3(II) is such a class. In [4], [5a], and [5b] it has been shown that if the solution

REGULARITY OF THE SOLUTION OF ELLIPTIC PROBLEMS WITH PIECEWISE ANALYTIC

DATA,

Ih THE TRACE SPACES

AND APPLICATION TO THE BOUNDARY VALUE PROBLEMS

WITH NONHOMOGENEOUS BOUNDARY

CONDITIONS*

I.

BABUKAf

AND B. Q. GUO

Abstract. Thispaper analyzesthe tracespacesofthe weighted space

(fl)

introducedby Babuka

andGuo [SIAMJ. Math. Anal.,19 (1988),pp.172-203].

Keywords, ellipticequation withpiecewise analyticdata,Dirichletproblem,cornersingularities

AMS(MOS)subject classifications. 35B65,35D10,35G15,35J05

1. Introduction. Elliptic boundary value problems with piecewise analytic data

aretypical inmanyfieldsof applications, for example,instructuralmechanics.These problems are then numerically analyzed in engineering by the finite-elementmethod. The design andperformance ofanumerical method directly dependson theclass of problemstowhich it is oriented.Thesmallerthe class is, themoreeffectivethenumerical method can be.

Hence,

it isimportanttocharacterizemathematicallya(minimal) class

that encompasses virtually all practicalproblemsin a field ofapplications. The space

3(II)

is such a class. In [4],

[5a],

and [5b] it has been shown that ifthe solution belongs to the space

3(1-1),

then the h-p versionof the finite-elementmethodhas an

exponential rate of convergence. The h-p version uses properly refined mesh and a

high degree of elements in contrast to the usual h-versionthat uses only low-degree elements.Forthe surveyof various theoretical and practical aspects ofthe h-pversion werefer the readerto

1]

andthe references giventherein.

In

[3]

the spaces

(l-l)

have beenanalyzed. Ithas been shownthat thesolution

ofthe elliptic boundaryvalue problems withpiecewise analytic databelongsto these spaces.

The present paper elaborates in detail on thestructure of thetraces offunctions

of

(1).

The results give easycharacterization ofthecasewhen the solutionbelongs to

3(fl).

In 2 wegive the preliminaries and basic definitions. Section3 defines the

model problem of second-order elliptic partial differentialequations. Section 4

intro-duces the space oftracesofu s

(fl)

on theboundary0fl. Itisalsoshownthatthese

traces can be extended into

(fl).

2. Preliminaries. Letflc_R

2,

_(xl,

x2)

xbeasimply-connected, boundeddomain withthe boundary 01"1

F

U

1

’i.

F

are analytic simple arcs called edges,

’,

{(p,(s),

,())

s [= [-1,

1]},

where p(sc), ffi() are analytic

functions

on [and >_{,>0. By}

r,

_we

denote the open arc,i.e., the image ofI

(-1, 1).

Let

A,

i=1,.

.,

M,

be thevertices * Receivedbythe editors April6, 1988; acceptedfor publication October 11, 1988.

"

Institute for Physical Science andTechnology, UniversityofMaryland, College Park, Maryland20742. Theworkof this authorwassupported bythe Office of NavalResearchunder contract N00014-85-K-0169. tEngineeringMechanicsResearch Corporation,Troy,Michigan.The workofthisauthorwassupported bythe National Science Foundation under grant DMS-85-16191 duringastayattheInstitute forPhysical ScienceandTechnology,University ofMaryland, CollegePark, Maryland20742.

764 I. BABUKA AND B. Q. GUO

of

II

and

F

AiAi+l, i.e., the edgeF islinking thevertices

A

and

A+I.

Forsimplicity

we will also write

A1

AM+I.

An example ofthe domain

II

under consideration is

given in Fig. 2.1. By to, i-1,...,

M,

we denote the internal angles of

II

at

A.

We willassumethat0

<

toi

=<

2,r. Wewill also consider the casewhentwo edgescoincide. Then weunderstand themin a "two-sided" sense. If all edges are straight lines then

we callthe domain

II

astraightpolygon.Otherwise wespeak aboutacurvilinearpolygon. If0<toi<2r, 1,’’.,

M,

we speak about a Lipschitzian domain. Let us assume

thatF

F

()

U

F()

whereF(

U

o

,

F(1)

F-

F

(),

(a)

U

o’

’,

whereQissome subsetofthe set

{1,

2,.

.,

M}

M

and

Q’= M

Q.

We assume for simplicity that fl is a simply-connected domain. The results we

present herearealsovalidwhen flis ann-connected,boundeddomain and itsboundary iscomposed of n-cues.

Denote I

{x]-I

<x<

1};

wealso writeI=

{Xl,

X2]--I

<X

<

1,X2

=0}c

2when

nomisunderstandingcan occur.

By

L2(),

Lp(), L2(I), Lp(I),

the usual spaces of p-integrable, 1

<

p

<

,

func-tions on or I are denoted. By

Hm(),

Hm(I),

m0 integer, we denote the usual

Sobolev space offunctions with square integrable derivatives of order less than or

equalto m onfl(respectively,

I).

The space

H()

isfurnished withtheusual norm

L2(

0llm

where a

(a, a2),

a

0 integer, 1,2,

a]

a

+

a2, and Du

Ox

Fuhermore, we let

As

usual, we write

H()

H(n)

{u

e

Hi(n) ].

0 on

F<)}.

Inan analogouswaywe define

H(I)

by

Du

u

=

du/dx

.

By

(x)=dist(x,

As)=]x-AsJ,

xe,

je, we denote the Euclidean distance between the point x and the

veex

As,

Fl(X)=x+l,

r(x)=x-1,

xL Let

(1,"

",

3M)

(respectively,

(,

32))

be an M-tuple of real numbers 0

<

<

1,

A3

A

A

1,.

,

M. Wewrite O

<

<

O2 (respectively,/3

</3)

ifal

< fli <

O2 (respectively,

fli

</3i), i=1,.

.,

M. For any integerfl, we write

fl

+

k {ill

+

k,. .,/3M

+

k}

such

that and M

/(x)--

II

Ir,(x)l

’/,

i=1 2 i=1

By

C(I’I),

C(fi),

C(I),

C([),

j>=Ointeger we will denote the set ofall functions

with continuousj-derivatives ongl,

fi,/,

/,

furnished withthe usual norm

[[.

[["

c(). Let

H"(gl),

m ->t_->O integers, be the completion of theset ofall infinitely

differentiable functions under thenorm

k=m

Ilull,’(m

Ilull

=

k=l for1_>-1, k=m

Ilull,,(.)

E

II%/lDulll

=

L2(fl) k=O

Ifm 1=0 we will write

H

’=

L(fl).

Analogously as beforewe define

lulz.’(m

X II%lDulll =

Lz()"

In asimilar way

H’I(I)

is defined

Ilull,’(,)

Ilull

=

.’-(i)+

E

II%+-llDulll(,)

for lN 1,

k=l k=m

ilull=

Hr’o(I)--

E

II%+lOul

k=0

Furthermore we introducethe space

(fl),

l=>0 integer that will.play animportant rolein this paper:

(’)

{U

It/E H’/([),

for any k >-

I,

II+_,loul

(.)

<--

Cdk-t(k-l)

!,

I1

k. c>0.

d ->_ 1independent of

k},

where C and d may dependon u. Ifwe wish to emphasize the dependence on d we will write

t,d(f/).

Analogously for -> 0 integer

(I)={ulu

H’t(

I),

for any k>-_1,

II$+_,u<)ll

_<,

<= Cd’-(k

l)!, C

>

0,d

>=

1independent of

k}.

Furthermore, forj 1,2,

ip(ll)

{u

Hi(II)IID"u(x)l

<--lal

k=j- 1,j,.

.,

C

> O,

d-_> 1independent of

k},

(I)

{u

e

HI(I)I lu()(x)l-<

ClP+l_+/2(x)l-dkk!,

766

.

BABUKA AND B. Q. GUO

Lety

U

io=

Pi.

Then we define

Hk-1/E(y),

k_-> 1 (respectively,

H3-1/2"l-1/2(’y),

k l

1)

integersasfollows" for any e

H-a/(y)

(respectively,

H-l/U’t-/(y))

there exists

fe

H()

(respectively,

H’*(O))

such that

fir

.

We define then

IIIIH-,,=<>

(respectively,

inf

Ilfll-m

(respectively,

Ilfll

By

-/(),

IN

1, we will denote the set ofthe traces on _{T of}functions from the

space

().

Let

F

be anedge of

;

thenby the assumption thereexists a one-to-onemapping

m

ofI onto

F

which is analytic. If

F

is a straight line, then we will assumethat

m

isthelinearmapping. Letu be defined on

F, U(x)

u(m(x))

be defined onL Then

we define

H(F)=(ulUHm(I)),

IlUllH’(r,--II UII-(I)

In the same way we define the spaces

H"(F,),

3(Fi),

(r,).

Let us remarkthat, as we defined it,

I1"

[[q(r,

depends on the mapping mi, i.e., it depends on the

parameterizationofthearc

F.

Nevertheless the space

H"I(F)

doesnotdoas wellas

(F)

(see

Lemma

4.6)

but

3(F)

couldbe dependentonmi. Letusnow state some lemmas thatwillbeused later.

LEMMA 2.1. Wehave

H’2(a)

with thecontinuous injection. See Lemma7 of[2].

LEMMA 2.2. Letu

H’2(f).

Then

(2.1)

(i)

(ii) Let

u(A)

O, 1,.

,

M. Then

(2.2)

See Lemma 8 of

[2].

I.uA 2.3.

(al=(a)

.a

(a=+(a,

0<+<1,

arbitrary.

See Theorems 2.2 and 2.3 of

[4].

LZMMA2.4. Letu

(f),

j >--_0; then u is analyticon

-

t.J

iM=l

Ai,

LZMMA2.5. Letr#1 and

F(x),

0

<

x

<

oe is

defined

by

F(x)=

f( t)

dt

for

r>1,

F(x)

f( t)

dt

for

r

<

1.

x-F_x_

<- x xf) dx. r-1 e>O Then See Theorem 330 of

[7].

3. The model problem and its properties. Let fl be the curvilinear or straight polygon and let L be astrongly elliptic operator

2 2

L(u)

,

(ai,

j(x)ux,)xj +

E

bi(x)ux,

+

c(x)u

i,j=l i=1

where

ai3(x)=

aj,i(x), bi(x),

c(x)

are analytic functions on 12 and for any and any x f let

2

E

a,,j(i>-

to(

+ )

i,j=l

with/Zo >

0.

Let

B(u,

v)

be a continuous bilinear form on

H(12)

H(I):

We assumethat

i,j=l i=1

inf sup

IlUlIttl(II) _{[I/3IIHI(I/)__.}

u_H(D)

andfor any ve

H(O),

v 0

IB(u, v)l

tZl>O

sup

[B(u, v)l

>

O.

u.H(D)

Assumenowthat

gtqa 3/2-1(Fl),

I= O, l,fe

(fl)

and considertheboundaryvalue

problem

(3.1a)

Lu

f

onfl,

(3.1b)

u=gtJ

onF

(,

Ou

gill

onF(1)

(3.1c)

wherewe denoted by

n

theconormal inthe usual sense. The solution ofourproblem

isunderstood in the usual sense. Thenwe have Theorem3.1.

TEOREM 3.1. There exists unique solution Uo

Hl(fl) of

theproblem

(3.1).

See Lemma 3.1

of

[3].

Letus mention some theorems addressing regularity of the solution Uo.

THEOREM3.2. Thereexists0<-

<

_{1, 1,}

,

M,

_dependingontheproblem (i.e., operator

L,

w,

etc.),

such that

if

f

(),

g[l]

E

/2-1(F(l)),

O,1,

/<fl

<1, then

Uo().

Proofis given in

[3].

THEOREM 3.3. Let f be a curvilinear) polygon (instead

of

straight polygonas in

Theorem

3.2)

andlet the assumptions

of

Theorem 3.2 _{hold. Then Uo}

(f).

Proofof the theorem is givenin

[4].

We have seen in [4], [5a], and [5b] that when the solution u ofthe problem

(3.1a)-(3.1c)

belongs to the class

3(f)

then the h-p version of the finite-element method converges exponentially.

Theorems 3.1 and 3.2 show that it is important to develop practical characteri-zationsof spaces

/2-(F),

0, 1,which canbeeasilyused in concrete cases toverify whether the inputdata, i.e.,

gt

belongtothedesired space. Wewillelaborate on this inthe nextsection.

768

.

BABUKA AND B. Q. GUO

4. Tracesand extensions of weighted Sobolevspaces. Characterization of thespaces

/2-(F).

Inthis section we willelaborateonthecharacterizationofthe space

/2-t(1-’),

0,1, whichleadsto aneasyverification inthe concretecases of applications. LEMMA4.1.

Let/3

(1, 2),

0

<

<

1/2

and g

H’(I).

en

(i) g

C([)

and

IIllcO(r>

CllgllA,’(,);

(ii)

[g(x)-g(-1)l

Cl/2_o(x)llg[l,k(i>,

Ig(x)-g(1)l

where Cis a constant independent

of

g(x)

(but

dependson

fl).

Proo

Obviously,

g(x)

g(

t)[

g’(

r)

dr

(4.1)

g,2(,)(,)

d,

(B(,))

-2_d,

Ilgll

_u.’(l)

(o(z))

-2_dr

whichshows that g iscontinuous on Using the imbeddingtheorem on

(-,

)=

I’,

we have

(4.2)

and weget immediately

Ilgll

C(I)

Cllgll

Further,

(4.1)

immediately leadsto (ii).

LEMMA 4.2. Let

fl

(ill,/32),

1/2

</3

<

1 and g

H’2(I).

Then

(i) g

C(I)

and

whereCis a constant independent

of

g(x).

Proof

Using

(4.1),

we get

Ig(x)

g(t)l

,x

g’() d and

g’

----<

C[[g’(0)l

/

Ilg"

Inthelastinequality weusedLemma2.5 and the factthat

1/2

</3 <1. The lemmanow

LEMMA 4.3. Letg

,d(I),

0</3

<

1. Then

for

k>₁

Ig(k)(x)l

_<_

C(k_,/2+o(x))-’(d,)kkt

where

dl=

"I’d, )’>1 is independent

of

g, k, d, and Cdepends on

,

but is independent

ofg, k.

Proof.

Let

I’

(-1/2, 1/2).

Then

for

any k->- 1 we have

IIg(k)llH,(r,)-- C(dp(-))-k-k!

dk

where/3

max

(/3,/32).

Henceby the imbeddingtheorem,we have

Ig(0)l

__<

Cdklk!

where

dl

-->

yd, y

>

-1(1/2)

>

1. Further, for k>-_1, we have that

ig,(x)l

<__

igk(0)[

/

g(k+l)(t)

dt

_-<

Ig(0)l+

(g(k+l)(t))2p213+k(t)

dt

*-+k(t)

dt --1

<-

Cdk![1

+

/_,/(x)]

<-CriCk!

(k-/2+

(x))-’

COROLLARY 4.4. Letg

1(I),

0</3

<1. Theng

1(I).

COROLLARY4.5. Let

g2o(I),

0</3

<1. Then

for

k>-2

Ig(x)l

<_-andg

(I).

LEMMA4.6. Let

m(x)

bea one-to-onemap

of

Ionto

I,

let

m(x)

be analyticon

[,

and let

Im’(x)l

>

0, x[. Assumethat

g(I),j=

1,2, and

define

v(x)=g(m(x)). Then v

(I),

j 1,2.

Proof.

Because

m(x)

isanalytic on I it canbe extended into the complex plane C on I=(z=x+iyl-l-8<x<l+,,lyl<8}, 8>0,

m(z)

is a one-to-one mapping of

[8

onto

*=I,,

’>0 and

Im’(z)l>ao>O,

z[.

Now let j=l and

XoI.

Then

for k>--1

Ig(xo)l

<__ andtheseries

g’(x)=

E

g(k+l)(Xo)(X--Xo)

k 1

k=0 k!

is absolutely convergent for

Ix-xol

a((Xo)/dl),

a<1. Hence also

g’(z)=

2

g(k+l)(Xo)(Z--XO)kk

k=O

convergesand

Ig’(z)l-<-

Cat+/(Xo)

for

IZ-Xol

<-

a(cI’(Xo)/d)

where Cisindependent of_Xo.Hence g(z) is aholomorphicfunctionand

v(z)=

g(m(z)) is holomorphic,too.

UsingCauchy’s theoremwe get immediately that for k

=>

1

Iv)(x)l

<

Cd4,-;

_

770 I. BABUKA AND B. Q. GUO

Remark 4.1. Lemma 4.6 shows that the space

(I)

is invariant withrespect to ananalyticmapping. Usingtheformulaof the nth derivative ofacomposite function

(see

formula 0.430 of

[8])

we can also show that

(I)

is an invariant space with

respect to an analytic mapping

m(x)

as in Lemma4.6.

Let

F

be an analyticarc.Thenwecould definethe spaces

(F)

and

(F)

with

respectto thelengthinstead as we did in 2 by usinga specificmapping. Thesetwo

definitions are then equivalent by Lemma4.6 and Remark 4.1.

LEMMA 4.7. Let

M(x),

xR

2,

M(x)= (Ml(x),

M2(X))

be a one-to-one mapping

of

onto and

IJ-l

_{_-< tz on}

,

whereJis the Jacobian

of

the mapping.Assume that M

(x)

canbe analytically extendedon l)

{x

R21

dist

(x, f)

<- 8

}

sothat it isa

one-to-one mapping

of

fi

onto

fi*,

f*

,.

Let u

(f),

j 1,2,

v(M(x))

u(x).

Then

v().

The proofisanalogous tothat ofLemma4.6, however,we mustapply the theory oftwo complexvariables.

LZMMA4.8. Letg

(I),

0</3 <

1,j 1,2. Then

g(I),

O<fi<l,

+

e, e

>

O arbitrary.

Proof.

Let us consider only the case j 1. The casej 2 is analogous. Because for k=>l

we get

Ig()(x)l

Cdkk!(k+13_l/2(X))

-1

(g(k)(X))2p2k+fi_l(X)

dx

<= Cdk(k!)

(I)_{fi_/3_}

1/2(X

dx

--1

C(e)dEk(k)

2.

WeseethatLemma2.3 hasacompletelyanalogousversionfortherelationbetween

(I)

and

(I).

THEOREM 4.1. Let u

H+2’2(),

k

O,

and

Fi

be a straight line edge

of

and

Ulr

g.

en

we have the following: (i) For

<

,,

fl+

<

1 andk 0

g,

HI’I(F,),

,

(,,1, ,,2),

,.

>

0.

,.

,._.-.

1).

1.

and

withC independent

of

k and d>- 1. (ii) For 0

<

fl,, fl,+l

<

1/2,

k

>=

1

g

H(F),

Hk+,2rF

g,e

,

,)

flid

e(fli+-l+1/2

1),

j=l 2,

(iii)

Vu()

and

,,(r,),

r,

+

1.

flid(fli+-Proof

Withoutany loss ofgeneralitywe can assumethat

Fi

1-’1

and

lPl

{Xl,

X21X

I,

x2

0},

A

(-1,

0), A2

(1, 0),

/

(/3a, fl).

Let k>_- 0 and Vk

(oku/oxk)k+.

Then for k_->2,

II

I1(

-i,/CI)k+

+

k D

-i,/CI)k+,_l

/--.2(fl) L:(,O.)

Using Lemma2.2, we get for k 1

BecauseofLemma2.1 u

C(I)),

andhence vo(Ai) 0, 1,2.

Hence,

using Lemma 2.2,we get

IVol.m

<-and hence forall k

=>

0

(4.3)

where C is independent of k. Thereforeby the imbedding theorem

Vk

C(fi),

k_->1.

Letus nowshow that

Vk(A)

0, 1,2, k 1.Assumeonthecontrary that

v(A)

>

O. Thenbecause Vk

C(fi)

we have

v(x)>e>O

forlx-Al<,

>0. Hence for k

=>

2 where

>=

e2

f

f

dP2--2

dx=O dx=

2v2

dx

and we have the desired contradiction. For k 1 we use Lemma2.2 and get

O0>

0213_1

dx>

82

022

dx

Ifu

(f)

then we get from

(4.3)

for k=>O

IIv,

llw,)-

<

Cdklk!.

Wehave

gk(xl)=Oku/oxkIr

k>0.=Then

gk(xl)=d-lk+ (X)Vk(X)lr

=d

-k+(xl)vk(x)

wherewe wrote

-1

k+(Xl)

and

Vk(Xl)

insteadof -1

k+(Xl,

0)

andVk(Xl,

0).

Assumefirst

that

1/2<fll,fl2<l.

Let

do={min=3,...,4

dist(A,F,)}

4-2.

Then we have for x

eFt,

(Xl)

-<-

(Xl)d

-,

and hence forj 1,2,.

.,

k

+

1,

j_,+,lg{J)(Xl)l

2

dx,

<=

Cj2

j-l+,[lvj-,I

"

2

s+j

-,

+

-2

--1

772 i. BABUKA AND B. Q. GUO

Using Lemma2.5,thefactthatj 1,.

,

k

+

1,

Vj_l(Ai)

0, 1,2 and

that/3

fll +

1>

1/2,

weget for some

dl

<

1

_+lg(x,)]

dx, Cd,

_15,_

--1

By

(4.3)

and the imbedding theorem we havefor 1

<

p

<

andj 1,.

.,

k

+

1 Hence forj=

I,...,

k+1,because

-fl

>-,

we get

L2q(I)

--1 -1

Because byLemma 2.1

we get

Ilgl]

L=r,<

Hence we have proven (i) and (iii) for

1/2</3i,/3i+1

<

1 and k->0.

Assume

now that

0</31,/32

<

1/2.

Wewill proceed analogously as before. Forj->2,we have

I

2 (j) 2 2 --2

(I)j-E+lg (Xl)l

dXl

<_Cj2

dXl

j-2+I[IVj-ll

2-2+j-1 +

dpj+lvj_l

2]

--1 <-_Cd-(2

_+,_(v_(x))2

dx

-1

wherewe onceused Lemma2.5 andthe factthat -1

+/,-

>-1/2.

Hence,

using

(4.3)

and realizing that -1

+/-/

>-1/2,

we getanalogously as before for j 2,..., k

+

1

%-=/1 Ig

><x,)l

=

dx,

Cd

u

+=,=,

Letus provenowthat

Wehave

vo(A1)= vo(A=)=0,

andhence

g,2

dx

=< Cdff

=

[=[vgl=+lvol=

-=+1]

dx

dff

2 2

Ivl

=

dx

--1 -1

wherewe have again used Lemma2.5. Because 0<

<)

and

weproceedasbefore and (ii) and (iii) follow easily.

Remark 4.2. In the

proof

of Theorem 4.1 it has been assumed that

fl,

(fl+_

-,

1),respectively,

fl.

(+-1

+,

1),

i.e., oftheopeninteal.Theproofdoes

not hold forthe closed inteal. Ithas been assumed in Lemma 4.9 that the edge

F

of the domain was straight. Let us now assume that

F=

re(I)

where m

=(,

)

are

LEMMA 4.9. Let the edge

F

of

the domainbe analytic. Thenpart (iii)

of

Theorem

4.1 holds.

Proof.

By Lemma2.3, u

(l).

Let

M(sc)

(p(:), p(sc)),

sc

I,

be themapping

ofI onto

F1.

Then we define

Then the mapping

M(,)=(M(,),M2(,))

is analytic on

I=

(,

-1

< <

1

+

,

]

< ),

>

0,

J]

<

,

]J-]

<

on

I

(where

JistheJacobian

of the mapping) and maps

I

onto the

(open)

neighborhood

S*

of

F.

Denoting

*=S*,

T=M-I(*),

we see that

r(x)=u(M-(x))

is defined on

T,

and r

(T)

by usingLemma4.7.Hence r

+(T),

e

>

0arbitrary,byLemma2.3. Hence for

<

fl,

fl+

<

1 wegetby Theorem 4.1(iii)

gi()

V(,

0)

,(i)

fli,

(fli+j-1

+

e

--

),

j 1 2.

Because e

>

0 is arbitrary,

id

(fli+-l-, ).

Analogously for0<fli,

fl+l

<,

gi()

,(I), ,.j

(fli+j-1

+,

1).

LEMA4.10.

Letg(I),O<<,O<2<l, g2(I),)<<l,O<2<

1. Let S

{

r,0 0

<

0

<

2

,

0

<

r

<

}

where r,

O)

arepolar coordinates with respect to

(-

1,

O)

and

( r)

r.

Define

Ui(r,

O)=

gi(-l

+

r),

V(r,

O)=

O[g,(-l

+

r)-g(-1)]

(by Theorems 3.1,3.2, gi

C(),

i=1,2, and hence gi(-1) is welldefined). Then

u,, v,

U2,

V2

I(S),

1-.

Proofi

Assume

first that

0<,<

and

g,(I).

Set

fl=,+

and

U,=

g(-1

+

r).

Thenfor k 2

r

rd dO

<= Cdk(k!)

2.

Henceby Theorem 1.1 of

[4]

we have fork_->2,

[a[

k

V,16.+ - ll

Cdgk!.

Furthermore,

Hence, U,

(S).

Nowlet

<

1

<1. Set

fl

,-.

As before,we have for k2 s

k

Ork

]

(r-2+)2r

dr dO

Cdk(k)

2

and weget

UI

’(s

<

m-

Hence,

U

e

(S).

Letusnowconsiderthefunction

V(r, 0).

Thenasbefore

774 I. BABUKA AND B. Q. GUO Furthermore, using Lemma2.5 and k->2, we get

[ OVI

r-2(rk-2+p)2r

_{dr dO}

/8-lgl

r-2(r-2+#)

2

ar

0

ar,,

/

rdrdO

Cdkll.(k-1)

2 IlSl

k-2+fllllL2(l)

2

Cd](k

).

Inthe last inequalitywe usedthe fact that

Ig(-’(0)l

Cd(k)

andrealizing that

aV/(ar

-

d0)

=0 for kj2 we have for

}a}=

k2 Fuhermore for0

<

1

<

and

I*

(-

1,

0)

we have

v,

’(s

c[

g

’

,/11L(,*

+

II(g,

()

g,

(-

1

))_,/

L(,*]

<

c[llg1’a, L(,*

+

_{(g,() g,(-1}

cllg,

ll.’(,).

In the last inequality we have used once more Lemma 2.5 and the fact that Quite analogouslywe prove

tat

Lh 4.11. Letg

(I),

0

<

+().

Proo

For k 1, --1

g(’(6_)

(’-’

_,++

ax

-1 !=0 N

Cdk

(g(O

((k

l)2

dx /=0

J

-1

Cd2k

(gO)

+,_((k_/)t)2

dx+

(g)E_(k)2

dx /=1 --1 Cd

(g’YCZ+,_,((k-

))

dx+

(g’)}(kY

dx

Cd’(k)

1=1

wherewehave used Lemma2.5 inthe above inequality. Fuhermore,

v2dx

g2@

_vdx

CIIgll%i’,

by Lemma4.1.

LEMMA 4.12. Let

ge(I),.g(+l)=0,1/2<<l,

0<T<1/2,

v=gP_v.

Then

for

fi+y>

1,

ve93+_(I)

and

for+y<l,

ve93+(I).

Proo

(a) Assume

first that

+

T

>

1. Thenfor k 2

(V(k))2++,-2

&<=

Cd2

(g(/))uu-,-(-,)+k+g+,-2((k-

1)1)

&

--1 /=2 --1

+(k)

g

*}_2dx+((k-1))

2

g’2}_

dx --1

Cd2k

(g(1))2}+,_2((k_/))2

dx I=2

+(k)

g’_

dx

Inthe lastinequality, Lemma 2.5 has beenused. Because

by,the

imbedding theorem

Ig’(0)l

cllgl}(

and using Lemma2.5 once moreyields

-

1

>-,

we get

’(I)

"

-1 -1

Hence,

Furthermore, as before 1

(v(k))Zk+g+V_

2dx <--_

Cdk(k!)

2.

J

Vt2(+T_

_dx<_C

I

g-’==r-,

_dx<-

CIIgll=,,

_<o. --1 --1 13 Because g

C(),

v

L2(I),.

(b)

Now assume

that/3

+

_T

<

1. Thenfor k

=>

2 weget exactly as beforethat

Furthermore,

(V(k))2)i+fi

+")’--2dx<

Cd21

k

(k!)

.

-1 Vt2dx<C g2^2

_r_dx+

--1 --1 --1

=<

C

g"

(I)_--T+

Because -3’

+

1

>

fi

by ourassumption we see that

v’2

dx<-

Cllglli.

-1

Using Lemma4.2, we also get

LEMMA4.13. Let 12 beacurvilinearpolygonwith theverticesAi, 1,.

.,

M. Let u

(12)

and wbesuch that

IU"

wl--<

c-.+1

!d

",

776 I. BABUgKA AND B. Q. GUO

Proof.

For k->2,

al

k,we have

IDllvl20l_+a

dx<= Cd=k

Iok-’ullO’wl

.2_2+

e

dx /=0 <-

Cd

E

((l+ 1)!

D- ul:(I)_:_,+

dx I=0 k <-

Cd

k

((l+

1)!)2((k

1

+/)!)2_<

Cdk-2((k_2)!)2.

I=0 Furthermore,

Ill

lDll)12

dx

C[fflDltl[21wl2

dxqIgl

lul2lDIwl2

dx]

<00 becausebyLemma 2.1 ue

C(fi).

It isvery easyto provethe followinglemma.

LEMMA 4.14. Let

ge?3(I),

0</<1/2.

Then

v=gPef:3}(I)

and

v(+l)=0.

Let

geOl,(I),

1/2</<1;

then

v=gPe}(I)

and

v(+l)

=0.

Proof.

The statement that

ve3(I)

canbe directly verified. By Lemma 4.1 v is continuouson

f.

If

v(-1)

#0,then

v2(x)

>

e

>

0forall

Ix

+

11

<

,

Hence,

g2

(v-1)2

>=

^2

( ),0</3<1/2

The proof of the eq-l, which contradicts the assumption that ge3 I

second part of the lemma is analogous.

LEMMA4.15. Letue

3(1)),

0</3 <1 andu=0at

Ai.

Then

u

-1 The proof follows easily using Lemma 2.2.

THEOREM 4.2. Let 12 be a straightpolygon with the edges

F,

i-1,...,

M,

and

letg e

(F1),

0<

,

<

1/2,

fl,

fl +1/2,

i=1,2(respectively,g e

3(F1),

1/2</

<1,

fl

=/3,-1/2,

1,

2)

andg(A) O, 1,2. Then there is u such that (i) ue

3(12),

with 0

<

fl

<

1,j 3," ",

M,

arbitrary; (ii)

U[r,=gandulr=Oforj=2,...,M.

M

Proof.

Let

O

H,=3

x

A[

2,

x e1) Denote

ff

g/O.

Then

obviously

g

e3

}(F1)

(respectively,

g

e

3}(F1)).

Now select 0< y<

1/2

such that 0</3

+

y<

1/2

(respectively,

0_</3

+

y,- 1

<

1/2).

Denote

ff

I-I=l

Ix

A,I

-,

_

where y (yl, 72,0,...,

0).

By Lemmas4.1 and 4.2

(A)-0,

i=1,2. Using Lemma4.11

(and 4.12)

we see that

e

3}+y(I)

(respectively,

ff

e

3+v_1(I)).

Let UeH

I(I)),

AU 0and U on

F1

and U 0 on

F,

j 2,

,

M.Function

U

exists and is uniquely determined. To see this consider q(x), x eF1, pe

C(F1),

(p(x) Ifor

Ix- AI

-<_e/2,i-1,2 andp(x)=Ofor]x-Ail>e,i=1,2 withesufficiently small. Wedefine

U--

UI--

U2

where AUi=0,

UieH(I)),

i=1,2,

Ullr--(1-q),

U21rl--,o

and

U=0

on

F

s,

j=2,...,M. Because

h=(1-o)eC(F1)

and hi(x)=0 for

obviously exists.

By

Lemma 4.10 there exists We

HI()

such that

W]rl

h2=o,

and

Wlrj=0,

j 2,.. ",M.

Hence,

U2

exists too. Function U has the following properties" (i) AU=0.

(ii)

U]r,

,

Ulrj

0,j 2,...,M. (iii) is analyticon

F

(not

on

F1).

(iv) In

.a-tqfl{xllx-Al<,},

i=1,2, with t sufficiently

sall,

there is

W

such that

W

e

3}(t2i,),

where

/3i

=/

+

3’

+1/2

(respectively,

/

fl

+

3’-1

+1/2)

and

Wlr,na,., ft.

(This followsfrom Lemma

4.10.)

Bythe selection of yi we

have/3-

>

1/2,

i=1,2. Nowusing the same

arguments

as

inthe proofofTheorem2.1 in

[4],

weconclude that U

(12),

where/ffi

_=/3

+

y

+1/2

(respectively,/3

=/1

+ Y,-1/2),

i=1,2, and 1

>/

>

1/2.

By

Lemma 4.13 we see that u

OvU

(12)

where fl,

,

+1/2

(respectively, /3i

fl-1/2),

1,2 and0

<

flj

<

1 arbitrary for j 3,.

.,

M. In addition,

Ulr

g and

ulrj=0,

j=2,...

,M.

Let us outline the main idea of the assertion that

U(f).

Let

S.,=

{ri,

0i[0<

ri

<

8,0<

0i < to}

fq12 where

(r, 0)

arethe polarcoordinates withthe origin in

A.

We select

8

<

1 suchthat

S.2,

f’)

Sj,2j

for <-j. Using Theorems5.7.1,5.7.1’, and 6.6.1 of [8], we conclude similarly, as in the proof of Theorem 2.1 of

[3],

that U

(-u/M_-I

Si,i/4

due to the analyticity of on

F-UI

Si,i/4.

Hence we have

to prove only that U

(Si.,/4).

Let

o C(+),

o(r)=l

for0_-<r=<1/2,

o(r)=0

forx=>l,

tp,(r)

qo(r/2t,)=q(r).

Denote v qU. Then v canbe understood to be defined onthe infinite sector

Q

{(r,

O,)lO<r,<,O<O,<o}

when extended by zero outside of

S,,

and we have

Now weprove that

g(S,a,/)

asin

[3].

Remark 4.3. We have assumed that either

1/2</

<

1. Obviously Theorem4.2 is correct if

J(F1)

only inthe neighborhood of

A1

and

(F1)

in the neighborhood of

A:.

Theorem 4.1 leads easily to the next

theorem.

TrEOREM 4.3. Let 12 beastraightpolygon with theedges Fi, 1,

,

Mandlet

g

,

(,)

/,

(/,,1,

fl,,2),

0

<

fli,1,

fl,,2 <

1/2

--/,1-’-i,1--1/2, --/,2--i,2-t"

1/2,

iQc{1,...,M}

or

g fli,1,

fli,2 <

1,

fl,l=/,,1-1/2, fl.2=/,,2-1/2,

iQc(1,...,M}.

Further,letgbecontinuouson y

U

,o

,.

Theng

/2(y) where/

max

(/-1.2,

for

A

),

if

1 Qor

e

Qthenwe

define

fl-1.2

O,respectively,

fl.l

O)

and0

<

fl

<1

arbitrarily

for

Ai

:

7.

Proof.

Becauseg is continuous on y we can construct apolynomial Pon12 such that g-P 0 at

A.

Then we can apply Theorem 4.2. l-]

Remark 4.4. It is obvious how the theorem may be modified when g

(F),

respectively, g

(F)

in the neighborhood of

A

only. See also Remark 4.3. Remark4.5. Theorems4.1 and 4.3 arecomplementary,which isanalogoustothe theorems oftrace andextension in usual Sobolev

spaces

onsmoothdomain. Namely,

F

ifg

,(,),

0</,a<1/2

(respectively,

g

.(F),

1/2<a<

1)j 1,2, thenwehavean

extension

byfunction G

(f),/3

fl,,1

+,

fl,+l

fl,.2

+

-

(spectively,

fl,

,.1

_1/2

i+1

i.2--1/2),

and if G

(I)

then

G[r,

g

,+(Ii),

for

1/2

</3,,/3,+1

<

1 (respectively, g

,+(F,),

/3,.1

fl,

+1/2,

fl,,2

fl,+l

+1/2

for 0

<

778 I. BABU;KA AND B. Q. GUO

THEOREM

4.4..,Let

12 be a

straight

polygon with the edges Fi, i=1,.-.,

M,

and

letg93(F),

0<3,<1/2,

i=1,2,

i=fli+1/2,

i=1,2(respectively, g

(F),

1/2</,

<

1,

fl

fl-1/2,

i=1,

2).

Then there is usuch thatwe havethefollowing" (i) u

2)(1))

with 0

<

flj <1,j-3,.

.,

Marbitrary.

(ii)

Ulr

‘=

gand

Ulr

j=O,

j=2,

..,

M.

Proof

By Lemma 4.14,

=g4(r),

respectively,

2)(F1)

and

(A,)=0,

i=2,3,and hence by Theorem4.2there is v

23(1)

such that v on

F1

and v=0 onFj,j 2,

,

M. By Lemma4.15thefunctionV(I)-1_{has the}desiredproperties. [3

Theorem4.4leads immediatelyto Theorem4.5.

THEOREM4.5. Letf be astraightpolygon with theedges

F,

1,...,

M,

and let i), ]i

(i,1, i,2),

0

<

3i,1, 3i,2

<

1/2

fl-i.l

i.,

+1/2,

fl-/.2=/i.2+1/2,

Qc {1,

M}

or

g6

2),(r,),

fl,

(fl,.1, fl,.2),

1/2

<

_i.1,

fli.2

(_1,

fl,l=/,,1-1/2,

fl,2=/,.2-1/2,

ieO{1,’..,M}.

Let

y=U,Q,. en

z(y)

where

,=max(,_l.2,,.1),_

A,_

y (ifi-lq or

Q

when we

define

-1.

O,

respectively,

.1

O)

and 0

<

i

<

1 arbitrarily

for

AT.

Remark 4.6. Itis obvious how Theorem4.4 has tobemodifiedwheng

(F),

respectively, g

(F)

in the neighborhood of

A

only. See Remark4.3.

Theorems 4.3 and 4.5 give the characterization ofthe boundary conditions that guarantees that thesolution ofanelliptic paialdifferential equationof secondorder

withanalyticcoefficients on a domainfl withpiecewise analytic boundarybelongsto

(fl)

or

()

(see

Theorems 3.2 and

3.3).

Intheconcrete casestheseconditions are usually very easyto check. Letusstate auseful lemma that characterizes the space

(I)

(respectively,

(I)).

LEMMA4.16. Let

a=

{z

x

+

iylx

I, lyl

p(x),

>

O)

and

G(z)

be a holomorphic

function

on

12

such that

for

/2--(/21, /22)

Letg(x) Re

G(z)[x

orIm

G(z)[I.

Then

for

u,

>

-1/2+

(j

1),/,

+

u,>

1/2+

(j

1),

0

</,

<

1, i=l,2, j=0,1,2

g(x)6

?O(I).

Proof

Bythe Cauchy formulawe have for k

>

0

Ig)(x)/__<

C4,(x)(4,(x))-k!

-.

Hence,

O,_l+t]g(k)(x)

dx<=(Ck!

a

Ov+/_ dx<=(Cld

!)2

-1

and hence forui

>

--12,

g

H(I).

Thelemmaisproved forj 1. The

proof

of thecase j 0 isanalogous.Letusconsider nowthecase j 2.Weseethatfor,i+/3

>

andk-> 2

,_+

a

g*)

(x)

ax<=(Ck!a-)

_+_+

dx<(Cldkkl)

2

Fuhermore,if

v>,

then also g

Hi(I).

Instead of

G(z)l

C(Re

z)

we can assume that

]G(z)-P(z)l

C(Re

z)

where

P(z)

is apolynomial.

Lemma 4.16 isvery useful in practice. For example, if g is analytic on

F

then g(x) can beextendedinto someneighborhood of andtherefore g

(I).

Lemma 4.16characterizes very well the structure of the spaces

(1)

(respectively,

(1)).

LEPTA4.17. Let

g(I),

0<<.

en

thereexists a>0 such that g can be analytically extendedonto and

(z)-g(-1)-g(1),(1-x)

(x+l)2,

C+/_(Rex)

(g C

([)

by Theorem

3.1).

Proof.

Since g

(I)

we haveby Lemma4.3 for k_->₁

[gk)(X)l

<_

Hencethe series

g’(x)

E

g(k+l)(Xo)(X--Xo)kkl,

XoeI k=O

is absolutely convergent for

Ix-Xol

<=1/2(dP(Xo)/d),

and hence also

O’(z)=

Z

g+l)(xo)(Z-Xo)

1

k=O k!

converges for

Iz

Xol<1/2(P(Xo)/d)

and

IG’(z)I<=Cp

-

+l/2(XO),

X0 Re

(z),

and C is

independent ofXo, whichyields the lemma. [3

Sofar we have assumedthat12is astraight polygon. Wedid not excludethecase where the internal angle is 2r; i.e., we did not exclude the slit domain. Let us now consider the curvilinear polygon andassume that it is a Lipschitziandomain. Letus

prove first Lemma 4.18.

LEMMA 4.18. Let

[’--{Xl,X2l-l<xl<l,O<x2<h(x),h(x)>o(Xl+l),

h(-1)=0,

a>0}.

Assume that q,(x,x2) is an analytic

function

on S= {Xl,

x21

(Xl

+ 1)2+ x-<-4}

such thatwehave the following"

(i)

ff/(Xl,

h(x,)) 0;

(ii)

Od//OXl(X, O) >

a

> O,

-1<- Xl<-1.

Define

F1

{X1,

X21-1

<

xl

<

1,x2

0},

F2

{x,

x21-1

<

X

<

1,x_

h(x)},

and let T fl(q_{$1 where$1}

{

_r,

0[0

_<0

<

27r,0

<

r

<

1

}

where

r,

O)

are

polar

coordi-nateswith respectto

(-1,

0)

and

T*=

$1-T. Letg

2)(F1),

0<

fl <1/2,/31

_=/32

(respec-tively,

g2=(F1),

1/2</3,<1,/31

=f12),

g,(-1)=0, i=1,2 and ep=r. Then there exists

780

.

BABUKA AND B. Q. GUO

(respectively,

V2

8

(

T),

V*2

(

T*),

1/2)

suchthat

V

g, and

V*

g,on

F1

and

V,

V*i

=0 on

F2CI

.

Proof.

Let

o(r, 0)

(r, O)/r.

Then

(r, 0)

O(Xl)

is analyticon

F1

and

(Xl)>

6>0; hence

-l(x)

is analytic on

x

too. In addition, =0 on

F2.

Furthermore,

IDO(Xl,X2)l

<-

Clal

!-Ildl’l

by Cauchy’s theorem on the theory of two complex variables. Define

1

glo-l(Xl).

Then

ffl

e

(F)

andby Lemma4.11there exists

U1

on

S1

such that

U1

e

($1),

=/+1/2

and

Ullrl

1.

Now define

V1

UI.

Using

Lemma 4.13, we conclude that

Vle(T)

(respectively,

(T*)),

Vllr,=gl

and

VIlF2

0. The proof that V2has desired propertiesis analogous.

LEMMA4.19. Let {Xl,

x2[-1

<

_Xl

<

1,hi(x1)

<

_x2

< hE(X), hl(Xl) < -a(Xl

+

1),

hE(X)> a(xl+l),

a>0, hi(-1)=0, hi(Xl) analytic

functions

on

I,

i-1,2} and

ri

{x,

x21-1

<

x

<

1,x2

h,(x)},

a.

=aN

S.,

S,,={r,

OlO<O<=2mO<r<rt,q>O},

where

(r, O)

arepolar

coordina.tes

with the origin at

(-1,

0). Letg e

}(F1),

0</3 (respectively, g2e

}(F1),

1/2</3i

< 1),/31

=/32.,

gi(-1)=0 and let r. Then thereexist rl

>

0 and V e

(’

rl

),

VI

e

(’

),

--

1/2

q-E, E

>

0 arbitrary (respectively, V2

(On),

V2

(l-l*n),

=

-1/2+

e)

such that

V[rlnfi,

g, and

V[r2nfi,

=0.

Proof

Because

hl(Xl)

is analytic on I it can be analytically extended onto

I

{-1

6

<

Xl

<

1

+

6}.

Thenthe mapping M"

(Xl,

x2)

-

(Yl,

Y2),

Yl Xl, Y2 x2-

hl(Xl)

isanalytic on

n,

r/=

6/2

and

M(fn)=

12_n. For r/1 sufficiently small we have

S.,

F*

U

F*

where

F*

{Yl,

Y2

l-1

<

Yl

<

--1

+

r/,y_

0},

F2*

{y,

Y21-1

<

y<-1

+

g/a, Y2

h*(y)=

h2(yl)-h(yl)},

and

h2(Yl) > al(Yl+ 1).

In addition, it is easy to see that

IJI,

IJ-1l

</x

<

a, where J is

the Jacobian of the mapping M. Because

h*(yl)

is analytic on -1

<-Yl

-< -1

+

1

we define O(yl,

Y2)=-y2+ h2*(yl)

and

O(Yl, Y2)

has the properties in Lemma4.18. Now using Corollaries 4.4, 4.5,

gle}(F1),

g2e}(F1),

and hence using Lemma 4.6,

gl(M-l(y))ly2=o

t(F1

*

),

g2(M-l(y))ly2__oe

}(F*).

Using Lemmas 4.8 and 4.18,we

obtain functions

Va

and

V*

(respectively,

V2

and

V2)on

nf’lS2

(respectively,

ln*

f’!

S,),

which belong to

3}+/2(12

_nf-I

S,)

(respectively,

23}+,/2(1*

_nf’l

S,)).

Now whenwe use Lemmas 4.7 and 2.3, ourlemma follows. [q

The lemma leads to the following theorem.

THEOREM4.6. Theorems 4.3 and4.5 hold also

for

aLipschitziancurvilinearpolygon when

fli

are replacedby

fl +

e, e

>

0 arbitrary.

Proof

Becausetheedgesare analyticcurves andg are analytic on

F

(but

noton

F)

we show similarly

(as

in the proof of Theorem

4.1)

that the solution u of the

Laplace equationbelongs to

23+(f).

This can be done identically as inthe proofs. of Theorems 3.3 and 3.4 of[6], showing that u

+(tq).

[3

Remark4.7. Comparing the respective theorems for straight andcurvilinear poly-gons, we see thatin the latter case we lose slightly in the regularity. It is not known

whether this loss canbe removed.

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[2] I. BABUKAANDM. R. DORR, Errorestimatesforthecombinedh andpversion_ofthefiniteelement

method, Numer.Math., 37(1981),pp. 257-277.

[3] I.BABU,KAANDB.Q.Guo,Regularity_ofthesolutionofellipticproblemswith piecewiseanalyticdata.

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[4]

.,

Theh-pversionoffiniteelementmethod withcurvedboundary, SIAM J.Numer.Anal.,25(1988),

pp. 837-861.

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[Sb]

.,

Theh-pversionoffiniteelementmethod,part 2:general results and applications,Comp.Mech.,

(1986),pp. 203-220.

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York, 1980.

[7] G.H.HARDY,J. E.LITTLEWOOD,ANDG.POLYA,Inequality,Secondedition, Cambridge University

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