Some topics in theory of 2nd order elliptic PDE

7/30/2019 Some topics in theory of 2nd order elliptic PDE

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SOME ASPECTS OF THE THEORY OF SECOND-ORDER ELLIPTIC

PARTIAL DIFFERENTIAL EQUATIONS

AGUS L. SOENJAYA1

Abstract. In this paper, we study some aspects of the theory of second-order el-

liptic PDE. In particular, we study the existence, uniqueness and regularity problem

for uniformly elliptic, second-order PDE with appropriate boundary conditions. Var-

ious methods such as the energy methods in Sobolev spaces and maximum principle

methods will be discussed. Some considerations of the related problems involving

eigenvalues and eigenfunctions are also given. Finally, we give a glimpse to some of

the theory for quasi-linear and nonlinear equations.

Contents

1. Introduction and Overview 12. Sobolev Spaces 23. Existence and Uniqueness of Weak Solutions 64. Regularity of Weak Solutions 105. Maximum Principles 136. Eigenvalues and Eigenfunctions Problems 167. Quasi-linear Equations and Schauder Approach 188. Aleksandrovs Maximum Principle 219. Conclusion 22References 22

1. Introduction and Overview

The Laplaces equation u = 0 is of fundamental importance in many fields ofscience, including electromagnetism and fluid dynamics. Indeed, in Mathematics itself,the study of potential theory and harmonic function is of great importance, and haslead to various important results in analysis and related areas.

Here, we will study the boundary-value problem (BVP):

Lu = f in Uu = 0 on U (1.1)where U is an open, bounded subset ofRn, and L is a second-order partial differentialoperator having the form

Lu := aijDiju + biDiu + cu =

ni,j=1

aij(x)uxixj +

ni=1

bi(x)uxi + c(x)u (1.2)

for given coefficient functions aij , bi, c (i, j = 1, . . . , n), and aij = aji .Moreover, we require the operator L to be (uniformly) elliptic.

1


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2 AGUS L. SOENJAYA

Definition 1.1. A partial differential operator L is (uniformly) elliptic if there existsa constant > 0 such that

n

i,j=1aij(x)ij ||

2 (1.3)

A simple example would be when aij = ij , bi = 0, c = 0, in which case L is .

The second-order elliptic PDE is therefore a generalization of Laplaces equation.To study this equation, we will consider the question of existence, uniqueness and

regularity of the solution. First, we ask whether the equation has solution (in theclassical sense). More generally, we will properly define the notion of weak solutionthat still satisfy the equation in a certain weak sense. Next, we ask whether suchsolution is in fact unique. Finally, it turns out that under some conditions, this weaksolution is in fact smooth enough to qualify as a classical solution. This will be exploredas well. We will also consider the problem related to eigenvalues and eigenfunctions ofsuch elliptic operator.

We will use various methods from measure theory, Lebesgue integration and func-

tional analysis to achieve these objectives. Unfortunately, due to the limitation of space,it is not possible to extensively review this background materials and the readers areinvited to consult [2] and [5].

2. Sobolev Spaces

To use functional analytic methods, it is necessary to study our problem abstractlyin a suitable space. Sobolev space is an appropriate space designed to do our analysis.Here, we introduce the notion of weak derivatives, define the Sobolev space and studyits properties.

Definition 2.1 (Weak Derivative). Suppose u, v L1loc(U), and is a multiindex. v

is the -th weak partial derivative of u, written Du = v, providedU

uD dx = (1)||U

v dx (2.1)

for all test functions Cc (U).

Definition 2.2 (Sobolev Space). The Sobolev space Wk,p(U) consists of all locallyintegrable functions u : U R such that for all multiindex with || k, Du existsin the weak sense and belongs to Lp(U).If u Wk,p(U), define its norm to be

uWk,p(U) = ||k U |Du|p dx1/p

if 1 p < ,||k ess supU |D

u| ifp = (2.2)

It can be checked that Wk,p(U) is a Banach space for k = 1, . . . and 1 p .Moreover, we will usually write Hk(U) = Wk,2(U), where the letter H is used since itis a Hilbert space.

It is often awkward to work directly with functions in Sobolev space since it can beill-behaved at times. Subsequently we will develop some approximation properties. Inthe following, we define U := {x U| dist(x,U) > } for U R

n open, and > 0.


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SOME ASPECTS OF THE THEORY OF SECOND-ORDER ELLIPTIC PDE 3

Definition 2.3 (Mollifier). Define a function C(Rn) by

(x) :=

Cexp

1

|x|21

if|x| < 1,

0 if |x| 1(2.3)

where the constant C is selected so that Rn (x) dx = 1.Now for each > 0, set(x) :=

1

nx

(2.4)

is called the standard mollifier. It is C, integrates to 1, and satisfies supp()

B(0, ). Now if f L1loc(U), define its mollification, f := f in U.

Mollifier is important in that it has the following properties.

Proposition 2.4. Letf be the mollification of f. Then

(1) f C(U).(2) f f a.e. as 0.(3) If f C(U), then f f uniformly on compact subsets of U.

(4) If 1 p < and f Lploc(U), then f f in Lploc(U).

Theorem 2.5. Suppose U is bounded, and suppose u Wk,p(U) for some 1 p < .Then there exist a sequence of functions {um} in C

(U) Wk,p(U) such that um uin Wk,p(U)

Proof. Define

Ui := {x U| dist(x,U) > 1/i} (i = 1, 2, . . .) (2.5)

so that U =i=1 Ui. Write Vi := Ui+3 Ui+1.

Choose any open set V0 U so that U =i=0. Let {i}

i=0 be a smooth partition of

unity subordinate to the open sets {Vi}i=0, i.e. 0 i 1, i Cc (Vi)

i=0 i = 1 on U(2.6)

Now choose any function u Wk,p(U). Then iu Wk,p(U) and supp(iu) Vi.

Next, fix > 0. Choose i > 0 small enough so that ui := i (iu) satisfies

ui iuWk,p(U)

2i+1(i = 0, 1, . . .)

supp(ui) Wi (i = 1, . . .)(2.7)

for Wi := Ui+4 Ui Vi (i = 1, . . .).Now, v :=

i=0 u

i C(U) since for each open set V U, there are at most finitelymany nonzero terms in the sum. Since u =

i=0 iu, we have for each V U,

v uWk,p(V)

i=0

ui iuWk,p(U)

Taking supremum over sets V U, we conclude v uWk,p(U) as required.

We will now discuss the problem of assigning boundary values along U to a functionu W1,p(U), assuming that U is C1. This will be useful for our analysis of BVP inthe context of Sobolev space later. The difficulty is that since U has n-dimensionalLebesgue measure zero, there is no direct meaning attached to the expression u re-stricted to U. The proof of the following is found in [3].


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4 AGUS L. SOENJAYA

Theorem 2.6 (Trace Theorem). Suppose U is bounded and U is C1. Then thereexists a bounded linear operator T : W1,p(U) Lp(U) such that

(1) T u = u|U if u W1,p(U) C(U), and

(2) T uLp(U) CuW1,p(U),

for all u W1,p(U), where C = C(p, U).

Definition 2.7. T u is said to be the trace of u on U.

Now, denote the closure of Cc (U) in Wk,p(U) by Wk,p0 , and so similarly write

Wk,20 (U) = Hk0 (U). The following gives a useful interpretation of this.

Theorem 2.8. Suppose U is bounded and U is C1 and u W1,p(U). Then u

W1,p0 (U) if and only if T u = 0 on U.

Estimates are very important in the theory of PDE. Next, we will introduce animportant inequality, called the Gagliardo-Nirenberg-Sobolev (GNS) inequality.

Definition 2.9. Suppose 1 p < n. The Sobolev conjugate ofp, denoted by p is such

that1

p=

1

p

1

n, p > p (2.8)

Theorem 2.10 (Gagliardo-Nirenberg Sobolev (GNS) Inequality). Suppose1 p < n.Then there exists a constant C = C(p, n) such that

uLp(Rn) CDuLp(Rn) (2.9)

for all u C1c (Rn).

Proof. First assume p = 1.Since u has compact support, for each i = 1, . . . , n, and x Rn, we have

u(x) = xi

uxi(x1, . . . , xi1, y1, xi+1, . . . , xn) dyi,

and so

|u(x)|

R

|Du(x1, . . . , yi, . . . , xn)| dyi (i = 1, . . . , n)

Consequently,

|u(x)|n

n1 ni=1

R

|Du(x1, . . . , yi, . . . , xn)| dyi

1n1

(2.10)

Integrating with respect to x1,

R

|u|n

n1 dx1 R

ni=1

R

|Du| dyi 1n1

dx1 (2.11)

=

R

|Du| dy1

1n1

R

ni=2

R

|Du| dyi

1n1

dx1

R

|Du| dy1

1n1

ni=2

R

R

|Du| dx1 dy1

1n1

,


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the last inequality follows from the general Holder inequality.Now integrate (2.11) with respect to x2,

R

R

|u|n

n1 dx1 dx2

R

R

|Du| dx1 dy2

1n1

R

ni=1,i=2

I1

n1

i dx2, (2.12)

with

I1 :=

R

|Du| dy1, Ii :=

R

R

|Du| dxidyi (i = 3, . . . , n)

Continue applying general Holder inequality and integrating with respect to x3, . . . , xnto find

Rn

|u|n

n1 dx ni=1

R

. . .

R

|Du| dx1 . . . d yi . . . d xn

1n1

=

Rn

|Du| dx

nn1

(2.13)

which is the required inequality for p = 1.Now consider the case when 1 < p < n. Applying estimate (2.13) to v := |u|, where

> 1 to be selected,Rn

|u|nn1 dx

n1n

Rn

|D|u|| dx =

Rn

|u|1|Du| dx (2.14)

Rn

|u|(1) p

p1 dx

p1p

Rn

|Du|p dx

1p

Now set = p(n1)np > 1 to get the required inequality.

The following theorem concerns compact embedding of the Sobolev space W1,p(U) inLq(U) for some appropriate value of q. This compact embedding property is essentialin our application of functional analytic methods to PDE theory later. The proof of

this result makes use of some further result in Sobolev space theory, GNS inequalityand Arzela-Ascoli theorem. Interested readers are referred to [3].

Theorem 2.11 (Rellich-Kondrachov Compactness Theorem). SupposeU is a boundedopen subset ofRn, and U is C1. Suppose 1 p < n. Then W1,p(U) Lq(U) foreach 1 q < p.

Another important inequality we will use later is the Poincares Inequality.

Theorem 2.12 (Poincares Inequality). Let U be a bounded, connected, open subsetof Rn, with a C1 boundary U. Assume 1 p . Then there exists a constantC = C(n,p,U) such that

u (u)ULp(U) CDuLp(U) (2.15)

for each function u W1,p(U), where

(u)U =1

|U|

U

u dy

where |U| is the Lebesgue measure of U.

Proof. We will argue by contradiction. Suppose the statement is false, then there existsa function uk W

1,p(U) for each integer k = 1, . . . satisfying

uk (uk)ULp(U) > kDukLp(U) (2.16)


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6 AGUS L. SOENJAYA

Normalize this by defining

vk :=uk (uk)U

uk (uk)ULp(U)(k = 1, . . .) (2.17)

Then (vk)U = 0 and vkLp(U) = 1, and (2.16) implies

DvkLp(U) 0 such that:

(1) |B[u, v]| uv for u, v H, and(2) u2 B[u, u] for u H.

Letf be a bounded linear functional on H. Then there exists a unique element u Hsuch that B[u, v] = f(v) for all v H.


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We will now make some estimates for the bilinear form introduced in Definition 3.10.

Proposition 3.4. For the bilinear formB[, ] introduced in Definition3.10, there existsconstants, > 0 and 0 such that:

(1) |B[u, v]| uH10

(U)vH10

(U), and

(2) u2

H10 (U) B[u, u] + u2

L2(U)

for all u, v H10 (U).

Proof. We have

|B[u, v]| ni,j=1

aijL

U

|Du||Dv| dx +ni=1

biL

U

|Du||v| dx + cL

U

|u||v| dx

uH10

(U)vH10

(U)

for some appropriate constant .Furthermore, by the ellipticity condition,

U |Du|2 dx Un

i,j=1

aijuxi

uxj

dx

= B[u, u]

U

ni=1

biuxiu + cu2 dx

B[u, u] +ni=1

biL

U

|Du||u| dx + cL

U

u2 dx

Using Cauchys inequality (with ), observe thatU

|Du||u| dx

U

|Du|2 dx +1

4

U

u2 dx (where > 0)

Choosing > 0 so small that ni=1 biL < 2 , and inserting into above, we have

2

U

|Du|2 dx B[u, u] + C

U

u2 dx

for some appropriate constant C. Then using Poincares inequality, it easily follows thatu2

H10

(U) B[u, u] + u2L2(U) for some appropriate > 0, 0 as required.

Note that when > 0, then B[, ] above does not satisfy the hypothesis of Lax-Milgram Theorem. We address this below and give our first existence theorem.

Theorem 3.5 (First Existence Theorem). There is a number 0 such that for each and each function f L2(U), there exists a unique weak solution u H10 (U) ofthe boundary-value problem

Lu + u = f inUu = 0 onU (3.2)Proof. Take from Proposition 3.10. Let , and define the bilinear form

B[u, v] := B[u, v] + u, v (u, v H10 (U)) (3.3)

which corresponds to the operator Lu := Lu+u. Then B[, ] satisfies the hypothesesof Lax-Milgram Theorem.Now fix f L2(U) and set f(v) = f, v, which is a bounded linear functional onL2(U), and thus on H10 (U). By Lax-Milgram Theorem, there exists a unique function


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8 AGUS L. SOENJAYA

u H10 (U) satisfying B[u, v] = f, v for all v H10 (U). Hence, u is the unique weak

solution of (3.5).

To have further existence theorem, the Fredholm theory of compact operators willbe useful.

Definition 3.6. (1) The operator L

, the formal adjoint of L, defined by

Lv := ni,j=1

(aijvxj)xi ni=1

bivxi +

c

ni=1

bixi

v, (3.4)

provided bi C1(U) (i = 1, . . . , n).(2) The adjoint bilinear form B : H H R is defined by

B[v, u] = B[u, v] (3.5)

for all u, v H10 (U).(3) We say that v H10 (U) is a weak solution of the adjoint problem

Lv = f in U

v = 0 on U

(3.6)

provided B[v, u] = f, u for all u H10 (U).

This leads us to the second existence theorem for weak solutions as explained below.

Theorem 3.7 (Second Existence Theorem). The following statements hold:

(1) Precisely one of the following statements holds:(a) either for each f L2(U) there exists a unique weak solution u of the BVP

Lu = f inUu = 0 onU

(3.7)

(b) or else there exists a weak solution u 0 of the homogeneous BVP

Lu = 0 inUu = 0 onU (3.8)(2) Furthermore, should assertion (1)(b) above hold, the dimension of the subspace

N H10 (U) of weak solutions of (3.8) is finite and equals the dimension of thesubspace N H10 (U) of weak solutions of

Lv = 0 inUv = 0 onU

(3.9)

(3) Finally, the BVP (3.7) has a weak solution if and only if f, v = 0 for allv N.

Proof. Choose = as in Theorem 3.5 and define the bilinear form

B[u, v] := B[u, v] + u, v (3.10)corresponding to the operator Lu := Lu + u. Then for each g L

2(U), there existsa unique solution u H10 (U) solving

B[u, v] = g, v (3.11)

for all v H10 (U). Write u = L1 g whenever this holds.

Observe next u H10 (U) is a weak solution of (3.7) if and only if

B[u, v] = u + f, v (3.12)


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for all v H10 (U), i.e. if and only if u = L1 (u + f).

Rewrite this equality to read u Ku = h for

Ku := L1 u (3.13)

and h := L1 f.

Now observe that K : L2

(U) L2

(U) is a bounded, linear, compact operator. Fromour choice of , note that if (3.18) holds, then

u2H10

(U) B[u, u] = g, u gL2(U)uL2(U) gL2(U)uH10 (U)

so that (3.13) implies KgH10

(U) CgL2(U) for g L2(U) for some constant C.

By Rellich-Kondrachov compactness theorem, H10 (U) L2(U). Hence K is a com-

pact operator. By Fredholm alternative in functional analysis,

(1) either for each h L2(U), the equation u Ku = h has a unique solutionu L2(U),

(2) or else the equation u Ku = 0 has nonzero solutions in L2(U).

Should assertion (1) hold, by arguments in (3.10)-(3.13), there exists a unique weak

solution of problem (3.7).On the other hand, should assertion (2) hold, then = 0, and the dimension of thespace N of the solutions of (2) is finite and equals the dimension of the space N ofsolutions of

v Kv = 0 (3.14)

Now (2) holds if and only if u is a weak solution of (3.7), and (3.14) holds if and onlyif v is a weak solution of (3.8).Note that (1) has a solution if and only if h, v = 0 for all v solving (3.14). Howeverfrom (3.13) and (3.14), we calculate

h, v =1

K f ,v =

1

f, Kv =

1

f, v

so BVP (3.7) has a solution if and only if f, v = 0 for all weak solutions v of (3.9).

Our third existence theorem concerns the spectrum of the operator L.

Theorem 3.8 (Third Existence Theorem). The following statements hold.

(1) There exists an at most countable set R such that the BVPLu = u + f inUu = 0 onU

(3.15)

has a unique weak solution for each f L2(U) if and only if .(2) If is infinite, then = {k}

k=1, a nondecreasing sequence with k +.

We call the (real) spectrum of the operator L.

Proof. Let be the constant in 3.10 and assume that > , where WLOG > 0.According to the Fredholm alternative, the BVP (3.15) has a unique weak solution foreach f L2(U) if and only if u 0 is the unique weak solution of the BVP

Lu = u in Uu = 0 on U

(3.16)

which is in turn true if and only if u 0 is the only weak solution ofLu + u = ( + )u in Uu = 0 on U

(3.17)


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10 AGUS L. SOENJAYA

BVP (3.17) holds exactly when

u = L1 ( + )u = +

Ku (3.18)

where we set Ku = L1 u. We have also shown that K : L2(U) L2(U) is bounded,

linear, compact operator.Now if u 0 is the only solution of (3.18), we have

+ is not an eigenvalue of K (3.19)

so BVP (3.15) has a unique weak solution for all f L2(U) if and only if (3.19) holds.Now the collection of eigenvalues of K comprises either a finite set or the values of asequence converging to zero. In the second case, we see that by (3.18), the BVP (3.15)has a unique weak solution for each f L2(U), except for a sequence k +.

4. Regularity of Weak Solutions

We will now study whether a weak solution u of the BVP (1.1) is in fact smooth(qualify as a classical solution). This is the regularity problem for weak solutions.The regularity problem is often difficult as we need to do hard estimates of a certainquantity. Here we will study some of the regularity theorems.

Theorem 4.1 (Interior H2-regularity). Suppose aij C1(U), bi, c L(U) for i, j =1, . . . , n, and f L2(U). Suppose further that u H1(U) is a weak solution of theelliptic PDE Lu = f in U. Then u H2loc(U), and for each open V U we have theestimate

uH2(V) C(fL2(U) + uL2(U)) (4.1)

where C = C(L,U,V).

Proof. Fix any open V U, and choose an open set W such that V W U.Then define a smooth function (called a cutoff function) such that

1 on V, 0 on Rn W,0 1.

Since u is a weak solution, B[u, v] = f, v for all v H10 (U). Consequently,

ni,j=1

U

aijuxivxj dx

A=

U

fv dx

B, (4.2)

where

f := f ni=1

biuxi cu. (4.3)

Now let |h| > 0 be small and choose k {1, . . . , n}, and then substitute

v := Dhk (2Dhku), where D

hku(x) :=

u(x + hek) u(x)

h, (h R, h = 0) (4.4)


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into (4.2). Now we will estimate the quantity A.

A = ni,j=1

U

aijuxi

Dhk (

2Dhku)xj

dx

=

ni,j=1

U

Dhk(aijuxi)(2Dhku)xj dx

=ni,j=1

U

aij,h(Dhkuxi)(2Dhku)xj + (D

hkaij)uxi(

2Dhku)xj dx

=ni,j=1

U

aij,hDhkuxiDhkuxj

2 dx

A1

+n

i,j=1U

aij,hDhkuxiDhku (2xj ) + (D

hkaij)uxiD

hkuxj

2 + (Dhkaij)uxiD

hku (2xj )

dx

A2

By the uniform ellipticity,

A1

U

2|DhkDu|2 dx (4.5)

Furthermore, we have

|A2| C

U

|DhkDu||Dhku| + |D

hkDu||Du| + |D

hku||Du| dx,

for some constant C. By Cauchys inequality with ,

|A2| U 2|DhkDu|2 dx + C W |Dhku|2 + |Du|2 dxChoosing = /2, and using the estimate

W|Dhku|

2 dx C

U

|Du|2 dx, (4.6)

hence we have

|A2|

2

U

2|DhkDu|2 dx + C

U

|Du|2 dx (4.7)

This estimate and our earlier estimate for A gives

A

2 U 2|DhkDu|2 dx CU |Du|2 dx (4.8)Now we will estimate B. We have

|B| C

U

(|f| + |Du| + |u|)|v| dx (4.9)

Similarly, using estimate (4.6), we haveU

|v|2 dx C

U

|Du|2 + 2|DhkDu|2 dx (4.10)


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12 AGUS L. SOENJAYA

Using Cauchys inequality with = /4, we get

|B|

4

U

2|DhkDu|2 dx + C

U

f2 + u2 + |Du|2 dx (4.11)

Finally, we combine estimate (4.8) and (4.11) to get

V

|DhkDu|2 dx

U

2|DhkDu|2 dx C

U

f2 + u2 + |Du|2 dx (4.12)

for k = 1, . . . , n and all sufficiently small |h| = 0.We have Du H1loc(U), and thus u H

2loc(U), with the estimate

uH2(V) C(fL2(U) + uH1(U)) (4.13)

Refining the above estimate by noting that V W U, then the same argumentshows

uH2(V) C(fL2(W) + uH1(W)) (4.14)

for some constant C depending on V, W, etc. Choose a new cutoff function satisfying

1 on W, supp U,0 1.Now set v = 2u in (4.4) and by some calculations, we get

U2|Du|2 dx C

U

f2 + u2 dx

so that

uH1(W) C(fL2(U) + uL2(U))

This inequality and (4.14) yield the desired result.

By induction, we have the following higher interior regularity theorem.

Theorem 4.2. Letm be a nonnegative integer, and supposeaij , bi, c Cm+1(U), wherei, j = 1, . . . , n and f Hm(U). Suppose u H1(U) is a weak solution of the ellipticPDE Lu = f in U. Then u Hm+2loc (U), and for each V U we have the estimate

uHm+2(V) C(fHm(U) + uL2(U)), (4.15)

where C = C(m,L,U,V).

Applying the above theorem continuously, we have the following theorem.

Theorem 4.3. Suppose aij, bi, c C(U), where i, j = 1, . . . , n, and f C(U).Suppose u H1(U) is a weak solution of the elliptic PDE Lu = f in U. Then u C(U).

We can extend the above estimates to study the smoothness of the solution up tothe boundary. The estimates are generally more intricate. We present some of thetheorems below, whose proof can be found in [3].

Theorem 4.4 (Higher Boundary Regularity). Let m be a nonnegative integer, andassume aij, bi, c Cm+1(U), where i, j = 1, . . . , n, and f Hm(U). Suppose thatu H10 (U) is a weak solution of the BVP

Lu = f inUu = 0 onU

(4.16)


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Assume finally U Cm+2. Then u Hm+2(U), and we have the estimate

uHm+2(U) C(fHm(U) + uL2(U)), (4.17)

where C = C(m,U,L).

The analogous theorem on infinite differentiability follows similarly.

5. Maximum Principles

Maximum principle methods are based on the observation that if a C2-functionu attains its maximum over an open set U at a point x0 U, then Du(x0) = 0 andD2u(x0) 0. Deductions based on this observation are therefore pointwise in character,and thus distinct from the methods outlined previously.

We begin with a statement of maximum principles, which are generalizations of thecorresponding statements in the case of Laplaces equation.

Theorem 5.1 (Weak Maximum Principle). Suppose u C2(U) C(U) and c 0 inU.

(1) If Lu 0 in U, then maxUu = maxU u.(2) If Lu 0 in U, then minUu = minU u.

Proof. First, let us suppose that we have strict inequality

Lu < 0 in U (5.1)

but there exists a point x0 U such that

u(x0) = maxU

u (5.2)

At this maximum point, we have Du(x0) = 0 and D2u(x0) 0.

Since the matrix A = (aij(x0)) is symmetric and positive definite, there exists anorthogonal matrix O = (oij) such that

OAOT = diag(d1, . . . , dn), OOT = I (5.3)

with dk > 0 for k = 1, . . . , n. Write y = x0 + O(x x0). Then x x0 = OT(y x0),

and so

uxi =nk=1

uykoik, and uxixj =n

k,l=1

uykyloikojl (i, j = 1, . . . , n)

Hence, at point x0,

n

i,j=1 a

ij

uxixj =

n

k,l=1

n

i,j=1 a

ij

uykyloikojl =

n

k=1 d

kuykyk 0 (5.4)

since dk > 0 and uykyk(x0) 0 (k = 1, . . . , n). Therefore at x0, we have

Lu = ni,j=1

aijuxixj +ni=1

biuxi 0

using (5.4). Hence (5.1) and (5.2) are incompatible, a contradiction.For the general case, write u(x) := u(x) + ex1 where x U, > 0 will be chosen


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14 AGUS L. SOENJAYA

below and > 0. The ellipticity condition implies aii(x) (i = 1, . . . , n, x U).Therefore

Lu = Lu + L(ex1)

ex1(2a11 + b1)

ex1(2 + bL) < 0

in U, by choosing sufficiently large. By the previous arguments, maxUu = maxU u

.Letting 0, we have the result.The next result also follows by replacing u with u.

We need the following technical lemma to prove the strong maximum principle.

Lemma 5.2 (Hopfs Lemma). Suppose u C2(U) C1(U) and c 0 in U. Supposefurther that Lu 0 in U and there exists a point x0 U such that

u(x0) > u(x) (5.5)

for all x U.

Assume finally that U satisfies the interior ball condition at x0, i.e. there exists anopen ball B U with x0 B . Then u(x

0) > 0, where is the outward unit normal

to B at x0.If c 0 in U, then the above conclusion holds also provided u(x0) 0.

Proof. Assume c 0 and u(x0) 0. WLOG assume B = B(0, r) for some radiusr > 0. Define

v(x) := e|x|2

er2

for > 0 as selected below. Using the ellipticity condition, we compute

Lv = e|x|2

n

i,j=1 aij(42xixj + 2

ij) e|x|2

n

i=1 2bixi + c(e

|x|2 er2

)

e|x|2

(42|x|2 + 2tr A + 2|b||x| + c)

for A = ((aij)) and b = (b1, . . . , bn).Next, consider the open annular region R := B(0, r) B[0, r/2]. We have

Lv e|x|2

(2r2 + 2tr A + 2|b|r + c) 0 (5.6)

in R, provided > 0 is sufficiently large.In view of (5.5), there exists a constant > 0 so small that

u(x0) u(x) + v(x) (x B [0, r/2]) (5.7)

Also, note that

u(x0) u(x) + v(x) (x B [0, r]) (5.8)

since v 0 on B [0, r].From (5.6), we observe that

L(u + v u(x0)) cu(x0) 0 in R,

and from (5.7),(5.8), we see that

u + v u(x0) 0 on R


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By weak maximum principle, u + v u(x0) 0 in R. But u(x0) + v(x0) u(x0) 0and so

u

(x0) +

u

(x0) 0

Consequently,

u

(x0)

u

(x0) =

rDv(x0) x0 = 2rer

2

> 0

as required.

Now we can prove the strong maximum principle.

Theorem 5.3 (Strong Maximum Principle). Suppose u C2(U) C(U) and c 0 inU. Suppose also U is connected, open, and bounded.

(1) If Lu 0 in U and u attains its maximum over U at an interior point, then uis constant within U.

(2) If Lu 0 in U and u attains its minimum over U at an interior point, then uis constant within U.

Proof. Write M := maxUu and C := {x U| u(x) = M}.If u M, set V := {x U | u(x) < m}.Choose a point y V satisfying dist(y, C) < dist(y,U), and let B denote the largestball with centre y whose interior lies in V. Then there exists some point x0 Cwith x0 B . V satisfies the interior ball condition at x0, and so by Hopfs Lemma,u(x

0) > 0. This is a contradiction since u attains its maximum at x0 U.

If the constant term c is nonnegative, we also have such maximum principle, theproof of which is similar.

Theorem 5.4. Supposeu C2(U)C(U) andc 0 inU. Suppose also U is connected,open, and bounded.

(1) If Lu 0 in U and u attains a nonnegative maximum over U at an interiorpoint, then u is constant within U.

(2) If Lu 0 in U and u attains a nonpositive minimum over U at an interiorpoint, then u is constant within U.

The following inequality relates the supremum and infimum of a solution of suchPDE. The proof can be found in [4].

Theorem 5.5 (Harnacks Inequality). Suppose u 0 is a C2 solution of Lu = 0 inU, and suppose V U is connected. Then there exists a constantC = C(V, L) suchthat

supV

u c infV

u

The maximum principles could also provide a simple pointwise estimate for solutionsof the equation Lu = f in bounded domains. This is particularly useful when nonlinearproblems are considered. In the following, let u+ = max{u, 0} and u = min{u, 0}.

Theorem 5.6. Let Lu f in a bounded domain U, c 0, and u C2(U) C(U).Then

supU

u supU

u+ + CsupU

|f|

(5.9)


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16 AGUS L. SOENJAYA

where is the smallest eigenvalue of L, andC is a constant depending only on diamUand = sup(|b|/).If Lu = f instead, then (5.9) becomes

supU

|u| supU

|u| + CsupU

|f|

(5.10)In particular, if U lies between two parallel planes d distance apart, then (5.9) and(5.10) is satisfied with C = exp[(+ 1)d] 1.

Proof. Let U lies in the slab 0 < x1 < d, and set an operator L0 := aijDij + b

iDi. For + 1, we have

L0ex1 = (a11 + b1)ex1 (2 )ex1

We let

v := supU

u+ + (ed ex1)supU

|f|

Then, since Lv = L0v + cv supU (|f|/), we have

L(v u)

supU

|f|

+

f

0 in U

and v u 0 on U. Hence, for C = ed 1 and + 1, we have the desired resultfor the case Lu f, i.e.

supU

u supU

v supU

u+ + sup

|f|

Replacing u with u, we obtain the last result.

6. Eigenvalues and Eigenfunctions Problems

In this section, we will consider BVP having the formLw = w in Uw = 0 on U

(6.1)

where U is an open, bounded region. We say that is an eigenvalue of L providedthere exists a nontrivial solution w of above BVP. We recall from Theorem 3.8 that theset of eigenvalues of L is at most countable.

Here, we have the following theorems as generalizations of the corresponding resultsin linear algebra, namely that a real symmetric matrix has real eigenvalues and anorthonormal basis of eigenvectors, and the Perron-Frobenius Theorem. We need thefollowing results from functional analysis.

Proposition 6.1 (Bounds on Spectrum). LetS : H H be bounded, symmetric, anddefine

m := inf uH,u=1

Su,u, M := supuH,u=1

Su,u

Let(S) be the spectrum of S. Then (S) [m, M] and m, M (S).

Theorem 6.2. Let H be a separable Hilbert space, and suppose S : H H is acompact and symmetric operator. Then there exists a countable orthonormal basis ofH consisting of eigenvectors of S.


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Now, for simplicity, we consider an elliptic operator having the divergence form

Lu = ni,j=1

aijuxi

xj

, (6.2)

where aij C(U), (i, j = 1, . . . , n). We suppose the usual uniform ellipticity condi-tion and aij = aji . The operator L is thus formally symmetric, and the correspondingbilinear form satisfies B[u, v] = B[v, u], (u, v H10 (U)).

Theorem 6.3 (Eigenvalues of Symmetric Elliptic Operators). The following statementshold:

(1) Each eigenvalue of L is real.(2) Furthermore, if we repeat each eigenvalue according to its (finite) multiplicity,

we have = {k}k=1, where 0 < 1 2 . . ., and k as k .

(3) There exists an orthonormal basis {wk}k=1 of L

2(U), where wk H10 (U) is an

eigenfunction corresponding to k, i.e.

Lwk = kwk inUwk = 0 onU (6.3)for k = 1, 2, . . ..

Proof. As in Theorem 3.7, S := L1 is a bounded, linear, compact operator from L2(U)to itself. We claim S is symmetric. Let f, g L2(U). Then Sf = u means u H10 (U)is the weak solution of

Lu = f in Uu = 0 on U

and likewise for the meaning of Sg = v.Thus

Sf,g = u, g = B[v, u]

and

f,Sg = f, v = B[u, v]

Since B[u, v] = B[v, u], we have Sf,g = f,Sg for all f, g L2(U), so that S issymmetric.Also, note that Sf,f = u, f = B[u, u] 0. Consequently, by Proposition 6.1 andTheorem 6.2, all the eigenvalues of S are real, positive, and there exist correspondingeigenfunctions which make up orthonormal basis of L2(U). Observe as well that for = 0, we have Sw = w if and only ifLw = w for = 1 , and the theorem follows.

We can in fact gain more information on the first eigenvalue of L. The proof of the

following is found in [3].Definition 6.4. Let 1 > 0 be the first (smallest) eigenvalue ofL. Call 1 the principaleigenvalue of L.

Theorem 6.5 (Variational Principle for the Principal Eigenvalue). The following state-ments hold:

(1) We have

1 = min{B[u, u] | u H10 (U), uL2 = 1} (6.4)


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18 AGUS L. SOENJAYA

This is the Rayleighs Formula, and is equivalent to the statement

1 = minuH1

0(U),u0

B[u, u]

u2L2(U)

(2) The above minimum is attained for a function w1, positive within U, which

solves Lw1 = 1w1 inUw1 = 0 onU

(3) If u H10 (U) is any weak solution ofLu = 1u inUu = 0 onU

then u is a multiple of w1.

7. Quasi-linear Equations and Schauder Approach

In the following, we will move on from the case of linear equations, and have a glimpse

on the theory for quasi-linear equations. In particular, we will study some results onthe existence of the solutions of a quasi-linear equation following Schauders theory.We will be working with Holder spaces and their variants, as defined below.

Definition 7.1. A domain Rn is said to have boundary of class Ck, for a nonneg-ative integer k and [0, 1] if for each point x0 , there exists a ball B := B(x0)and an injective mapping of B onto a domain D Rn such that

(B ) Rn+ := {x Rn : xn > 0};

(B ) Rn+;

Ck,(B), 1 Ck,(D).

Furthermore, a domain is said to have a boundary portion T of class Ck, iffor each x0 T, there is a ball B := B(x0) in which the above conditions are satisfied,

and such that B T.

Definition 7.2 (Global Holder Space). Let be a bounded domain in Rn, k be anonnegative integer, and (0, 1]. The global Holder space Ck,a() is the set of allfunctions in Ck() for which the following quantity

[Dku]; := supx,y, ||=k

|Du(x) Du(y)|

|x y|

is finite. In this case, we say that the k-th derivatives of u are Holder continuous on with exponent . We define a norm on Ck,() by

uk,; = uk; + [Dku];

where

uk; =kj=0

sup

|Dju|

Definition 7.3 (Local Holder Space). The local Holder space Ck,() is the set of allfunctions in Ck() whose derivatives ofk-th order are Holder continuous with exponent

on any compact subset of . We denote by Ck,0 () the set of functions in Ck,()

having compact support in .


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Remark 7.4. The space Ck,() is a nonreflexive, nonseparable Banach space. The in-

clusion Ck,() Ck,() whenever k+ > k+ holds for domains with sufficiently

smooth boundary, but does not hold in general.

Remark 7.5. It is also often convenient to introduce non-dimensional norms on the

above spaces in the following way. Let d := diam . Define

uk; :=kj=0

dj sup

|Dju|

and

uk,; := uk; + d

k+[Dku];

Subsequently, assume is a bounded domain in Rn with C2, for some (0, 1), and L is an elliptic partial differential operator, i.e. Lu = aijDiju + b

iDiu + cusatisfying

aij

(x)ij ||2

for all x , Rn

where > 0. Assume also that for a further positive constant we have

aij0,;, bi0,;, c0,;

We have the following global estimate, whose proof can be found in [4].

Theorem 7.6 (Schauders Estimates). The following holds:

(1) If c 0, then for any f C0,() and any C2,(), the Dirichlet problemLu = f inu = on

has a unique solution u C2,().(2) If u C2() C0() is any solution of the above Dirichlet problem, but not

necessarily with c 0, then u C2,(), and

u2,; C(u0; + 2,; + f0,;)

where C := C(n,, , , ). If, in addition, , Ck+2,() and f Ck+2() and

aijk,; bik,; ck,; ,

then u Ck+2,() and

uk+2,; C(u0; + k+2,; + fk,;)

where C := C(n,k,, , , ).

Remark 7.7. If c 0, then u0; can be estimated in terms of f and . Indeed, wehave

sup

|u| sup

|| + Csup

|f|/

where C := C(n, diam, sup |b|/) (see [4] for details).


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20 AGUS L. SOENJAYA

Next, we will look at how the question of existence of solutions to a quasi-linearequation can be reduced to that of the linear case by using the previous Schauderstheory.

Subsequently, assume that aij , b C0,( R Rn), where [aij ] is symmetric andpositive definite, and Rn is a bounded domain with C2,. Assume also

(0, 1), and C2,

(). We want to solve the following Dirichlet problemaij(x,u,Du)Diju + b(x,u,Du) = 0 in u = on

(7.1)

We are able to use the linear theory outlined previously to solve this problem. First fix (0, 1). Then for any v C1,(), consider the associated linear problem

aij(x,v,Dv)Diju + b(x,v,Dv) = 0 in u = on

(7.2)

It can be checked that since aij , b C0,( RRn) and v C1,(), the coefficientsin (7.2) belong to C0,( R Rn). Furthermore, by positive-definitiness of [aij],there exists positive constants and , depending on v, such that

||2 aij(x,v,Dv)ij ||2 for all x , Rn

Therefore by Schauders theory, there exists a unique solution u C2,() of equation(7.2). Denote u := T v.

Then we have constructed a continuous mapping T : C1,() C2,(). Further-more, since the embedding C2,() C1,() is compact by Arzela-Ascoli theorem,T can be regarded as a continuous compact mapping from C1,() into itself. Anyfixed point of T is clearly a solution of (7.1). Now note that since the range of T is asubset ofC2,(), any fixed point u ofT belongs to C2,() automatically. However,by the same argument as above, now with = 1, we see that the coefficients of (7.1)belong to C0,(). Therefore by Schauder theory again, we have u C2,(). Hence,

the solvability of (7.1) in u C2,

() reduces to proving that T has a fixed point inC1,().It is therefore appropriate at this stage to note the following useful fixed point the-

orems for our case, beside the well-known Brouwers fixed point theorem.

Theorem 7.8 (Lerray-Schauder). Let T be a continuous compact map of a Banachspace X to itself, and suppose that there is a constant M such thatxX < M wheneverx = T x for some [0, 1]. Then T has a fixed point.

For proof of the above fixed point theorems, see [4]. By the argument preceding theabove theorem, we then immediately have the following existence result.

Theorem 7.9. Suppose there exist constants (0, 1) andM > 0 such thatv1,;

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Some topics in theory of 2nd order elliptic PDE