rlth*0, if hef(O), h*0

Under the assumption

sup _ll a,llo,rr,r+1y <

-,

Lemma 6.3 and Rellich's compactness theorem guarantee the ^existence

of

a sub-sequence

zl,

converging

in

Z,(O(^S+1)) and hence

in H?#(q to

an element

u$?#@).

Clearly, rz ^satisfies (6.12) and (6.13)

with .q;:0,

^{as well as}

(6.16)

(ulLh)s,j2:

0

for

all

h€rf@).

AIso,

z

^is

in

"r(O):

With X

^{and u2 as in the proof of Lemma 6.3,}

f ^::

^PIL)u,

belongs

to 9(R')

and by Lemma 6.2 (i) there exists a unique solution ,?€"/(R')

of

P(DA: i.

Putting

u,,

:: (l-X)un,, i,, : _{f lqu;}

and continuing these functions by zero, we see from interior regularity results, that

,tT

^ll(t

*

lxlz)r

(l-i)il,,

"" :

o'

Hence, by (6.5),

u,

^tends

to fr in

H?k@). But then

uz:fr.(J(R");

thus uU(A1.

It

^remains to lead the assumption

,lim ^ll

z,llo,e1s+r):-Exterior boundary value problems for elliptic equations

to a contradiction. For this we consider the normalized sequence w,

:

_{ll uoll o, a(s + ry} -14,,.

As above we can conclude that a subsequence 1e;, conv€rges

in

Hffi(CI) to a solu-tion

w(J(Q) of P(L)w:0,yrw:0,

which satisfies (6.16). By the choice

of r/

this implies

w:0,

which contradicts the normalization

of

^wo,.

Let us now consider problem (6.12;6.13) with nonzero

g;. It

is possible to construct a function

wct*!-(Q

^satisfying

(6.17)

^Tiw

: 9i' i :

^O'

"'' ffi-l'

supp wnsupp rlr

:

0.

Green's identity ^gives

for u(.,{*(A)

(r

1r1wlv)o, o

: _{2) G}

1lr, ryo.,.

Hence,

if

^(6.14) is valid, we have

(f-P(L)wlo)0,, : 0 for all

u(,/r*(Q).

By the first part

of

the proof there is a solution

{i<J@) of P(L)fr:f _-P(L)w, y.ifr:O.

Moreover, the function

fi

satisfies (6.16). Then the function

u:: il*w

solves (6.12; 6.13) and, in addition, by (6.17), satisfies (6.16).

Since condition (6.14)

is

also necessary

for

the solvability, Theorem 6.4 is proved.

From the proofofTheorem 6,4 one easily deduces a pseudoinverse for problem (6.12;6.13)

o-'

^: x (a)

x'fi' ^{p'- -' -'t' (t)} *

J (a) j:o

which is continuous as a mapping from

H!,,(qXIIf=or pzm-i-rtz(f)

_{into äffi(O).}

For this we choose

a

nonnegative testfunction

E€0(A)

^{such that the mapping}

.,(*(Q)*K19), u-eu is

injective. Denoting

by

^uL,...,Dd

a

basis

of ,f*(A)

for which

(rpuilui\s,e: öii,

the mapping

Q:

K(o)xfi' g'^-t-'t2(f) _*1((o)x fi' g'*-i-'21r1,

j:0 j:r

e(f; s): (f-

_i:t

* «f,,.;,,,o-Z'k,lr,u,)o,,)vu,;

_j:o ^s)

is a

continuous projection

of K(O)XII!;LHzn-i-u2(f)

onto the range

of

the mapping

g

_i

J(O)*K@)X ilf;'

^Hzn-i

^-uz(I'),

^where

39

(6.18)

9u - ^(P(L)u)

^{loul., .}

..,y*-fllr).

(6.19)

40 J. SnnnNEN and K. J. Wrrscrl

A

pseudoinverse

of I ^is

^{given by assigning}

^to

^any

(f;go,.",8^-t)

the unique solution

u€J(A) of

P(L)u

: f, !iu : _8i, i :

^{O, ...,}

^m-1,

with the property (6.16).

Hete (f

_;80,

...,8.-r):Qff,80, "',8,-r)'

We have

to

show the continuity

of 9-r.

^Since

Q

^is continuous,

9-r

^is

continuous

if

llullr*,o(R)

=

^{g(ttr Q)u|| 0, o(s)}

⁺ 2t

^{lly iullr,,-} j -Lp, r) J:0

holds

for

any

u€.I(A)

with the properties (6.16-6.18). Estimate (6.19) has to be shown for any Å,

§,

^{and the constant}

^{c:c(R, S)}

^{may not depend}

^{on a. If}

^(6.15)

were wrong, ttrere would exist numbers Ä, ,S>0 and a sequence uo(J(Q) ^such that

lluollzm,e(R)

: l,

l.-3 ^(tt

r

@)uoli,, ,,(s)

.Zilv

iuollr,,- i -ttz,r)

:

0

and that (6.15) are valid. Without ^loss of generality, we may suppose that ^.R

>'S* l.

Then by Lemma 6.3 and Rellich's compactness theorem there ^exists a subsequence

uk,,

cotyerging to some

a in ^Hffi(O).

^{As in the proof of Theorem 6.4, we can}

conclude,

that u

^belongs

to

.,4r(A) and ^{hence vanishes} by (6.16). This contra-dicts (6.20).

We have shown that for any exterior domain

O

with ^smooth boundary ^the Dirichlet operator, as defined by (6.18), is a weakly ^indexed operator with index

0

and admits a pseudoinverse

0-,

^which

^is

continuous as

a

mapping from

K@)xJIf;t gzn-i-rtz(f) _into

_H?*@) and for which

Q:gth-' ^it

^{a continuous}

projection

of

iK(O) onto the ^range

of 9.

^Since the adjoint Dirichlet problem ^is a problem of the same kind, this is true

for 9* too.

^{Therefore all} assumptions

(A1)-(A6)

are valid

in

the case under discussion and one can conclude that Theorem 3.6

is

applicable

for

exterior boundary value ^problems

with

respect to the operator

A-

P(L).

Remark. To

solve problem

Q.l5),

one could ^proceed as

in

the proof of Theorem 6.4, using estimate (6.5)

with

1l; ^replaced

by Bi.

However, this would yield neither the index of the problem nor the finiteness of the codimension of ^the range.

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J. Saranen

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