 (A2 + r2) r2

for some A > 0. One can verify that u(r) = [n(n

is a solution, and

by the uniqueness of the ODE problem, this is the only solution. Therefore, we conclude that every positive solution of (8.10) must assume the form u(x) = for A > 0 and some x° (=-

[n(n-2)A2]P n2 2

(A2 + Ix - x° I2)

Rn. This completes the proof of the Theorem.

Proof of Theorem 8.2.3. i) The general idea in proving this part of the Theorem is almost the same as that given for Theorem 8.2.2. The main difference is that we have no decay assumption on the solution u at infinity, hence the Method of Moving Planes cannot be directly applied to u. So we first make a Kelvin transform to define a new function v(x) =

Then obviously, v(x) has the desired decay rate Ixln_2 at infinity, but has a

possible singularity at the origin. It is easy to verify that v satisfies the same equation as u does, except at the origin:

Av+vT(x)=0, xERn\{0},n>3.

(8.20)

We will apply the Method of Moving Planes to the function v and show that v is radially symmetric and monotone decreasing about some point. If the

244

8 Methods of Moving Planes and Moving Spheres

center point is the origin, then by the definition of v, u is also symmetric and monotone decreasing about the origin; if the center point of v is not the origin, then v has no singularity at the origin, and hence it has the desired decay at infinity. Then the same argument in the proof of Theorem 8.2.2 would imply that it is symmetric and monotone decreasing about some point. Define va(x) = v(x'), vJA(x) = va(x) - v(x).

Because v(x) may be singular at the origin, correspondingly wa may be singular at the point xa = (2A, 0, .. 0). Hence instead of on Ea, we consider wa on Ea = Ea \ {x ,}. And in our proof, we treat the singular point carefully. Each time, we show that the points of interest are away from the singularities so that we can carry on the Method of Moving Planes to the end to show the existence of a a° such that wao (x) _- 0 for x E Eaa and that v is strictly increasing in the x1 direction in Eao. As in the proof of Theorem 8.2.2, we see that v), satisfies the same equation as v does, and -Owa = To' (x)wa(x), where *a(x) is some number between vA(x) and v(x). ,

Step 1. We show that for A sufficiently negative, we have

wa(x) > 0, Vx E EA.

(8.21)

By the asymptotic behavior v(x)

1xln-2,

we derive immediately that, at a negative minimum point x° of wa, Y',T-1(x°)

the power of

of I

(1x1n-2)T-1

- IxIV

is greater than two, and we have the desired decay rate

x

for c(x) := as mentioned in Corollary 7.4.2; hence, we can apply "Decay at Infinity" to w), (x). The difference here is that wa has a singularity at xa, and hence we need to show that the minimum of wa is away from xa. Actually, we will show that

If infta w,\(x) < 0, then the infimum is achieved in Ea \ B1(xa). To see this, we first note that for x E Br (0), v(x) > min v(x) = co > 0 8B1(0)

(8.22)

8.2 Applications of the Maximum Principles Based on Comparisons

245

due to the fact that v(x) > 0 and Ov < 0. Then let A be so negative that v(x) < co for x E Bi(x,\). This is possible because v(x) -4 0 as xI -> oo. For such A, obviously wA(x) > 0 on B1(xa) \ {xA}. This implies (8.22). Now similar to Step I in the proof of Theorem 8.2.2, we can deduce that wa(x) ? 0 for A sufficiently negative. Step 2. Again, define A. = sup{A I WM(x) > 0, Vx E ti,, µ < A}.

We will show that

If ao < 0, then w ,, (x)

0, Vx E Eao.

Define IDa = T the same way as in the proof of Theorem 8.2.2. Suppose that wao (x) L0; then, by the Maximum Principle, we have wao(x) > 0

,

for x E Eao.

The rest of the proof is similar to step 2 in the proof of Theorem 8.2.2 except that now we need to take care of the singularities. Again, let Ak N ao be a sequence such that wak(x) < 0 for some x E EAk. We need to show that for each k, infE w,\, (x) can be achieved at some point xk E Eak and that the k

sequence {xk} is bounded away from the singularities xak of wa,,. This can be seen from the following facts:

a)

There exists e > 0 and S > 0 such that wao(x) > e for x E Bs(x),0) \ {xa0}.

b)

lima-.ap infXEBk(XA) wa(x) > infZEBS(%Ao) wao(x) > C.

Fact (a) can be shown by noting that 7-v,\, (x) > 0 on Eao and Away < 0, while fact (b) is obvious.

Now through a similar argument as given in the proof of Theorem 8.2.2 one can easily arrive at a contradiction. Therefore, wao (x) - 0.

If A, = 0, then we can carry out the above procedure in the opposite direction, namely, we move the plane in the negative x, direction from positive infinity toward the origin. If our planes TA stop somewhere before the origin, we derive the symmetry and monotonicity in xl direction by the above ar-

gument. If they stop at the origin again, we also obtain the symmetry and monotonicity in the xr direction by combining the two inequalities obtained

246

8 Methods of Moving Planes and Moving Spheres

in the two opposite directions. Since the x1 direction can be chosen arbitrarily, we conclude that the solution u must be radially symmetric about some point. ii) To show the non-existence of solutions in the case p < nn+2, we notice that after the Kelvin transform, v satisfies

Ov+ xn+2ly(n-2) vP(x) = 0 , x E R" \ {0}. Due to the singularity of the coefficient of vP at the origin, one can easily see that v can only be symmetric about the origin if it is not identically zero; hence, u must also be symmetric about the origin. Now given any two points x1 and x2 in Rn, since equation (8.10) is invariant under translations and rotations, we may assume that the origin is at the midpoint of the line segment x1x2. Then from the above argument, we must have u(x1) = u(x2). It follows that u is constant. Finally, from the equation (8.10), we conclude that u - 0. This completes the proof of the Theorem.

8.2.3 Symmetry of Solutions for -Au = e" in R2 When considering prescribing Gaussian curvature on two-dimensional compact manifolds: if the sequence of approximate solutions "blows up", then by re-scaling and taking the limit, one would arrive at the following equation in the entire space R2:

4u+expu=0, xER2 fR2 exp u(x)dx < +oo.

(8.23)

The classification of the solutions for this limiting equation would provide essential information on the original problems on manifolds (and it is also interesting in its own right). It is known that 32A2 OA,

(x) = In (4 + A21x _ x°12)2

for any A > 0 and any point x° E R2 is a family of explicit solutions. We will use the Method of Moving Planes to prove the following theorem:

Theorem 8.2.4 Every solution of (8.23) is radially symmetric with respect to some point in R2 and hence assumes the form of (x). To this end, we first need to obtain some decay rate of the solutions near

infinity.

8.2 Applications of the Maximum Principles Based on Comparisons

247

Some Global and Asymptotic Behavior of the Solutions The following Theorem gives the asymptotic behavior of the solutions near infinity, which is essential to the application of the Method of Moving Planes.

Theorem 8.2.5 If u(x) is a solution of (8.23), then as lxj -4 +oo, 1 (x ))

i - Zn fR, exp u(x) dx < -4

uniformly.

This Theorem is a direct consequence of the following two lemmas.

Lemma 8.2.2 (W. Ding) If it is a solution of

-Au=e", xER2, and

fR2 expu(x)dx < +oo, then

fR, exp u(x)dx > 8ir.

Proof. For -oo < t < oo, and letting Ot = {x I u(x) > t}, one can obtain f.at

u(x)dx = - 12, An = foot 1pulds

-dtIOtl =

Jan, Foul

By the Schwartz inequality and the isoperimetric inequality, ds

Jan, IvuI Jan

foul > lantl2 > 4trlQtl.

Hence

-(d IOt1) - fn, expu(x)dx > 47rIJ?tl, so

dt(fn

fn, expu(x)dx < -8trjQtlet.

expu(x)dx)2 = 2expt'

Integrating from -oo to oo gives

(J expu(x)dx) U2R

f

2

< -87r

JR'

which implies fR2 expu(x)dx > 81r as desired.

exp u(x)dx,

8 Methods of Moving Planes and Moving Spheres

248

Lemma 8.2.2 enables us to obtain the asymptotic behavior of the solutions at infinity.

Lemma 8.2.3. If u(x) is a solution of (8.23), then as IxI -r +oo,

1

IxI

-4

27r fR2 expu(x)dx uniformly.

Proof. By a result of Brezis and Merle [BM], we see that the condition fR2 exp u(x)dx < oo implies that the solution u is bounded from above. Let

w(x) =

,r

fR2(ln Ix - yI - ln(Iyl + 1)) expu(x)dy.

Then it is easy to see that ,nsw(x) = exp u(x) , x E R2, and we will show w(x) In IxI

-4

1

fR2

21

exp u(x)dx uniformly as IxI -4 +oo.

(8.24)

To see this, we need only verify that

I:= fR2

In Ix - yl - ln(Iyi + 1) - In lxl eU(y)dy->O In IxI

as IxI-ioo. Write I = I,+I2+I3i where 11, I2 and I3 are the integrals on the three regions

D,={yIIx-yI<<1}, D2 = {yIIx - yI > 1 and IyI
D3={yIIx-yI>1 and IyI>K}, respectively. We may assume that IxI > 3. a) To estimate I,, we simply notice that 11 < C

eu(y)dy

J Ix-yl<1

1

- In IxI J

In Ix Ix-yl<<1

Then by the boundedness of eu(y) and fR2e°(y)dy, we see that IL->0 as IxI->oo.

b) For each fixed K in region D2, we have, as IxI-100,

yIe°(y)dy.

8.2 Applications of the Maximum Principles Based on Comparisons

249

In Ix-yl-ln(Gyi+1)-InIxi-+0 In Ixi and hence I2->0.

c)To see 13-+0, we use the fact that for Ix - yI > 1

In Ix-yl -ln(Iyi+1) -lnixiI
Then let K-*oo. This verifies (8.24). Consider the function v(x) = u(x) + w(x). Then

0 and

v(x)
The Method of Moving Planes and Symmetry In this subsection, we apply the Method of Moving Planes to establish the symmetry of the solutions. For a given solution, we move the family of lines orthogonal to a given direction from negative infinity to a critical position and then show that the solution is symmetric in that direction about the critical position. We also show that the solution is strictly increasing before the critical position. Since the direction can be chosen arbitrarily, we conclude

that the solution must be radially symmetric about some point. Finally by the uniqueness of the solution of the following O.D.E. problem

ON + 'U'(T) = f (U) u'(0) = 0 U(O) = 1

we see that the solutions of (8.23) must assume the form ¢a,zo (x).

Assume that u(x) is a solution of (8.23). Without loss of generality, we show the monotonicity and symmetry of the solution in the x1 direction.

For a e Rr, let E'X = {(XI, x2) I xl < Al and

TA = ar'a = {(x1, x2) I x1 = A}. Let

xa = (2a - x1,x2)

250

8 Methods of Moving Planes and Moving Spheres

be the reflection point of x = (X1, X2) about the line TA. (See the previous Fig. 8.1.)

Define

WA(x) = u(xA) - u(x).

A straightforward calculation shows ,n,wA(x) + (exp i/'A(x))wa(x) = 0,

(8.25)

where 0A(x) is some number between u(x) and uA(x). Step 1. As in the previous examples, we show that, for a sufficiently negative, we have WA (X) ? 0, VX E EA.

By the "Decay at Infinity" argument, the key is to check the decay rate of c(x) := -e'f'aW at points x° where w.\(x) is negative. At such points, u(x°A) < u(x°), and hence OA(x°) < u(x°).

By virtue of Theorem 8.2.5, we have (8.26)

Notice that we are in dimension two, while the key function qs = given _F in Corollary 7.4.2 requires 0 < q < n - 2 and hence does not work here. As a modification, we choose ¢(x) = lnOxI - 1).

Then it is easy to verify that

-1

A(P

0 (x) _

Ixi(IxI - 1)2ln(IxI -

I).

It follows from this and (8.26) that

eOx(s) +

(x°) < 0 for sufficiently large x°j.

(8.27)

This is what we desire. Then, similar to the argument given in Subsection 8.2.2, we introduce the function WA(x) _

(())

which satisfies L\WA + 2pwA

+

wA = 0.

(8.28)

8.3 Method of Moving Planes in a Local Way

251

Moreover, by the asymptotic behavior of the solution it near infinity (see Lemma 8.2.3), we have, for each fixed A, u(xa)

'wa(x) =

]n(jxj - u(x) 1)

-+0 ,

as IxI-+oc.

(8.29)

Now, if wa(x) < 0 somewhere, then by (8.29), there exists a point x°, which is a negative minimum of wa(x). At this point, one can easily derive a contradiction from (8.27) and (8.28). This completes Step 1. Step 2 . Define

a° = sup{A I wµ(x) > 0, Vx E Eµ, µ < Al. We show that

If a° < 0, then wA. (x)

0, Vx E Eao.

The argument is entirely similar to that given in Step 2 of the proof of Theorem 8.2.2 except that we use ¢(x) = ]n(-x1 + 2). This completes the proof of the Theorem.

8.3 Method of Moving Planes in a Local Way 8.3.1 The Background In Chapter 6, we introduced the prescribing scalar curvature equation on S": -Inlu+n(n-2)u=

4

n-2 R(x)u, u>0, xES".

4(n - 1)

(8.30)

It is the so-called critical case where the lack of compactness occurs. In this case, the well-known Kazdan-Warner condition gives rise to many examples of R in which there is no solution. A tremendous amount of effort has been put into determining the existence of the solutions, and many interesting results have been obtained ( see [Ba] [BC] [Bi] [BE] [CL1] [CL3] [CL4] [CL6] [CY1] [CGY] [ES] [KW1] [LiY2] [LiY3] [SZ] (Linl] and the references therein).

One main ingredient in the proof of existence is to obtain a priori estimates

on the solutions. For equation (8.30) on S", to establish a priori estimates, a useful technique employed by most authors was a 'blowing-up' analysis; unfortunately, it does not work near points where R(x) = 0. Due to this limitation, people had to assume that R was positive and bounded away from 0. This technical assumption became somewhat standard and has been used

252

8 Methods of Moving Planes and Moving Spheres

by many authors for quite a few years. For example, see articles by Bahri [Ba], Bahri and Coron [BC], Chang and Yang [CY1], Chang, Gursky, and Yang (CGY], Li [LiY2] [LiY3], and Schoen and Zhang [SZ].

Then for functions with changing signs, are there any a priori estimates? In [CL6], we answered this question affirmatively. We removed the above well-known technical assumption and obtained a priori estimates on the solutions of (8.30) for functions R which are allowed to change signs. The main difficulty encountered by the traditional 'blowing-up' analysis is near the point where R(x) = 0. To overcome this, we used the `Method of Moving Planes' in a local way to control the growth of the solutions in this region and obtained an a priori estimate. In fact, we derived a priori bounds for the solutions of more general equations

-Lu + n(n4 2) u = R(x)tP, u > 0, x E S"

(8.31)

for any exponent p between 1 + a and A, where A is an arbitrary positive number. For a fixed A, the bound is independent of p. Proposition 8.3.1 Assume R E C2,"(S") and IVR(x)I > /3o for any x with IR(x)I < So. Then there are positive constants e and C depending only on Qo, So, M, and IIRIIc2.n (sn), such that for any solution it of (8.31), we have it < C in the region where IR(x) I < e.

In this proposition, we essentially required R(x) to grow linearly near its zeros, which seems a little restrictive. Recently, in the case p = nn+2, Lin [Linl] weakens the condition by assuming only polynomial growth of R near its zeros. To prove this result, he first established a Lionville type theorem in R" and then used a `blowing up' argument. Later, by introducing a new auxiliary function, we sharpened our Method of Moving Planes to further weaken the condition and to obtain a more general result:

Theorem 8.3.1 Let M be a locally conformally flat manifold. Let

P= {xeMIR(x)=0}. OR

Assume that r E C2," and R E C2,°(M) satisfying < 0, where v is the outward normal (pointing to the region where R is negative) of P, and R(x)

IVR(x)I

1

continuous near r,

= 0 at P.

(8.32)

8.3 Method of Moving Planes in a Local Way

253

Let D be any compact subset of M and let p be any number greater than 1. Then the positive solutions of the equation

-Lou + R0(x)u = R(x)up, X E M

(8.33)

are uniformly bounded near r n D. Here, Lo is the Laplace-Beltrami operator associated with a base metric g, and R4(x) the corresponding scalar curvature.

One can see that our condition (8.32) is weaker than polynomial growth restriction. To illustrate, one may simply look at functions R(x) that grow like exp{-dx) }, where d(x) is the distance from the point x to P. Obviously, these kinds of functions satisfy our condition but are not polynomial growth. Moreover, our restriction on the exponent is much weaker. Besides being

n+2

n-2' p can be any number greater than 1. 8.3.2 The A Priori Estimates Now, we estimate the solutions of equation (8.33) and prove Theorem 8.3.1. Since M is a locally conformally flat manifold, a local flat metric can be chosen so that equation (8.33) in compact set D C M can be reduced to

-emu = R(x)up, p > 1

(8.34)

in compact set K in R". This is the equation we will consider throughout the section.

The proof of the Theorem is divided into two parts. We first derive the bound in the region(s) where R is negatively bounded away from zero. Then based on this bound, we use the `Method of Moving Planes' to estimate the solutions in the region(s) surrounding the set P.

Part I. In the region(s) where R is negative In this part, we derive estimates in the region(s) where R is negatively bounded away from zero. It comes from a standard elliptic estimate for subharmonic functions and an integral bound on the solutions. We prove Theorem 8.3.2 The solutions of (8.34) are uniformly bounded in the region {x E K : R(x) C 0 and dist(x, P) > b}. The bound depends only on b, the lower bound of R.

To prove Theorem 8.3.2, we introduce the following lemma of Gilbarg and Trudinger [GT].

254

8 Methods of Moving Planes and Moving Spheres

Lemma 8.3.1 (See Theorem 9.20 in /GTJ). Let u E W2,"(D) and suppose au > 0. Then for any ball B2r(x) C D and any q > 0, we have sup u <

C(1 rn

Br(=)

(u+)9dV)4,

(8.35)

JB2r(-)

where C = C(n, D, q).

By virtue of this Lemma, to establish an a priori bound of the solut ons, one only needs to obtain an integral bound. In fact, we prove

Lemma 8.3.2 Let B be any ball in K that is bounded away from F; then there is a constant C such that updx <

(8.36)

JB

Proof. Let Sl be an open neighborhood of K such that dist (OSl, K) > 1. Let w be the first eigenfunction of -,L in fl, i.e.,

ow = Aiw(x), x c fl

I op(x) = 0,

x E BQ.

11

if R(x) > 0 if R(x) = 0

Let

sgnR =

0

-1 if R(x) < 0.

Let a = p 21 Multiply both sides of equation (8.34) by p°IRJ'sgnR and .

then integrate. Taking into account that a > 2 and o and R are bounded in fl, through a straightforward calculation, we have In

wvaI RI'+updx < Ci

fo wa-2iR1a-2udx < C2 J

2iRI°-2udx.

Applying the Holder inequality to the right-hand-side, we arrive at

'paIRl1+°updx
(8.37)

Now (8.36) follows accordingly. This completes the proof of the Lemma.

The Proof of Theorem 8.3.2. It is a direct consequence of Lemma 8.3.1 and Lemma 8.3.2

Part II. In the region where R is small

8.3 Method of Moving Planes in a Local Way

255

In this part, we estimate the solutions in a neighborhood of F where R is small.

Let u be a given solution of (8.34), and xo E F. It is sufficient to show that u is bounded in a neighborhood of xo. The key ingredient is to use the Method of Moving Planes to show that, near xe, along any direction that is close to the direction of 7R, the values of the solution u are comparable. This result, together with the bounded-boundedness of the integral f u' in a small neighborhood of xo leads to an a priori bound of the solutions. Since the Method of Moving Planes cannot be directly applied to u, we construct some auxiliary functions. The proof consists of the following four steps. Step 1. Transforming the region

Make a translation, a rotation, or, if necessary, a Kelvin transform. Denote

the new system by x = (x1, y) with y = (x2, - ,x). Let the x1 axis point to the right. In the new system, xo becomes the origin, the image of the set .(L+ :_ {x I R(x) > 0} is contained in the left of the y-plane, and the image of r is tangent to the y-plane at the origin. The image of P is also uniformly - -

convex near the origin.

Let x1 = 0(y) be the equation of the image of I' near the origin. Let 81D <_ {x I x1 = 0(y) + e, xi > -2e}.

The intersection of 81D and the plane {x E Rn I x1 = -2e} enclose a part of the plane, which is denoted by 02D. Let D be the region enclosed by two surfaces 81D and 82D (See Fig. 8.4).

Fig. 8.4. Moving Planes in a Local Way

8 Methods of Moving Planes and Moving Spheres

256

The small positive number c is chosen to ensure that (a) 81D is uniformly convex, R (b) (c)

ax, R(x)

< 0 in D, and

is continuous in D.

Step 2. Introducing an auxiliary function Let in = maxa,D it. Let v. be a C2 extension of it from 01D to the entire 8D, such that 0 < ii < 2m, and lvul < Cm. Let w be the harmonic function

Ow=0,xED lw = u', x c 0D. Then by the Maximum Principle and a standard elliptic estimate, we have

0 < w(x) < 2m

and IIwIIc=(D) <- C1m.

(8.38)

Introduce a new function v(x) = u(x) - w(x) + Com[e + 0(y) - x1] +m[e + 0(y) - x1]2 which is well defined in D. Here Co is a large constant to be determined later. One can easily verify that v(x) satisfies the following r Lsv

v(x) = 0,

f (x, v) = 0, x E D x E 81D,

where

'b(y) = -ComAO(y) - mA[e + 0(y)]2

- 2m,

and

f (x, v) = 2mx100(y)+R(x){v+w(x) -C0m[e+0(y)-x1]-m[e+0(y)-x1]2) p. At this step, we show that

v(x) > 0

Vx E D.

We consider the following two cases.

Case 1) 0(y) + s < x1 < 0(y) + E. In this region, R(x) is negative and bounded away from 0. By Theorem 8.3.2 and the standard elliptic estimate, we have

8.3 Method of Moving Planes in a Local Way

au 8x1

257

<- C2m,

for some positive constant C2. It follows from this and (8.38), 8x1

<(Cl+C2-Co)m-2m(e+(k(y)-xi).

Noticing that (e + 0(y) - xi) > 0, one can choose Co sufficiently large, such

that Ox1 <0. Then, since

v(x) = 0 Vx E 81D, we have v(x) > O.

Case 2) -2e < xi < ¢(y) + 2. In this region, again by (8.38), we have

v(x) > -w(x) +Com2 > -2m+Com2. Again, choosing Co sufficiently large (and depending only on e), we arrive at v(x) > 0. Step 3. Applying the Method of Moving Planes to v in the x1 direction

In the previous step, we showed that v(x) > 0 in D and v(x) = 0 on 01D. This lays a foundation for the moving of planes. Now we can start from the rightmost tip of region D and move the plane perpendicular to the x1 axis toward the left. More precisely, let

Ea={xEDIxl>A), TA = {xER"I x1=A}. Let xA = (2A - x1, y) be the reflection point of x with respect to the plane

TA. Let va(x) = v(xa)

and wa(x) = va(x) - v(x).

We are going to show that W'\ (X) > 0 V x E Ea.

(8.39)

We want to show that the above inequality holds for -cl < A < e for some 0 < cl < e. This choice of el is to guarantee that when the plane Tl, moves to A = -el, the reflection of Ea about the plane Ta still lies in region D. Now, (8.39) is true when A is less but close to e, because Ea is a narrow region and f (x, v) is Lipschitz continuous in v. For detailed arguments, the readers may see the first example in Subsection 8.2.1.

8 Methods of Moving Planes and Moving Spheres

258

We now we decrease A; that is, move the plane TA toward the left as long as the inequality (8.39) remains true. We show that the moving of planes can be carried on provided f(x,v(x)) < f(x),v(x)) for x = (xl,y) E D with xl > A >

(8.40)

In fact, it is easy to see that vA satisfies the equation ,Lva + z/i(y) + f (xa, va) = 0,

and hence wa satisfies -L WA = f (X A, vA) - f (X A, v) + Ax A, v) - .f (x, v)

= R(xa)C), (x) (v), - v) + f (xa, v) - f (x, v) = R(XA)CA(x)wa + f(xa,v) - f(x,v)

Here we have applied the Mean Value Theorem, CA(x) = XP-1(x), and eA(x) is valued between

{v(x) + w(x) - Come + 0(y) - xl] - m[e + 0(y)

- xl]2}

and

{v,\(x) + w(x) - Corm.[e + O(y) - xl] - rn[e + O(y) - xi]2}. If (8.40) holds, then we have

-Ow;, - R(x')CA(x)wa(x) > 0.

(8.41)

This enables us to apply the Maximum Principle to wA. Let ao = inf{A

j

wµ(x) > 0,Vx E E1,P > A}.

If ao > -el, then since v(x) > 0 in D, by virtue of (8.41) and the Maximum Principle, we have

wa° (x) > 0, Vx E Ea° and

,< 0, Vx E

Then, similar to the argument given in the examples in Subsection 8.2.2, one can derive a contradiction.

It is elementary to see that (8.40) holds if Of(Xv) < 0 in the set

{x E D I R(x) < 0 or xl > -2ci}.

To estimate a , we use 1

8.3 Method of Moving Planes in a Local Way

aX

259

+Com+2m(e+QS(y)-xr)1}.

First, we notice that the uniform convexity of the image of P near the origin implies

,Lo(y) < -ao < 0

(8.42)

for some constant ao. We consider the following two possibilities. a) R(x) < 0. Choose Co sufficiently large, so that aw

+Com>0.

ax,

Noticing that a R < 0 and (e + (y) - x1) > 0 in D, we have Of <0. ax-1 -

b) R(x) > 0, x = (x1, y) with x1 > -2e1. In the part where u > 1, we use it to control

/

aR

aw

1R(x)/axl p[axl +Com+2m(e+0(y)-x1)1. More precisely, we write

of _ axl

p-1 OR 6x1

u

=

(

x { u+

OR aw R(x)/ax) p[axl +Com+2m(e+O(y) - xl )1 }

Choose el sufficiently small, such that I

-RIHIVRI.

Then by condition (8.32) on R, we can make R(x)/9' arbitrarily small by a small choice of e1, and therefore arrive at a CO. In the part where it < 1, by virtue of (8.42), we can use 2mLO(y) to control

Rp[ axl On'

-x1)1.

Here again we use the smallness of R. So far, our conclusion is this: The Method of Moving Planes can be carried on up to A = -El. More precisely, for any A between -el and c, the inequality (8.39) is true.

8 Methods of Moving Planes and Moving Spheres

260

Step 4. Deriving the a priori bound Inequality (8.39) implies that, in a small neighborhood of the origin, the function v(x) is monotone decreasing in the xr direction. A similar argument

will show that this remains true if we rotate the xl-axis by a small angle. Therefore, for any point xo E P, one can find a cone 4zp with xo as its vertex and staying to the left of xo, such that v(x) > v(xo)

(8.43)

dx E 4yp.

Noticing that w(x) is bounded in D, (8.43) leads immediately to

u(x) + C > u(xo)

Vx E 4so.

(8.44)

More generally, by a similar argument, one can show that (8.44) is true for any point xo in a small neighborhood of P. Furthermore, the intersection of the cone 4.a with the set {xIR(x) > 2 } has a positive measure, and the lower bound of the measure depends only on bo and the C' norm of R. Now the a priori bound of the solutions is a consequence of (8.44) and an integral bound on u, which we have derived from Lemma 8.3.2. This completes the proof of Theorem 8.3.1.

8.4 Method of Moving Spheres 8.4.1 The Background In this section, we revisit the prescribing Gaussian and scalar curvature equations

-Au + 2 = R(x)e°lsl, x E Sz.

n(n4 2)u(x)

=

R(x)un2(x),

(8.45)

n > 3, x E S", u > 0.

4(n - 1)

(8.46)

We will use the Method of Moving Spheres to establish some stronger necessary conditions. Both equations are so-called `critical' in the sense that lack of compactness occurs. Besides the obvious necessary condition that R(x) be positive somewhere, there are other well-known obstructions found by Kazdan and Warner [KW1] and later generalized by Bourguignon and Ezin [BE]. The conditions are:

Jsf X (R) dVg = 0,

(8.47)

8.4 Method of Moving Spheres

261

where dVy is the volume element of the metric e" go and u^=ago for n = 2 and for n > 3, respectively. The vector field X in (8.47) is a conformal Killing field associated to the standard metric 9o on Sh. In the following, we call these necessary conditions "Kazdan-Warner type" conditions.

These conditions give rise to many examples of R(x) for which (8.46) and

(8.45) have no solution. In particular, a monotone rotationally symmetric function R admits no solution. Then for which R can one solve the equations? What are the necessary and sufficient conditions for the equations to have a solution? These are very interesting problems in geometry.

In recent years, various sufficient conditions were obtained by many authors (For example, see [BC] [CY1] [CGY] [CY2] [CY3] [CL1O] [CD1] [CD2] [CC] [CS] [ES] [Ha] [Ho] [KW1] [KW2] (LiY1) [LiY2] [Mo] and the references therein.) However, there are still gaps between those sufficient conditions and

the necessary ones. Then one may naturally ask: "Are the necessary conditions of Kazdan-Warner type also sufficient?" This question has been open for many years. In [CL3], we gave a negative answer to this question. To find necessary and sufficient conditions, it is natural to begin with functions having certain symmetries. A pioneer work in this direction is [Me] by Moser. He showed that, for even functions R on S2, the necessary and sufficient condition for (8.45) to have a solution is that R be positive somewhere. Note that, in the class of even functions, (8.47) is satisfied automatically. Thus one can interpret Moser's result as follows: In the class of even functions, the Kazdan-Warner type conditions are the necessary and sufficient conditions for (8.45) to have a solution. An interesting question was asked by Kazdan [Ka]: What are the necessary and sufficient conditions for rotationally symmetric functions?

To be more precise, for rotationally symmetric functions R = R(0), where 0 is the geodesic distance from the north pole, the Kazdan-Warner type conditions take the following form: R > 0 somewhere and R' changes signs.

(8.48)

Now, is this a sufficient condition? In [XY], Xu and Yang proved:

Proposition 8.4.1 Let R be rotationally symmetric. Assume that i) R is non-degenerate ii)

R' changes signs in the region where R > 0. Then problem (8.45) has a solution.

(8.49)

262

8 Methods of Moving Planes and Moving Spheres

The above results and many other existence results tend to lead people to believe that what really counts is whether R' changes signs in the region where R > 0. And we conjectured in [CL3] that for rotationally symmetric R, condition (8.49), instead of (8.48), would be the necessary and sufficient condition for (8.45) or (8.46) to have a solution. The main objective of this section is to present the proof of the `necessary' part of the conjecture, which appeared in [CL4]. We use the Method of Moving Spheres. Instead of showing the symmetry of the solutions, we obtain a comparison formula which leads to a direct contradiction. In an earlier work [P]., a similar method was used by P. Padilla to show the radial symmetry of the solutions for some nonlinear Dirichlet problems on annuli. The following are our main results.

Theorem 8.4.1 Let R be continuous and rotationally symmetric. If R is monotone in the region where R > 0, and R 0 C,

(8.50)

Then problems (8.45) and (8.46) have no solution at all. This result can be generalized to a family of non-symmetric functions.

Theorem 8.4.2 Let R be a continuous function. Let B be a geodesic ball 1. Assume that centered at the North Pole i) R(x) > 0, R is non-decreasing in xn}1 direction, ii) R(x) < 0, for x E Sn \ B; Then problems (8.45) and (8.46) have no solution at all.

for x E B

Combining our Theorem 8.4.1 with Xu and Yang's Proposition 8.4.1, one can obtain a necessary and sufficient condition in the non-degenerate case:

Corollary 8.4.1 Let R be rotationally symmetric and non-degenerate in the sense that R" y6 0 whenever IT = 0. Then the necessary and sufficient condition for (8.45) to have a solution is R > 0 somewhere and R' changes signs in the region where R > 0.

As we proved in Chapter 6, in higher dimensions, condition (8.49) is also sufficient for equation (8.46) to have a solution under an additional flatness assumption on R(x).

8.4.2 Necessary Conditions In this subsection, we prove Theorem 8.4.1 and Theorem 8.4.2.

8.4 Method of Moving Spheres

263

For convenience, we make a stereo-graphic projection from S" to R". Then it is equivalent to consider the following equation in Euclidean space R" :

-Lu = R(r)e", x E R2;

(8.51)

-,n u=R(r)uT, u>0, xER", n>3;

(8.52)

with appropriate asymptotic decay of the solutions at infinity

u--4lnIxI forn=2; u_CIxl2-nforn>3

(8.53)

for some C > 0. Here 4 is the Euclidean Laplacian operator, r = IxI and r = '±2 is the so-called critical Sobolev exponent. The function R(r) is the projection of the original R in equations (8.45) and (8.46); and this projection

does not change the monotonicity of the function. The function R is also bounded and continuous, and we assume these throughout the subsection. For a radial function R satisfying (8.50), there are three possibilities: i) R is non-positive everywhere, ii) R 0 C is nonnegative everywhere and monotone, iii) R > 0 and non-increasing for r < ro, R < 0 for r > ro; or vice versa. The first two cases violate the Kazdan-Warner type conditions, hence there is no solution. Thus, we only need to show that in the remaining case, there is also no solution. Without loss of generality, we may assume that:

R(r) > 0, R'(r) < 0 for r < 1; R(r) < 0 for r > 1.

(8.54)

The proof of Theorem 8.4.1 is based on the following comparison formulas.

Lemma 8.4.1 Let R be a continuous function satisfying (8.54). Let it be a solution of (8.51) and (8.53) for n = 2, then u(Ax) > u(IAI2) - 41n IxI

Vx E Br(0), 0 < A < 1.

(8.55)

Lemma 8.4.2 Let R be a continuous function satisfying (8.54). Let u be a solution of (8.52) and (8.53) for n > 2, then u(Ax) >

IxI2_"u(IA X12

)

Vx E Br (0), 0 < A < 1.

(8.56)

We first prove Lemma 8.4.2. A similar idea will work for Lemma 8.4.1.

264

8 Methods of Moving Planes and Moving Spheres

Proof of Lemma 8.4.2. We use a new idea called the `method of moving spheres.' Let x E Br(0), then Ax E Ba(0). The reflection point of Ax about the sphere 8Ba(0) is ,2. We compare the values of u at those pairs of points Ax and yi2. In step 1, we start with the unit sphere and show that (8.56) is true for A = 1; this is done by applying a Kelvin transform and the strong Maximum Principle. In step 2, we move (shrink) the sphere 8Ba (0) towards the origin. We show that one can always shrink 8Ba (0) a little before reaching the origin. By doing so, we establish the inequality for all x E Bl (0) and A E (0,1]. Step 1. In this step, we establish the inequality for x E Br(0), A = 1. We show that

U(X) > Ixl2-nu(II2) b'x E Bi(0).

(8.57)

Let v(x) = IxI2-su(m) be the Kelvin transform of it. Then it is easy to verify that v satisfies the equation

v=R(1)v' r

(8.58)

and v is regular. It follows from (8.54) that Lu < 0, and nsv > 0 in Br(0). Thus

-0(u-v)>0. Applying the Maximum Principle, we obtain u > v in B1(0)1

(8.59)

which is equivalent to (8.57).

Step 2. In this step, we move the sphere 8Ba(0) towards A = 0. We show that the moving cannot be stopped until A reaches the origin. Again this is done by applying the Kelvin transform and the strong Maximum Principle. Let u), (x) = a%-ru(Ax). Then

-qua = R(Ax)u')(x).

(8.60)

Let vA(x) = IxI2-'u,\(' ) be the Kelvin transform of ua. Then it is easy to verify that -Ova = R(A)va(x). (8.61) Let wa = ua - va. Then by (8.60) and (8.61),

Lwa + R(A) {Gawa = [R( ) - R(Ar)]ui r

r

,

(8.62)

8.4 Method of Moving Spheres

where V)a is some function with values between rua 1 and account assumption (8.54), we have

R(A) - R(AT) < 0 T

265

rva-1. Taking

into

for r < 1, A < 1.

It follows from (8.62) that Owa +CA(x)WA < 0

(8.63)

where CA(x) is a bounded function if A is bounded away from 0. It is easy to see that for any A, strict inequality holds somewhere for (8.63). Thus, by applying the strong Maximum Principle, we know that the inequality (8.56) is equivalent to wa > 0. (8.64) From step 1, we know (8.64) is true for (x, A) E Bi(0) x {1}. Now we decrease A. Suppose (8.64) does not hold for all A E (0, 1] and let Ao > 0 be the smallest number such that the inequality is true for (x, A) E B1 (0) x ]A,,1]. We will

derive a contradiction by showing that for A close to and less than A0, the inequality is still true. In fact, we can apply the strong Maximum Principle and then the Hopf lemma to (8.63) for A = A. to get: wao > 0 in B1 and

OWA °

< 0 on 8B1.

(8.65)

These combined with the fact that wa - 0 on .B1 imply that (8.64) holds for A close to and less than A0.

Proof of Lemma 8.4.1. In step 1, we let v(x) = u( p) - 4ln x1.

In step 2, let

ua(x)=u(Ax)+2lnA, va(x)=uA( -[ )-41njxj. Then arguing as in the proof of Lemma 8.4.2 we derive the conclusion of Lemma 8.4.1.

We are now ready to prove Theorem 8.4.1.

Proof of Theorem 8.4.1. Taking A to 0 in (8.55), we get In xl > 0 for xj < 1, which is impossible. Letting A-+0 in (8.56) and using the fact that u(0) > 0, we obtain again a contradiction. These complete the proof of Theorem 8.4.1.

Proof of Theorem 8.4.2. Noting that in the proof of Theorem 8.4.1, we only compare the values of the solutions along the same radial direction, one can easily adapt the argument in the proof of Theorem 8.4.1 to derive the conclusion of Theorem 8.4.2.

8 Methods of Moving Planes and Moving Spheres

266

8.5 Method of Moving Planes in Integral Forms Let R" be the n-dimensional Euclidean space, and let a be a real number satisfying 0 < a < it. Consider the integral equation u(x) =

,a(y)(n+a)l(n-a)dy.

1

R^

(8.66)

Ix-7/I"-a

It arises as an Euler-Lagrange equation for a functional under a constraint in the context of the Hardy-Littlewood-Sobolev inequalities. In [L], Lieb classified the maximizers of the functional and thus obtained the best constant in the H-L-S inequalities. He then posed as an open problem the classification

of all critical points of the functional-the solutions of the integral equation (8.66).

This integral equation is also closely related to the following family of semi-linear partial differential equations (8.67)

(-d)°1271 =

In the special case n > 3 and a = 2, it becomes -,AU = n(n+2)/(n-2).

(8.68)

As we mentioned in Subsection 8.2.2, solutions to this equation were studied by Gidas, Ni, and Nirenberg [CNN]; Caffarelli, Cidas, and Spruck [CGS]; Chen and Li [CL1]; and Li [Li]. Recently, Wei and Xu [WX] generalized this result to the solutions of (8.67), with a being any even number between 0 and n.

Apparently, for other real values of a between 0 and n, equation (8.67) is also of practical interest and importance. For instance, it arises as the EulerLagrange equation of the functional

I(u)=J nI(-A)aul2dx/(J nluluadx)-. R

R

The classification of the solutions would provide the best constant in the

inequality of the critical Sobolev embedding from H 2' (R") to L- (R"):

(

R^

IuI

adx

R^

As usual, here (-S)n/2 is defined by Fourier transform. We can show that all solutions of partial differential equation (8.67) satisfy our integral equation (8.66) and vice versa (see [CLO]). Therefore, to classify the solutions of this

8.5 Method of Moving Planes in Integral Forms

267

family of partial differential equations, we only need to work on the integral equations (8.66). In order for either the Hardy-Littlewood-Sobolev inequality or the above mentioned critical Sobolev embedding to make sense, it must be in L (RT). Under this assumption, we can show that (see [CLO]) a positive solution it is in fact bounded and therefore possesses higher regularity. Furthermore, by using the Method of Moving Planes, we can show that if a solution is locally L ^ a then it is in Ln2- (R"). Hence, in the following, we call a solution it "regular" if it is locally L.2-'.. It is interesting to notice that if this condition is violated, then a solution may not be bounded. A simple example is it which is a singular solution. We studied such solutions in [CLO1]. We will use the Method of Moving Planes to prove the following theorem:

Theorem 8.5.1 Every positive regular solution u(x) of (8.66) is radially symmetric and decreasing about some point x° and therefore assumes the form C(

t

t2+lx-x°l2

)(n-°)/2

(8.69)

with some positive constants c and t.

For more results on integral equations and systems, please see [CJLL], [CL131, [CL16], [CL17], [CL18], [CLO2], [H], [HWY], [LQ], [LZ], [MC], [MCI],

[MC2] [MZ], and the references therein.

To better illustrate the idea, we will first prove Theorem 8.5.1 under the stronger assumption that it E L^ (R"). We will then show how the idea of this proof can be extended under the weaker condition that it is only locally ] in L-.

For a given real number A, define Ea={x=(xl,...,xn)Ix1Ga},

and let xa = (2A - X1, X2,

,xn) and ua(x) = u(xa). (Again, refer to Fig.

8.1.)

Lemma 8.5.1 For any solution u(x) of (8.66), we have

n(x)-Z'a(x)=JEa

(

1

x - ym-a)

U,\(Y) ) dy, (8.70)

It is also true for v(x) = xn_au(IxI2), the Kelvin-type transform of u(x), for any x

0.

8 Methods of Moving Planes and Moving Spheres

268

Proof. Since Ix - ya1 = Ixa - yl, we have

±

1

u(x)

-yIn-aua(y)n-ddy, and

L'a ITA 1

ff u(x) )=JEa

1

yl-`"u(y)-ady+

Ea x -

Ixa-yIn_°u(y)"

dy+

1

Ea

dy. Ix-yI"_aua(y)"

This implies (8.70). The conclusion for v(x) follows similarly since it satisfies the same integral equation (8.66) for x # 0.

Proof of the Theorem 8.5.1 (Under the Stronger Assumption u E . (R")).

Lam'

Let wA(x) = ua(x) - u(x). As in the previous subsections, we will first show that, for A sufficiently negative, W), (X) > 0,

Vie E L.

(8.71)

Here, and in the following, when we write an inequality or equality for an LP function, we mean it holds almost everywhere in the given set. Then we can start moving the plane Ta = 8E',\ from near xl = -oo to the right as long as inequality (8.71) holds. Let ao = sup{A

I

wµ(x) > 0, Vie E Eµ,µ S A}.

We will show that ao < oo,

and wxp(x) - 0.

Unlike partial differential equations, here we do not have any differential

equations for wa. Instead, we will introduce a new idea - the Method of Moving Planes in integral forms. We will exploit some global properties of the integral equations.

Step 1. Start Moving the Plane from near xl = -oo. Define

Ea = {x E

W'\ (X) < o}.

(8.72)

This is the set where inequality (8.71) is violated. We show that for sufficiently negative values of A, EX must be empty. By Lemma 8.5.1, it is easy to verify

that

8.5 Method of Moving Planes in Integral Forms

U(X) - U,\(x)

JEA-

Ix

269

A- yn-° ) (uT (y) -'aa(y))dy

1 - yin-°

1

1

Ea Ix -

yhn-°

[U-(Y) - ua(y))dy

1

r

YI"_°

0a 1(y)Iu(y)

ua(y))dy

JEa ix -

Ea Ix -

1

yIn-a

ur-'(y)[u(y) - ua(y)]dy,

where OA(x) is valued between u,\(x) and u(x) by the Mean Value Theorem,

and since on Ea , u,\ (x) < u(x), we have ?P,\ (x) < u(x).

We first apply the classical Hardy-Littlewood-Sobolev inequality (its equivalent form) to the above to obtain, for any q > nIIWAII4(EA) <

CIIu.T-1 waIILn9/(n+ac)(EA

=C \J EA-

IuT-'(x)I

)

Iwa(x)l

dx

and then apply the Holder inequality to the last integral in the above. First choose

s=

n+aq n

so that 719

n+aq n+aq

Then choose

aq to ensure

-+-=1. r s 1

1

We thus arrive at n

II'WAIIL4(Ea)



`E

IIWAIIL9(Ea)

uT+'(y)dyln A

IIWAIIL=(Ea).

(8.73)

1

By the condition that u E LT+1(Rn), we can choose N sufficiently large, such that for A < -N, we have C

J ( uT+'(y)dy} lJEA

<

270

8 Methods of Moving Planes and Moving Spheres

Now (8.73) implies that Iw,\I6(Ea) = 0, and therefore Ea must be measure zero and hence empty by the continuity of wa(x). This verifies (8.71) and hence completes Step 1. Readers can see that here we have actually used Corollary 7.5.1, with 12 = Ea, a region near infinity. Step 2. We now move the plane Ta = {x I xr = Al to the right as long as (8.71) holds. Let A. be as defined before; then we must have A. < oo. This can be seen by applying a similar argument as given in Step 1 from A near +oo. Now we show that

wa jx) = 0 , V X E Eao.

(8.74)

Otherwise, we have wao(x) > 0, but wao(x) 0 0 on Eao; we show that the plane can be moved further to the right. More precisely, there exists an e

depending on n, a, and the solution n(x) itself such that w,\ (x) > 0 on Ea for all A in [A0, A. + e).

From the first inequality of (8.73), we have IIWAIIcq(Ea)

C {JE ur+i(y) dy1 T

(8.75)

We will show that condition u E LT+r(Rn) ensures that one choose e sufficiently small so that for all A in [A0, A0 + c),

(8.76)

C{fL-uT+i(y)dy{,. a

J

Now, by (8.75) and (8.76), we have IIWAIIL,iEa) = 0, and therefore Ej must be measure zero. Hence, for these values of A > A0, we have W.\ (X) > 0,

a.e. Vx E Ea.

This contradicts with the definition of A0. Therefore, (8.74) must hold. To complete the proof, we now need only prove inequality (8.76). For any small ri > 0, we can choose R sufficiently large, so that /fR^\Bx(o) nr+i (y)dy) J

< t).

(8.77)

We fix this R and then show that the measure of Ea f1BR(0) is sufficiently small for A close to A0. By Lemma 8.5.1, we have in fact wA0(x) > 0 in the interior of Eao. For any 6 > 0, let

8.5 Method of Moving Planes in Integral Forms

271

E6 = {x E EAa n BR(0) I WA. (x) > S} and Fa = (EA. n BR(O)) \ Ea.

Then obviously limoµ(Fs) = 0.

6-4

For A > A,, let DA = (EA \ EA.) n BR(0).

Then it is easy to see that

(Ea n BR(O)) c (Ea n E5) U Fa U D.

(8.78)

Apparently, the measure of DA is small for A close to A,. We show that the measure of Ea n Ea can also be sufficiently small as A close to A0. In fact, for any x E Ea n E6, we have W'\ (X) = u(xA) - u(xAo) + u(x"°) - u(x) < 0.

Hence u(xAo)

- u(xA)

> wAj(x) > J.

It follows that (Ea n E5) C Ga - {x c BR(0)I u(xAo) - u(x') > b}.

(8.79)

By the well-known Chebyshev inequality, we have

µ(Cs) < g +r

ca

lu(xA°)-u(xA)IT+rdx <

b +i+i

faa(c)

Iu(x)0)_u(xA)IT+rdx.

For each fixed 6, as A close to A,, the right hand side of the above inequality can be made as small as we wish (See exercise (ii) below). Therefore by (8.79) and (8.78), the measure of E n BR(0) can also be made sufficiently small. Combining this with (8.77), we arrive at (8.76). This completes the proof of the theorem.

Exercise 8.5.1 (i) Actually, one can show that the solution u(x) is continuous (See Chapter 3, or (CLO], or [CL13]), and hence wA(x) is continuous in both x and A. Under this continuity assumption, provide a different but simpler proof for inequality (8.76).

(ii) Prove that u(xA°)

BR(o)

- u(x?)IT+rdx-r0,

Hint: Approximate it by smooth functions.

as a-+ao.

272

8 Methods of Moving Planes and Moving Spheres

Proof of Theorem 8.5.1 (Under the Weaker Assumption that u is only Locally LT+i) Outline. Since we do not assume any integrability condition of u(x) near infinity, we are not able to carry on the Method of Moving Planes directly on u(x). To overcome this difficulty, we consider v(x), the Kelvin type transform of u(x). It is easy to verify that v(x) satisfies the same equation (8.66) but has a possible singularity at the origin, where we need to pay special attention. Since u is locally LT+r, it is easy to see that v(x) has no singularity at infinity, i.e., for any domain ,fl that is a positive distance away from the origin, vT+i(y) dy < oo.

(8.80)

12

Let A be a real number and let the moving plane be xr = A. We compare v(x) and va(x) on Ea \ {xa}, where xs is the reflection of the origin about the plane TA. The proof consists of three steps. In step 1, we show that there exists an N > 0 such that for A < -N, we have v(x) < VA (X), Vx E ZA \ {x,\}.

(8.81)

Thus we can start moving the plane continuously from A < -N to the right as long as (8.81) holds. If the plane stops at xl = a° for some a° < 0, then v(x) must be symmetric and monotone about the plane xr = a°. This implies that v(x) has no singularity at the origin and u(x) has no singularity at infinity.

In this case, we can carry on the moving planes on u(x) directly to obtain the radial symmetry and monotonicity. Otherwise, we can move the plane all

the way to xr = 0, which is shown in step 2. Since the direction of xl can be chosen arbitrarily, we deduce that v(x) must be radially symmetric and decreasing about the origin. We will show in step 3 that, in any case, u(x) can not have a singularity at infinity, and hence both u and v are in LT+r (R" ).

Step 1 and Step 2 are entirely similar to the proof under the stronger assumption: Simply replace it by v and La by LA \ {xA}. We therefore only show Step 3 here. Step 3. Finally, we show that it has the desired asymptotic behavior at

infinity, i.e., it satisfies

u(x)=0(Ix1

).

Suppose, on the contrary, that xr and x2 are any two points in R", and let x° be the midpoint of the line segment xlx2. Consider the Kelvin type transform centered at x°: 1 x-x° v(x) Ix - x°Ina u(Ix-x°I2

Then v(x) has a singularity at x°. Carrying on the arguments as in Steps 1 and 2, we conclude that v(x) must be radially symmetric about x°, and

8.5 Method of Moving Planes in Integral Forms

273

in particular, u(x') = u(x2). Since xl and x2 are any two points in R", it must be constant, which is impossible. Similarly, the continuity and higher regularity of it follows from standard theory on singular integral operators. This completes the proof of the Theorem.

A

Appendices

A.1 Notation A.1.1 Algebraic and Geometric Notation 1. A = (aid) is a an m. x n matrix with ijth entry aij. 2. det(ai5) is the determinant of the matrix (aij). 3. AT = transpose of the matrix A.

4. R" = n-dimensional real Euclidean space with a typical point x

(xi,...

X")-

5. Q usually denotes an open set in Rn 6. S2 is the closure of Si.

7. 8dl is the boundary of Si.

in R,

8. n

x'y=Exiyi, ixI=

l

4=r

1x - yI is the distance of the two points x and y in R". 10. B,(x°) = {x E R" Ix - x°I < r} is the ball of radius r centered at the point x° in R". 11. S" is the sphere of radius one centered at the origin in Rn+l, i.e., the boundary of the ball of radius one centered at the origin: 9.

B1(0)_{xER1+lI1xI <1}.

276

A Appendices

A.1.2 Notation for Functions and Derivatives 1. For a function u : ii C

we write

u(x) = u(xl,...,xn)

,

x E S?.

We say it is smooth if it is infinitely differentiable.

2. For two functions u and v, it - v means it is identically equal to v. 3. We set to define u as equaling v. 4. We usually write u2 for aT. , the first partial derivative of it with respect to xi. Similarly a2u ux.x

a3u ux.x,xk

axiax7

- axiax;axk

5. alalu(x) :_ axi'...axn"

Dau () x where a = (al,

-

,

a multi-index of order Ice] = al + a2 + ...a,.

6. For a non-negative integer k,

Dku(x) :=

{Dau(x) I jai = k),

the set of all partial derivatives of order k. If k = 1, we regard the elements of Du as being arranged in a vector

Du := (ux, , ... , ux.,) _ vu, the gradient vector. If k = 2, we regard the elements of D2u being arranged in a matrix 82u

D2u :_

ax, ax

a2.

02u

the Hessian matrix. 7. The Laplacian

n

Au

uxixi i=1

is the trace of D2u.

a'u

ax, ax,

A.1 Notation

277

8.

IDkuI :_ E IDavI2 (Ial=k

A.1.3 Function Spaces 1.

C(Q) :_ {u :.ft-iR1 I u is continuous } 2.

C(f2) :_ {u E C(S2) I it is uniformly continuous} 3.

IIkIIc(n) := .sup In(x)I El?

4.

Ck(S2) :_ {u : ,fl-rR1 it is k-times continuously differentiable } I

5.

Ck(t) :_ {u E Ck(,fl) I Dau is uniformly continuous for all Ic < k} 6.

IDaullc(n)

IIU C' (n) IaI5k

7. Assume .f2 is an open subset in R° and 0 < a < 1. If there exists a constant C such that

Iu(x)-u(y)I <- CIx - yIa

,

Vx,ye.fl,

then we say that it is Holder continuous in Q. 8. The ath Holder semi-norm of it is [u]co.a(n)

sup

(y)I }

.

l Iu(x)- yI

9. The ath Holder norm is IITIIc0.a(n)

IIuIIc(.a) + [U]ca.aln)

A Appendices

278

10. The Holder space C',"(12) consists of all functions u E Ck(t) for which the norm

Ilulick-(n) :_ E IID''niJC(n) + > [DIu]co..(b) X01
IQ1=k

is finite.

11. CO°(Q) is the set of all infinitely differentiable functions. 12. Co (12) denotes the subset of functions in Ck(12) with compact support in 12.

13. LP(Q) is the set of Lebesgue measurable functions u on (2 with 1/P

IIUIILP(n)'

(fnlu(x)IPdx)


14. L°°(,?) is the set of Lebesgue measurable functions u with IIHIIL°°(n) := ess sup Jul < oo. Q

15. Lc ,((2) denotes the set of Lebesgue measurable functions u on (2 whose Pth power is locally integrable, i.e., for any compact subset K C £2,

L 16. Wk,P(12) denotes the Sobolev space consisting of functions u with

Ilullwk,P(n) := E

(fR

\ 1/p

ID"ulPdx I

< co.

/

17. Hk,P(Q) is the completion of C°°(Q) under the norm Ilullwk,P(n). 18. For any open subset P c Rn, any non-negative integer k and any p > 1, it has been shown (See Adams [Ad]) that Hk.P(Q) = Wk'P(,(2).

19. For p = 2, we usually write Hk(D) = Hk,2(.fl).

A.2 Notation and Basic Facts from Riemannian Geometry

279

A.1.4 Notation for Estimates 1. In the process of deriving estimates, we use C to denote various constants that can be explicitly computed in terms of known quantities. The value of C may change from line to line in a given computation. 2. We write

u = O(v) as x->x , provided there exists a constant C such that 1u(x)1
3. We write

u = o(v) as x-4x°, provided

n(x)I -+0 v(x)1

as x->x°.

A.2 Notation and Basic Facts from Riemannian Geometry 1. M denotes a differentiable manifold. 2. TPM is the tangent space of M at point p. 3. TyM is the co-tangent space of M at point p. 4. A Riemannian metric g can be expressed in a matrix 911 .. 91. .

g=

.

9n1

gnn

)

I

where in local coordinates,

9j(P)=<1 Bx:)P,(!

)P>p.

5. (M, g) denotes an n-dimensional manifold M with Riemannian metric g. In local coordinates, the length element is 1/2

ds :_

I/

\

E gigdxidxj =1

J

280

A Appendices

and the volume element is

det(gjj)dxi ... dx .

dV :=

6. K(x) denotes the Gaussian curvature, and R(x) the scalar curvature of a manifold M at point x. 7. For two differentiable vector fields X and Y on M, the Lie Bracket is

[X, Y] := XY - YX. 8. D denotes an affine connection on a differentiable manifold M.

9. A vector field V(t) along a curve c(t) E M is parallel if

= 0.

10. Given a Riemannian manifold M with metric <, >, theredC exists a unique connection D on M that is compatible with the metric <, >, i.e.,

< X, Y >=(t)= constant for any pair of parallel vector fields X and Y along any smooth curve c(t). 11. In local coordinates (xi, - - , x.), this unique Riemannian connection can be expressed as

Doaa I] aa

k

k

where

are the Christoffel symbols.

12. We denote

17i=D T.a7

.

13. The summation convention. We write, for instance,

xi a

Xiaxia

axi

r.kaxk

r,.kdx" k

and so on.

14. A smooth function f (x) on M is a (0, 0)-tensor. defined by

of = vifdxi =

of axi

7f is a (1, 0)-tensor

dxi.

v2 f is a (2, 0)-tensor:

02f = o, (ax dxi) ®dxj

=(aaj -r,'ak /ll dxi®dxj. a

A.2 Notation and Basic Facts from Riemannian Geometry

281

In the Riemannian context, V2f is called the Hessian of f and denoted by Hess(f). Its ijrh component is

(02f)=; =

a2f

ak.

r"a

15. The trace of the Hessian matrix ((p2f)i;) is defined to be the LaplaceBeltrami operator A.

0=

1 a( Isl

giia)I ax;

i,;-1 axi

where gi is the determinant of (gi). 16. We abbreviate:

pupvdV9 := .1M

JM

< Vu, Vv > dV9,

where

< VU, VV >= gijl7iuvjv = 923

On av

is the scalar product associated with g for 1-forms. 17. The norm of the kth covariant derivatives of u, Ipkul is defined in a local coordinates chart by IvkuI2 _ gill ... gikik(,71a)i1..1k(Ok1l)Jl..jk

In particular, we have

to ul2 = Iv 1

2 I

=

ij

i;

ij

On On

s (ou)=(ow); = s oiwo;u = s axi ax;,

18. Ck(M) is the space of CO° functions on M such that Jr

IplulpdVg
M

19. The Sobolev space Hk,P(M) is the completion of Ck(M) with respect to the norm IIUIIHk.n(M) _

(f j=o

M

1/p

IvlulpdV9

282

A Appendices

A.3 Common Inequalities and Their Proofs 1

1

p

q

Young's Inequality. Assume 1 < p,q 0, it holds: ab <

-or

p

+

Then for any

b9

(A.1)

q

Proof. We will use the convexity of the exponential function e". Let xl < x2 be any two point in R1. Then since 1 + 1 = 1, 1n xi + 1x2 is a point between a xl and x2. By the convexity of c', we have < 1ex' + 1ez'. P

(A.2)

q

Taking xr = In ap and x2 = In b4 in (A.2), we arrive at inequality (A.1).

Cauchy-Schwarz Inequality. Ix' yI 5 IxI IyI

,

V x, y E R".

Proof. For any real number A > 0, we have 0 < Ix ± Ay12 = IxI2 ± 2Ax y + A2IyI2.

It follows that

fx y <

a IxI2 + 2IYI2.

The minimum value of the right hand side is attained at A = ll. At this value of A, we arrive at the desired inequality.

Holder's Inequality. Let ,fl be a domain in R. Assume that u E LP(Q) andvEL9(Q)with 1
q

in IuvIdx 5 IItIILP(n)IIVIILq(n).

Proof. Let f =

u

iP(n) and g = vlh4(n>' Then obviously, II!IILP(n) =1= II9IILa(n)-

It follows from Young's inequality (A.1),

f Ifgldx