HOMOGENIZATION OF BIHARMONIC EQUATIONS IN DOMAINS

PERFORATED WITH TINY HOLES

CHAOCHENG HUANG

DEPARTMENT OF MATHEMATICS AND STATISTICS

WRIGHT STATE UNIVERSITY

DAYTON OHIO USA

Abstract This paper deals with the homogenization of the biharmonic equation

u f in

a domain containing randomly distributed tiny holes with the Dirichlet boundary conditions The

size of the holes is assumed to be much smaller compared to the average distance between any

two adjacent holes We prove that as the solutions the biharmonic equation converges to

the solution of

u u f where depends on the shape of the holes and relative order of

with respect to

Introduction Let be a bounded domain in R

N

For any

denotes

a perforated domain formed by randomly removing many tiny holes depending on

from Consider the biharmonic equation

u

f in

with the Dirichlet boundary conditions

u

u

n

in

u

g

u

n

h in

We are concerned with the limiting behavior of the solution u

of problem

as Many authors studied homogenization of the biharmonic equations see

and the references cited there All the previous results were established for the

case that the holes are distributed periodically and that the size of the holes is of the

same order as that of the period It was shown cf that as the solution

u

converges to to order

the higher order asymptotic expansion was also studied

in

In this paper we are interested in the case when the holes are distributed ran

domly and when the volume fraction of the holes approaches zero as the number

increases to innity Our objective is to determine the order of the volume fraction of

the holes in order to obtain a nonzero limit of the solution u

of problem

as the number of the holes approaches Let be the average distance between

two adjacent holes the precise denition will be stated in the next section de

pending on the size of the holes We shall consider the case that

as

The main results are the follows

i If N then the solution u

of always converges to no matter

how small the volume fraction of the holes is ii if N and the limit

exists and is nite where

N

N

for N

ln

for N

then the solution u

converges to the solution of

u u f in

u g

u

n

g

n

in

where depends on the shape and distribution of the holes iii if N and

g then the limit of the solution u

is

The results show that the critical order of the size of holes is

N

N

for N

and exp

for N

For the analogous problems of the second order equations with periodically dis

tributed holes the case when the size of the holes is much smaller than the period has

been studied extensively in literatures In and the authors considered the

problem for the Laplace equation The homogenization for the wave equation in that

case was studied in The problem for the NavierStokes equation was analyzed in

These papers showed that the critical order for the second order equations is

N

N

for N and exp

for N For homogenization of other

equations and general references we refer to

The proof of our results is based on the energy method The main idea follows

those in and Since the dierent natures between the fourth order equations

and the second order equations the treatment here technically is dierent from the

papers mentioned above In contrast with and where certain periodic struc

tures are assumed we consider the more general situation that holes are randomly

distributed with a density function Our paper is also the rst approach to explore the

relationship between the orders of the equations and the critical orders of the holes

that will produce nontrivial homogenized equations The paper is organized as fol

lows In x we shall state the problem more precisely and introduce some notations

The case when N is also studied in that section In x we study the problem in a

ball containing one hole and derive some estimates Section is devoted to the study

of the homogenized equations and the results of convergence under various norms for

N The case N is treated in x

Problem Setting and Preliminaries Let and T be two bounded C

domains in R

N

locally lying in one side of their boundaries We assume that

A B

T B

where B

r

B

r

o the ball centered at origin with radius r For any and

a set of nite points fx

i

i m

g such that

A

x

i

x

j

for i j

We dene the holes T

i

by

A T

i

x

i

T

Set

n

i

T

i

where

fi T

i

g This means that we exclude the situation that the holes

intersect with the boundary We point out that this condition for

is set up

only for simplicity All the results can be extended to the case that the holes touching

the boundary We also assume that there exists a bounded density function P x

such that

A

m

P

i

N

x

i

RR

P xx dx

for any C

the set of all continuous functions with compact supports in

Note that under the above assumptions the number of the holes m

is approximately

c

N

for some constants c

One example that satises the above assumptions is the case that the holes are

distributed periodically In that case we may take

x

i

k

k

k

N

k

j

Z

and P x is a constant

Consider the Dirichlet problem

u

f in

u

u

n

on

u

g

u

n

g

n

on

where n is the exterior normal to the boundary To dene a weak solution of

we need to introduce some function spaces Let H

m

be the usual Sobolev

space of all functions that have squareintegrable derivatives up to mth order and

H

m

loc

the set of all functions that belong to H

m

for any

such that

The norm inH

m

is denoted by jjjj

H

m

Denote byH

m

the closure of C

the space of all smooth functions with compact supports in Let

H

fv H

v L

g

with the norm

jjvjj

H

jjvjj

H

jjvjj

L

We know that H

is a Hilbert space and that the closure of C

with

the norm jj jj

H

coincides with H

For any boundary portion

denote by H

m

the jj jj

H

closure of all smooth functions that have the

compact supports in A weak solution of is dened as a function

u

H

such that u

g H

and

ZZ

u

vdx

ZZ

fvdx

for any v H

Lemma Under the assumption AA if f L

and g H

then there exists a unique weak solution u

of problem Furthermore for

any subdomain

the following estimates hold with a constant C

independent of but depending on

jju

jj

H

C

jjf jj

L

When g holds for

Proof Existence and uniqueness are wellknown Assume rst that g

Taking v u

in we obtain

jju

jj

L

jjf jj

L

jju

jj

L

Using the relation

ZZ

jj u

jj

dx

ZZ

u

u

dx

it follows that

jj u

jj

L

jju

jj

L

jjf jj

L

Inequality then follows from the Poincar e!s inequality For the general case that

g we choose a cuto function such that in a neighborhood of and

in

The desired estimates follow by choosing the test function v

u

in

Note that since the weak solution u

H

the extension

"u

x

u

x for x

for x T

i

is in H

By the L

p

estimates of the harmonic equations we obtain

Corollary Let u

be the solution of problem Then for any

jj"u

jj

H

jju

jj

H

c

jjf jj

L

If g then the inequality holds for

It follows that there exists a subsequence of f"u

xg

that converges weakly in

H

to a function say u Another direct consequence of is the following

result

Corollary Let N and "u

be the extended weak solution dened in

Then as "u

uniformly in any

If in addition g

then the convergence is uniform in

Proof By and the Sobolev embedding it follows that "u

is H#older

continuous for some and that its H#older norm is uniformly bounded

Hence

j"u

xj j"u

x "u

x

i

j c

for any x B

x

i

Let

be the characteristic function of

m

S

i

B

x

i

Then

means that j"u

xj c

for any x

Since

P weakly in L

we

obtain "u

uP Therefore u since P

In sequel we shall consider the case N

Local Problems Let T be a domain satisfying A For any consider

the following problem

W in B

n T

W

W

n

on T

W

W

n

on B

By Lemma there exists a unique weak solution to the above problem We extend

this weak solution by in T Denote byW

the extension of this solution From

it is easy to see that W

H

B

Note that we are interested in W

only for large

We begin with the estimates for H

norm

Lemma Let W

be the extension of the solution of Then there

exists a constant c independent of such that the following estimates hold

for W

i jj

W

jj

L

B

c

ii jj W

jj

L

B

c

iii jjW

jj

L

B

c

Proof For simplicity we take Let C

R

N

such that in B

in R

N

nB

By taking v W

in we obtain

ZZ

B

nT

jW

j

dx

ZZ

B

nT

W

dx

It follows that

ZZ

B

nT

jW

j

dx c

ZZ

B

nT

jj

dx c

ZZ

B

nB

jj

dx c

By rescaling we set V x W

x for x in B

The above estimate implies that

ZZ

B

jV j

dx

N

ZZ

B

nT

jW

j

dx c

N

Since V on B

by L

Estimates for the second order elliptic equations we

obtain that

kV k

H

B

c

N

The assertions follow by rescaling back

We next study the behavior of W

near B

By the regularity theory for the

higher order elliptic equations $ we know that W

is smooth up to the boundary

portion B

this is also a consequence of later in this section We shall

frequently use the following particular form of RayleighGreen!s identity

ZZ

B

nT

W

vdx

Z

B

W

v

n

ds

Z

B

W

n

vds

for any function v H

B

nT such that v

v

n

on T To show we take a

smooth function such that in B

outside B

Then vv H

B

nT

Hence by we obtain

ZZ

B

nT

W

vdx

ZZ

B

nT

W

vdx

ZZ

B

nB

W

vdx

From the RayleighGreen!s identity

ZZ

w %vdx

Z

w

%v

n

ds

Z

w

n

%vds

ZZ

w %vdx

that holds for any function w H

%v H

where

is any Lipschitz

domain the identity follows by taking w W

%v v and

B

nB

To proceed studying the behavior of the function W

near the boundary B

we

need the Green!s function G

N

for the biharmonic equation in B

with the Dirichlet

boundary condition For N the Green!s function is

G

N

x y

N N

N

jx yj

N

jyjjx yj

N

N

N

jxj

jyj

jyj

jx yj

N

For N we have

G

x y

jxj

jyj

jyj

jx yj

ln

jyjjx yj

jx yj

Here in and y

jyj

y is the re&ection of y with respect to B

and

N

is the surface area of the unit sphere in R

N

Note that the Green!s function G

N

x y

satises the Dirichlet boundary condition

G

N

x y

G

N

x y

nx

x B

for any y B

By the RayleighGreen!s identity we can represent any smooth

function w that has the compact support as the integral

wx

ZZ

B

G

N

x y

wydy

Lemma Let N and W

be the solution of Then the

following estimates hold for any x

B

n B

B

i there exists a constant c independent of such that

jW

xj

c

N

$

ii there exists a constant c independent of and a positive function that

decreases in such that

W

x

T

N

x

jxj

N

jxj

N

c

jxj

N

where

T

ZZ

B

nT

jW

yj

dx

Proof Choose a cuto function such that in B

in

R

N

n B

Set v W

Then v C

B

H

since W

x is smooth for

x

B

n

T and v satises

v F in B

where

F W W

W

W

trace

W

Sincex for x B

nB

it is easy to see that the function F is supported only

in B

n B

By the interior estimates for the higher order elliptic equations Lemma

and by the Sobolev embedding we know that in B

n B

all the derivatives of

W

up to the third order are bounded by a constant independent of Therefore

jF xj c for any x For convenience here and in what follows c is always reserved

as a universal constant Using the Green!s representation formula we can write

vx

ZZ

B

G

N

x yF ydy

In particular for x B

nB

the above equality reads as

W

x

ZZ

B

nB

G

N

x yF ydy

since supp F B

nB

Hence for x B

nB

W

x

ZZ

B

nB

x

G

N

x yF ydy

W

x

ZZ

B

nB

x

x

G

N

x yF ydy

From the explicit formulas and we can compute G

N

directly It

follows that for N

x

G

N

x y

jx yj

N

N

N

x y

where

jx yj

N

N

N

jyj

N

jyj

N

jyj

N

jx yj

N

x

i

y

i

y

i

Njyj

N

N

jyj

N

jx yj

N

and the convention of summation of the repeated indices is applied For jyj

max

jxj we can bound

x

G

N

by

j

x

G

N

x yj

c

N

since

jx yj

x

jyj

y

c

The assertion i follows from and Next by direct computation we

obtain

j

x

x yj

c

N

for jyj max

jxj Therefore from and the boundedness of

F we deduce that

W

x

N

ZZ

B

nB

x y

jx yj

N

F ydy

x y

N

x

jxj

N

ZZ

B

nB

F ydy

x y

N

ZZ

B

nB

x y

jx yj

N

x

jxj

N

F ydy

where

j

x yj

ZZ

B

nB

j

x yF yjdy

c

N

for jyj max

jxj The third term in the last equality of can

be estimated by

N

ZZ

B

nB

x y

jx yj

N

x

jxj

N

F ydy

jxj

N

for jyj max

jxj where as Finally from

we obtain

ZZ

B

nB

F ydy

ZZ

B

vydy

Z

B

v

n

ds

Z

B

W

n

ds

Z

B

W

n

W

ds

where we used the fact that W

on B

It follows from the RayleighGreen!s

identity that noting that

W

n

on B

ZZ

B

nB

F ydy

ZZ

B

nT

jW

yj

dy

T

The proof is complete

From Lemma

T dened in is uniformly bounded We next show

that the limit of

T exists as

Lemma Let N Then

T T as where T

depends only on the domain T

Proof Let W

be the solution of for large By ii of Lemma

jj

W

jj

L

B

c Since on T W

W

n

by the Poincar e inequality we

obtain that for any r

jjW

jj

H

B

r

Cr

where Cr is independent of but depends on r By the diagonal argument we

may select a subsequence

k

and a function W H

loc

R

N

n T such that

W

k

W as

k

where the convergence is interpreted in the sense that for any r W

k

weakly

converges to W in H

B

r

n T Since W

is biharmonic in B

n T one can see that

W is also biharmonic outside T FurthermoreW

W

n

on T and

W is

bounded in L

R

N

n T since jj

W

jj

L

B

c

Now for any H

loc

R

N

nT such that

n

on T outside B

r

by

the denition we have for

k

r

ZZ

B

k

nT

jW

k

xj

dx

ZZ

B

k

nT

W

k

It follows that for large

k

k

T

ZZ

B

k

nT

W

k

dx

ZZ

B

r

nT

W

k

k

ZZ

B

r

nT

W

ZZ

R

N

nT

W

We now take for any xed

l

x

W

l

x if x B

l

n T

if x R

N

nB

l

Then by the matching Lemma H

loc

Hence

k

T

k

ZZ

R

N

nT

W W

l

dx

l

ZZ

R

N

nT

jW j

dx T

It remains to show that T and that the whole sequence

T T To

this end it su'ces to show that the weak limit W is unique

We observe that the Green!s functions and their gradients satisfy the

following estimates for x B

y B

nB

jG

N

x yj

c

jxj

N

c

N

j

x

G

N

x yj

c

jxj

N

c

N

for N It follows from the expressions that for x B

jW

x j

c

jxj

N

c

N

jW

xj

c

jxj

N

c

N

for N SinceW

k

W strongly in H

B

r

nT for any r it follows that for almost

all x B

jW x j

c

jxj

N

$

j W xj

c

jxj

N

We now claim that W is the only biharmonic function in R

N

n T that satises

W

W

n

on T and $ Otherwise suppose that

"

W is another

biharmonic function having the above properties Then V W

"

W satises

jV xj

c

jxj

N

j V xj

c

jxj

N

for large x Then for any smooth function that has the compact support in R

N

ZZ

R

N

nT

V V

It follows that

ZZ

R

N

nT

jV j

c

ZZ

R

N

nT

jV

j j V

jjV jdx

Therefore by the Cauchy!s inequality

ZZ

R

N

nT

jV j

dx c

ZZ

R

N

nT

jV jj

j jV jj j

j V jj jjj

dx

We choose such that

in B

r

in R

N

nB

r

jj

c

r

c

r

From it follows that

ZZ

B

r

nT

jV j

dx

ZZ

R

N

nT

jV j

dx

c

r

N

Letting r we obtain V in R

N

n T Consequently V since V

on the boundary Hence the whole sequence fW

g converges to W in the sense we

explained before It immediately follows that

T T Finally if T

then W Since W

W

n

on T we can extend it by into T The

extended function is harmonic in R

N

and is bounded by $ HenceW constant

A contradiction to $ and the fact W on T

Homogenized Problem I N Let u

be the solution of and

u be a weak limit of a subsequence of "u

the extension of u

by in the holes T

i

In

this section we shall study the homogenized problem for N Set

N

N

Theorem Assume AA N g C

and that the limit

lim

N

N

exists If then the whole sequence "u

converges weakly to u in H

Fur

thermore u satises

u T Pu f in

u g

u

n

g

n

on

If then for any subdomain

the whole sequence "u

converges to

weakly in H

To prove the theorem we need to construct a test function for any Let

W

be the solution of Under the assumption A the balls fB

x

i

g are

disjoint Dene a function w

in by

w

x

W

x x

i

for x B

x

i

n T

i

for x T

i

otherwise

where

as One can see easily that w

H

Lemma Let N and w

be the function dened in Then

kw

k

L

c

kw

k

L

c

w

L

c

Consequently w

weakly in H

Proof By Lemma we obtain noting that jm

j c

N

kw

k

L

m

X

i

ZZ

B

x

i

nT

i

W

x x

i

dx

m

X

i

jT

i

j

cm

N

kW

k

L

B

cm

c

kw

k

L

m

X

i

ZZ

B

x

i

nT

i

W

x x

i

dx

m

N

kW

k

L

B

c

$

and

kw

k

L

m

X

i

ZZ

B

x

i

nT

i

W

x x

i

dx

m

N

kW

k

L

B

c

The assertions and follow immediately Again from we have

the estimate kw

k

H

c

The assertion follows from the equivalence

of two norms kk

H

and kk

H

since w

H

Proof of Theorem Consider rst the case that In this case

we claim that as claimed in Lemma in the case g k"u

k

H

is uniformly

bounded and

ku

k

H

c

kfk

L

kgk

c

Indeed we choose the function v u

w

g It is easy to verify that v is eligible as

a test function in It follows from that

ZZ

ju

j

dx

ZZ

u

w

g dx

ZZ

f u

w

g dx

By Lemma we obtain

ku

k

L

c

kfk

L

kgk

c

ku

k

L

Next from the identity

ZZ

vvdx

ZZ

jrvj

dx

and using we have

kru

k

L

c

kfk

L

kgk

c

ku

k

L

c kgk

c

Since "u

g H

by applying Poincar e!s inequality to "u

g we derive from

and that

kru

k

L

ku

k

L

c

kfk

L

kgk

c

follows from the equivalence between H

norm and H

norm for

"u

g

By we know that there exists a subsequence of "u

that weakly converges

to a function u It now su'ces to show that any weak limit u satises By

for any C

we have

ZZ

"u

w

dx

ZZ

fw

dx

This equality can be rewritten as

ZZ

w

"u

dx

ZZ

"u

w

dx

ZZ

fw

dx

where

ZZ

"uw

w

"u "u

w

dx

Since "u

is uniformly bounded in H

and w

weakly in H

we know that

as and that

ZZ

w

"u

dx

ZZ

u dx

ZZ

fw

dx

ZZ

fdx

It remains to study the limit of the second term on the lefthand side of We

now apply the identity with v "u

It follows that

ZZ

B

x

i

nT

i

"u

w

dx

Z

B

x

i

w

"u

n

ds

Z

B

x

i

w

n

"u

ds

For any x B

x

i

by Lemma we have

jw

xj

W

x x

i

c

where

Next we rescaling "u

by setting vx "u

x

i

x for x B

By

applying the trace inequality

Z

B

jvj

ds c

ZZ

B

jvj

v

dx

we derive that

Z

B

x

i

j"u

j

ds

c

ZZ

B

x

i

j"u

j

"u

dx$

Combining and $ the rst term on the righthand side of can be

estimated by

Z

B

x

i

w

"u

n

ds

c

Z

B

x

i

"u

n

ds

c

N

B

Z

B

x

i

"u

n

ds

C

A

c

N

B

ZZ

B

x

i

j"u

j

"u

dx

C

A

c

B

N

ZZ

B

x

i

j"u

j

"u

dx

C

A

We now evaluate the second integral on the righthand side of By ii of

Lemma for x B

x

i

rw

x

W

x x

i

T

N

N

x x

i

jx x

i

j

N

where

j

j

c

N

N

c

It follows that

Z

B

x

i

w

n

"u

ds

T

N

Z

B

x

i

"u

ds

i

where

j

ij

Z

B

x

i

j

"u

j ds

c

Z

B

x

i

j"u

j ds

Combining

we derive that

ZZ

"u

w

dx

m

X

i

ZZ

B

x

i

nT

i

"u

w

dx

T

N

m

X

i

Z

B

x

i

"u

ds

where

j

j

B

N

ZZ

B

x

i

j"u

j

"u

dx

C

A

m

X

i

i

c

k"u

k

H

m

X

i

i

By assumption A for any smooth function

m

X

i

Z

B

x

i

ds

N

m

X

i

N

x

i

m

X

i

Z

B

x

i

x x

i

ds

N

ZZ

P xxdx as

Approximating "u

by smooth functions we thus obtain

m

X

i

Z

B

x

i

"u

ds

N

ZZ

Pudx as

From

also implies that

m

X

i

i c

It follows from and Lemma that

ZZ

"u

w

dx T

ZZ

Pudx

Letting in we obtain from and that

ZZ

u dx T

ZZ

Pudx

ZZ

f

for any C

This is the weak formulation of

It remains to prove the case that In this case we dene

%w

x

w

x

where w

is dened in It follows from that

k %w

k

L

k %w

k

L

as

%w

L

c

We note that when g "u

in general is not uniformly bounded in H

By Lemma however it is locally uniformly bounded Let

is any

subdomain of and C

By replacing w

with %w

in it follows that

ZZ

"u

%w

dx

Since "u

is uniformly bounded in H

there exists a subsequence that converges

weakly in H

to a function u H

Moreover the estimates

remain true except that the constant c must be replaced by c

which depends on

Therefore we can evaluate the above integral by the same methods as in deriving

By modifying we obtain

ZZ

"u

%w

dx T

ZZ

Pudx

Hence

ZZ

Pudx

for any C

It follows that u since P and

can be chosen

arbitrarily The proof is complete

From the above results we know that "u

u weakly in H

if is nite One

may ask if the convergence is also strong In general it cannot be strongly convergent

But if we add a corrector w

u the convergence is strong

Theorem Let N g C

and u be the solution of Suppose

that where is dened in Then

ZZ

"u

w

u

p

dx

where p

N

N

Proof For any C

such that g on we write by using the weak

formulation for u

ZZ

j"u

w

j

dx

ZZ

"u

"u

w

dx

ZZ

w

"u

w

dx$

ZZ

f "u

w

dx

ZZ

w

"u

w

dx

ZZ

w

u "u

w

dx

where

ZZ

"u

w

w

w

"u

w

dx

ZZ

w

"u

w

dx

ZZ

w

"u

w

dx

Since w

it is easy to see that as By repeating the process of

deriving we obtain

ZZ

w

"u

w

dx T

ZZ

P u dx

since "u

w

u weakly in H

It follows from $ that

ZZ

j"u

w

j

dx

ZZ

f u dx

ZZ

u dx

T

ZZ

P u dx

ZZ

ju j

dx T

ZZ

P u

dx

From the Sobolev imbedding for any C

kk

L

p

c kk

H

p

N

N

kk

L

p

c kk

H

p

N

N

Therefore

ZZ

jw

u j

p

dx c

ZZ

jw

j

p

j uj

p

dx

c

ZZ

jw

j

p

j uj

p

jw

j

p

j uj

p

dx

c k uk

p

L

pq

c k uk

p

L

pq

c k uk

p

L

pq

where

q

N

N

q

q

N

N

We thus obtain since pq p

pq

p

pq

ZZ

j"u

w

uj

p

dx

ZZ

j"u

w

j

p

dx

ZZ

jw

u j

p

dx

c

ZZ

j"u

w

j

dx

A

p

c k uk

p

H

Letting and using we derive that

lim

ZZ

j"u

w

uj

p

c k uk

p

H

The assertion follows by letting u and the L

p

Estimates for the second order

elliptic equations

In the special case that u is a C

function this is true for example if T f

are regular enough the Theorem can be strengthened such that holds for

p

Theorem Let N g C

and u be the C

solution of

Then

k"u

w

uk

H

c k"u

w

uk

H

c

Proof Taking u in $ and using the weak formulation for u we have

ZZ

j"u

w

uj

dx T

ZZ

uP "u

w

u dx

ZZ

w

u "u

w

u dx

It is easy to see that dened in can be estimated by

jj c k"u

w

uk

H

c kw

k

L

c kw

k

L

k"u

w

uk

L

We now estimate the second integral in the righthand side of By

in each B

x

i

nT

i

we have

ZZ

B

x

i

nT

i

w

u "u

w

u dx

Z

B

x

i

w

n

u "u

w

u ds

Z

B

x

i

w

n

u "u

w

u ds

By and the trace inequality analog to $ we obtain similar to

Z

B

x

i

w

n

u "u

w

u ds

c

Z

B

x

i

j u "u

w

uj ds

c

N

B

Z

B

x

i

j u "u

w

uj

ds

C

A

c

N

B

ZZ

B

x

i

j u "u

w

uj

ds

C

A

c

N

B

ZZ

B

x

i

u "u

w

u

ds

C

A

By the estimates analog to and

the second integral in the righthand

side of can be bounded by

Z

B

x

i

w

n

u "u

w

u ds

c

Z

B

x

i

ju "u

w

uj ds

c

N

B

ZZ

B

x

i

ju "u

w

uj

dx

C

A

c

N

B

ZZ

B

x

i

j u "u

w

uj

dx

C

A

where we also used the trace formula like $ and the fact that is bounded

Substituting and into we derive that

ZZ

w

u "u

w

u dx

m

X

i

ZZ

B

x

i

nT

i

w

u "u

w

u dx

$

c

ku "u

w

uk

H

u "u

w

u

L

where we have used the inequality

m

X

i

a

i

m

m

X

i

a

i

Note that ku "u

w

uk

H

is uniformly bounded Combining and

we nd

ku "u

w

uk

L

c k"u

w

uk

H

c kw

k

H

c

By and the fact that H

is isometric to H

it follows that

ku "u

w

uk

H

c k"u

w

uk

H

c

Homogenized Problem II N In this section we study the caseN

In this case the Green!s function behaves like

ln

as Hence

we need dierent approaches to treat the limiting equation We rst look for a

biharmonic function V

in B

nB

for such that

V

in B

nB

V

V

n

on B

V

V

n

on B

Existence and uniqueness follows from Lemma In fact such a solution can be

constructed explicitly One may verify that for jxj r r

V

x

r

ln

r

r

where

ln

We observe that for large c

ln j

j c

ln for some constant c

c

By direct computation we obtain

V

x

r

jV

xj

c

r

r

jV

xj

c

r

r

We now for convenience extend the function V

into B

by setting V

in B

The extended function is still in H

B

One can see that for large the following

estimates hold

kV

k

L

B

c

c

ln

kV

k

L

B

c

p

ln

c

p

ln

We next dene a test function w

as in by

w

x

V

x x

i

for x B

x

i

otherwise in

where

Set

ln

Lemma Let

Then

kw

k

L

c

ln

ln

kw

k

L

c

ln

ln

Consequently if

is uniformly bounded then w

weakly in H

Proof The proof is based on the estimates similar to $ and It follows

from and that

kw

k

L

cm

kV

xk

L

B

c

jln j

c

jln lnj

w

L

cm

V

x

L

B

c

jln lnj

The assertion follows immediately

We now in the position to prove the main result in the case N

Theorem Let N u

be the solution of with g C

and "u

be the extension of u

by in T

i

If

lim

then the whole sequence "u

u weakly in H

as Moreover u satises the

following homogenized problem

u

Pu f in $

u

u

n

on

If then u

Proof Consider rst the case that As in the proof of Theorem we

can show that "u

is uniformly bounded in H

Since the rest of the proof is local we

may assume without loss of generality that g For any C

we take w

as a test function Note that B

x

i

T

i

B

x

i

and that the function w

is

biharmonic only in B

x

i

Hence in this time we need to handle some extra terms

As in we can write the variational formulation as

ZZ

w

"u

dx

ZZ

"u

w

dx

ZZ

fw

dx

where

ZZ

"uw

w

"u "u

w

dx

Consider rst the case that Then by Lemma as Also

and hold By note that w

in B

x

i

we have

ZZ

B

x

i

"u

w

dx

ZZ

B

x

i

nB

x

i

"u

w

dx

Z

B

x

i

w

"u

n

ds

Z

B

x

i

w

n

"u

ds

Z

B

x

i

w

"u

n

ds

Z

B

x

i

w

n

"u

ds

I

i

I

i

I

i

I

i

We now proceed to estimate each I

k

i

By for any x B

x

i

w

x

V

x x

i

def

where

j

j

c

ln

c

ln

ln

c

It follows that

m

X

i

I

i

m

X

i

Z

B

x

i

w

"u

n

ds

m

X

i

Z

B

x

i

"u

n

ds

m

X

i

ZZ

B

x

i

"u

dx

Combining with we obtain

m

X

i

I

i

c

ZZ

j"u

j dx c

k"u

k

H

To estimate I

i

we observe that for x B

x

i

from

w

x

V

x x

i

def

while

c

ln

Therefore

m

X

i

I

i

m

X

i

Z

B

x

i

w

"u

n

ds

m

X

i

Z

B

x

i

"u

n

ds

m

X

i

ZZ

B

x

i

"u

dx

ZZ

S

m

i

B

x

i

"u

dx

It follows from the Cauchy inequality and that

m

X

i

I

i

c

ln

ZZ

S

m

i

B

x

i

j"u

j dx

c

ln

m

i

B

x

i

ZZ

j"u

j

dx

A

c m

ln

k"u

k

H

c

k"u

k

H

We next estimate I

i

Again from for x B

x

i

w

n

n

V

x x

i

n

jx x

i

j

def

"

while

"

c

ln

It follows that

m

X

i

I

i

m

X

i

Z

B

x

i

w

n

"u

ds

"

m

X

i

Z

B

x

i

"u

ds

Consider the following Neumann problem

q in B

n

on B

where q

jB

j

Existence of a smooth solution for general Neumann problem

is followed by variation inequality But in this particular case one can easily verify

that the function

x

r

r

r jxj

provide a smooth solution We now set for x B

x

x

Then

satises

q

in B

n

on B

$

Using the identity with w

v "u

we obtain

Z

B

x

i

"u

ds

q

ZZ

B

x

i

"u

dx

ZZ

B

x

i

"u

dx

Substituting this into we derive

m

X

i

I

i

q

"

m

X

i

ZZ

B

x

i

"u

dx

"

m

X

i

ZZ

B

x

i

"u

dx

J

J

Since

x

x

by we obtain

jJ

j

"

m

X

i

B

B

ZZ

B

x

i

j

j

dx

C

C

A

B

B

ZZ

B

x

i

j"u

j

dx

C

C

A

c

ln

m

X

i

kk

L

B

B

B

ZZ

B

x

i

j"u

j

dx

C

C

A

c

ln

k"u

k

L

B

where we used To estimate J

we observe that for any function H

B

such that in B

by rescaling the following Poincar e!s inequality holds

ZZ

B

x

i

dx c

ZZ

B

x

i

jj

dx

Using this inequality twice we deduce that

ZZ

B

x

i

j"u

j

dx c

ZZ

B

x

i

j "u

j

dx c

ZZ

B

x

i

"u

dx

Therefore

jJ

j

c

ln

m

X

i

ZZ

B

x

i

j"u

j dx

c

ln

m

X

i

B

B

ZZ

B

x

i

j"u

j

dx

C

C

A

c

ln

m

X

i

B

B

ZZ

B

x

i

"u

dx

C

C

A

c

ln

k"u

k

H

Combining and

we nd

m

X

i

I

i

I

i

I

i

c

ln

k"u

k

H

Finally we evaluate I

i

For x B

x

i

we have by

w

n

n

V

x x

i

Hence

m

X

i

I

i

m

X

i

Z

B

x

i

w

n

"u

ds

m

X

i

Z

B

x

i

"u

ds

Using the arguments as we did in showing we can easily obtain

m

X

i

Z

B

x

i

"u

ds

ZZ

Pudx as

Note that

ln

Therefore it follows from that

m

X

i

I

i

ZZ

Pudx

Combing this with and we obtain from

ZZ

"u

w

dx

ZZ

Pudx

The assertion for follows

The case can be handled in the way similar to the proof of Theorem

by taking %w

w

as the test function in instead of w

We skip the

details

We conclude this paper by the corrector results for N

Theorem Let N Then the assertion in Theorem remains

true for any p Furthermore if u C

then holds for p

Acknowledgments

The author would like to express his deep appreciation to Professor Avner Fried

man for many valuable discussion

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