Z angew Math Phys (ZAMP)

43(1992)

0044-2275/92/061023-15 $ 1.50 + 0.20 9 1992 Birkh~iuser Verlag, Basel

On strong solutions of Poisson's equation in Beppo Levi spaces By Paul Deuring and Werner Varnhorn, Fachbereich Mathematik, Technische Hochschule Darmstadt, Schlossgartenstr. 7, D 6100 Darmstadt, Deutschland 1. Introduction

Let G ___Rn (n-> 2) be an unbounded open set having a compact complement and a smooth boundary 3G of class C 2. In G we consider Poisson's equation concerning some scalar function u: -Au =f

in G,

ulna = ~ .

(1.1)

H e r e f i s given in G and ~ is the boundary value prescribed on OG. As usual, A denotes the Laplacian in N". It is well known that in unbounded domains the treatment of differential equations causes special difficulties, and that the usual Sobolev spaces wm'q(G) are not adequate in this case: Even for the Laplacian in E" we find [8] that the operator A: wrn,q([]~n) ~ wrn-2,q(~ n) is not in general a Fredholm operator, as it is in the case of bounded domains [19]. Therefore, in exterior domains the equations (1.1) have mostly been studied in connection with weight functions: Either (1.1) has been solved in weighted Sobolev spaces directly [9, 15, 17], or it has first been multiplied by some weights and then been solved in standard Sobolev spaces [24]. The aim of the present note is to prove the solvability of (1.1) in Beppo Levi spaces z2'q(G) (1 < q < o0) of the following type [4, 14]: Let Lq(G) be the space of functions defined almost everywhere in G such that the norm

Ilfllq,G=(fGLf(x) lqdX) 89 is finite. Then L2'q(G) is the space of all functions being locally i n Lq(G) (see the notations below) and having all second order distributional derivatives in Lq(G). We show that for f given in Lq(G) and some boundary value ~ W 2- l/q'q(~G) there always exists a solution u ~ L2"q(a), which we call a strong solution (compare [21], where similar spaces are used for the corresponding weak problem). Concerning the uniqueness of this solution we prove that the space of all u ~L2"q(G) satisfying (1.1) with f = 0 and

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q~ = 0 has the dimension n + 1, independent of q. This result also holds for the case n = 2. To start with our notation, throughout this paper we define

G = ~n\G b (17 > 2), where ~b -r Gb is an open b o u n d e d set with closure Gb and b o u n d a r y ~?Gb = 0G of class C 2

(1.2)

(compare [1, p. 67]). Thus, in particular, G and Gb need not to be connected. If Br = B~(0) denotes an open ball with center at zero and sufficiently large radius r > 0 such that

Ix - y[ >- [x[/2

OG___B~ and

for a l l y e 0 G ,

xeN~\Br,

(1.3)

then we define G~ = G ~ B~. Here I" ] is the Euclidean norm in ~". All functions appearing in this paper are assumed to be real valued. Let D ~ ~" be any open non-empty set with a compact b o u n d a r y ~?D o f class C 2, or let D = ~ . Besides the spaces Lq(O) w e need the well known function spaces Cm(D) (m e No = {0, 1, 2 , . . . }; C~ = C(D)), C~176 C~(D), and the space C;~ = {f~ I f e C~(R")}. We call a function u locally in Lq(D) (1 < q < o0) and write u e Lfo~(b) if u e Lq(D c~ B) for every ball B ~_ ~". N o t e that this space in general does not coincide with the space Lfoc(D) = {u: D ~ R [ u ~ Lq(B) for every ball B with B ___O }. By wm'q(D) (m =-O, 1, 2; w~ =. zq(o)) w e mean the usual Sobolev space of functions u such that D~'u ~Lq(D) for all multiindices c~ = (~l, 9 9 9 0~,) e Ng with ~1 + " 9 9+ (~n ~-~ m [ 1]. Here we used

D~u = D~ID~ . . . . D~"u, (i =

1 , . . .,

Di = ~/~X i

n ; x = ( x 1. . . .

, Xn) E ~n).

The spaces W~'cq(D) are defined analogously. We need the fractional order space W 2- 1/q'q(OD), which contains the trace UOD of all u ~ w2"q(o) [1, p. 216]. The norm in W 2 - l/q'q([3D)is denoted

by 11"12-l/q,q,3D The term V u = ( D j u ) j = l ...... is the gradient of u and V 2 u = (D~Dju)ij= 1..... , means the system of all second order derivatives of u. F o r these terms we define the seminorms

ILVullq,D=

(

I[OkU[Iq,D) 1/q

[IV2Ullq,D=

k=l

(

)l,q

~ [[OjOkl.g[Iq,D j,k=l

and introduce for m = 1, 2 and 1 < q < oe the Beppo Levi spaces

tm,q(o) = {bl E Z~oc(~)

[ llV'~ullq,D <

o0}.

(1.4)

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Finatly, concerning the norms and seminorms, we sometimes omit the domain of definition if it is obvious and use II Itq or [r 112- Jq,q instead of II IIq, or tl r12-1/q,q, OG, for example.

2. Potential theory Besides the Poisson equation (1.1) we also consider the special case of Laplace' equation with Dirichlet b o u n d a r y condition -Au=0

in G,

utaG=qb.

(2.1)

If 9 e C(c~G) is given, and, in addition for the case n = 2, some constant a e ~ is given, then there is at most one function u E C2(G) ~ C ( G ) satisfying (2.1) and the decay conditions

u(x)-aln]x I=O(1) Vmu(x) = O(]x[2 . . . .

u(x)=O(lxl z-n)

(n=2),

(n->3), (2.2)

) (n >- 2; m = 1, 2),

as Ix[--+ oo. This follows easily from the m a x i m u m principle for harmonic functions (n > 3) and Kelvin's transformation of harmonic functions being b o u n d e d at infinity (n = 2; compare [18, p. 523]). To show the existence of such a solution and, moreover, to develop a suitable representation, in the following we use methods of potential theory. Let En (n -> 2) denote the fundamental solution of the Laplacian such that - A E , ( x ) = 3(x), where 6 is Dirac's distribution in Nn. It is well k n o w n that E2(x) -

lnixl co2

(n = 2),

En(x) -

[xl2-"

(n - 2)con

(n >- 3),

(2.3)

where con is the area of the unit sphere in ~n (n --- 2). Let us define the single layer potential

(En|

= t" E,,(x - y)| doG

doy (x ~ ~n),

the double layer potential

(DnO)(x) = - [ " ON(y~En(X -- y)O(y) doy (x ~ ~'~), doG and the normal derivative of the single layer potential

(/-/no)(x)

-["

~3N(~)E,,(x --y)|

cloy (x ~ U).

deG Here and in the following, N = N(z) is the outward (with respect to the b o u n d e d set Gb = ~ n \ d ) unit normal vector in z e ~G, U is a suitable

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neighbourhood of aG, ~ ~ ~G is the unique orthogonal projection of x ~ U onto ~G, and | E C(~G) is the u n k n o w n source density. The existence of (DnO)(x) and (H'~O)(x) in x ~ ~G, in particular, follows from the estimate (3.6) below (note that (3.6) also holds for n =2). Clearly, the above potentials E~| and D~| are smooth harmonic functions in x ~ 0G, and we have the continuity properties e = (En|

(En|

E"|

on o a

(2.4)

as well as the j u m p relations ono

-- ( D n |

nn|

__ ( n n o ) e

e = (on| = (nn|

i-

on|

=

nn|

=

89 - ~10

o n 0G, on

~3G.

(2.5) (2.6)

Here the index e stands for the limit from outside, and the index i for the limit from inside. 2.7 Theorem. Let G _ En (n - 2) be an unbounded set as in (1.2) with boundary OG of class C 2, and let 9 ~ C(~G) be given. In addition, if n = 2, let a s R be given. Then there is one and only one function u e C~(G)nC(G) satisfying (2.1) and the decay conditions (2.2). This solution admits in G the following representation: If n -> 3, then for any with 0 < ~ ~ R we have u = Dn| - ~E~|

(2.8)

where | e C(OG) is the uniquely determined solution of the boundary integral equation = - - ~10 -k D n | - - ~ E n | on OG. (2.9) Ifn=2,

thenfor any~,~with0<~R,

0~/~[~wehave (2.10)

u = --aoo2E21 + D 2 0 -- ~ E 2 0 * - ~bo.

Here a e R is the above given constant appearing in (2.2), E21 is the single layer potential with constant density q' = 1, b| = f

.)aG

O(y) doy

is some constant, and | |

= |

is defined by (2.11)

- bo/(meas(OG)),

where | ~ C(~G) is the uniquely determined solution of the boundary integral equation + a ~ o 2 E 2 1 = - - ~i0

+ 020

-- ~E2|

* --/3bo

o n c3G.

(2.12)

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Proof. Let us first assume n-> 3. Then by (2.4) and (2.5) the ansatz (2.8) leads to the second kind Fredholm boundary integral equation (2.9) for the u n k n o w n source density | ~ C(~G). To see that (2.9) is uniquely solvable for all boundary values @ ~ C(0G), let 0 -r ~ e C(~G) be a solution of the homogeneous adjoint integral equation 0-----

- - ~1

+ H n ~ P - ~En~P

on 0G.

(2.13)

By (2.4) and (2.6) this means on c3G e(g~'~P) = (H"q0 i = -

(ONE'S)',

(2.14)

and Green's first identity yields

f~ [V(En~)[2 dx = ~ (EnW)(ONEnW)i b G

do =

G

IE~

do.

Because a > 0 we find EnLP = 0 in (~b. This implies (End) e = 0 using (2.4), and the uniqueness statement above yields E"~P = 0 in G, too. Thus E " ~ = 0 in the whole An, hence H ~ P = 0 in G and in Gb, and we obtain ~g = 0 by (2.6), as asserted. This shows that (2.9) is uniquely solvable in C(6G) for all n - 3. Now let n = 2. Then the above ansatz (2.10) satisfies the decay condition (2.2). In particular, because bo. = SaG | d% = O we obtain

E2O*(x)

= (bo,)(ln[x[)coy' + E2O*(x) = ('021

t" |

lnlx/(x -- Y)[ cloy =

o(1)

2a G

as ]x[ ~ m. Here again, (2.4) and (2.5) lead to the second kind Fredholm boundary integral equation (2.12). To see that (2.12) has a unique solution | for all b o u n d a r y values @eC(~3G) and all a e N , let 0 r q~ ~ C(c3G) solve the homogeneous adjoint integral equation 0 = - s q ? + H 2 ~ - e(E2tp) * -

flbw

on aG.

(2.15)

Because for any constant c e N we have - 1 / 2 c + OZc - - 0 [18, p. 511], and because c* = 0 (see (2.11) for the definition o f c*) implies EZc * --- 0, we find 0 = (c, - 89

+ H2q j - ,(E2W) * -/~b~v > =

-fl,

where here (@, ~o) = ~aG @(Y)cP(Y)doy denotes the corresponding duality. It follows bw = 0 and W * = Lp. Using (2.4) and (2.6), from (2.15) we find on 0G ~(E2qj), = ( H 2 W ) / _ - _ ( ~ N E 2 t I 1) i, and Green's first identity yields

feb [V(E2qJ)12dx = --~ ~c (E2~)(E2U?)* d~ -~ ~G '(EZqJ)*[~d~

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Paul Deuring and Werner Varnhorn

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Here the last equality follows f r o m ~G (E2~) * do = 0. Since ~ > 0, this implies (E2q/)* = 0 on ~G, and in Gb we obtain E 2 ~ = CE with the constant Ce = (meas(OG)) -~ f

3aG

E2q/ do.

On the other hand, the function E2~Ij - - C E is a solution of the exterior Dirichlet problem (2.1) with q~ = 0, and it satisfies the decay condition (2.2) for a = 0 (note that E2~--=o(1) as Ix[--*oo since b~,=0). The above mentioned uniqueness statement yields E2qJ--CE=O in (~, hence E 2 ~ = CE = const, in the whole R 2. N o w the j u m p relations (2.6) imply --0, and the theorem is proved. 2.16 Remark. The ansatz (2.8) has been used for the Helmholtz equation in [2, 13]. N o t e that only one single layer potential is sufficient, even if the complementing b o u n d e d open set Gb is not connected (compare [12, p. 137]). A n ansatz similar to (2.10) has been used in [11] for the case of the stationary Stokes equations.

3. Estimates in Sobolev norms

Let us start with the following uniqueness statement: 3.1 Lemma. Let G ___En (n -> 2) be an u n b o u n d e d set as in (1.2) with b o u n d a r y 0G of class C 2, and let 9 e W 2- 1/q'q(OG) be given. If n = 2, let a ~ R be given, in addition. Then there is at most one function u ~ L2'q(G)~C~ satisfying the Laplace equation (2.1) and the decay conditions (2.2) with the constant a given above if n = 2. Proof. Let u = u ~ - - u 2 be the difference of two solutions u ~ and u 2 with the required decay properties (2.2) above. Define the b o u n d e d set Gr = G n Br with (1.3). Then from the local regularity theory we conclude that Dju ~ CI(G,.) (j = 1 , . . . , n). Thus in Gr we m a y apply Greens first identity, obtaining

fc ,Vu,2 dx = fo r

(~NU)UdO,

(3.2)

Or

because the b o u n d a r y integral over OG vanishes. Here N denotes the outward (with respect to Gr) unit normal vector on the b o u n d a r y ~Br and 9NU is the normal derivative of u. N o w due to the decay properties of u, the right h a n d side in (3.2) tends to zero as r ~ oe. This is obvious if n -> 3. For n = 2, using the expansion theorem for harmonic functions at infinity

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Strong solutions of Poisson's equation in Beppo Levi spaces

1029

[18, p. 523], we find u ( x ) = O ( 1 ) and V u ( x ) = O(]x] -2) as [ x [ ~ o o , which implies the assertion above, too. It follows Vu = 0 in G, because u vanishes on the b o u n d a r y ~?G. This proves the uniqueness. To show the existence of solutions in the Beppo Levi space L2"q(G) we first treat the case n = 2: 3.3 Theorem. Let G ~ ~2 be an u n b o u n d e d set as in (1.2) with boundary OG of class C 2, and let q~ E W 2 - 1/q'q(OG) and a e R be given. Then there is one and only one function u ~ L2'q(G) ~ C ~ 1 7 6 satisfying (2.1) and the decay conditions (2.2) for n = 2. Proo~ Because n = 2 we have ( 2 - 1 / q ) q = 2 q - l > l = n - 1 , and Sobolev's L e m m a [1] implies ~b e C(SG). Thus we can apply Theorem 2.7, obtaining a uniquely determined function | C(~?G) satisfying the b o u n d a r y integral equation (2.12). The function u ~ C~176 ~ C(G) defined by (2.10) fulfills (2.1) as well as the decay condition (2.2) for n = 2 , as shown above. Because the uniqueness has been established in L e m m a 3.1, it remains to show u ~ L2"q(G). To do so, let us recall the definition Gr = G n Br (see (1.3)). We obtain U ~ W 2 - l/q'q(~Gr) , because u e C a ( G ) implies u e W 2 - 1/q'q(~Br) (see [6, p. 238]), and because u = qJ e W 2 - 1/q,q@G) on OG. Due to u e Coo(G~) n C(G~) this implies u ~ w2"q(Gr) (see [6, p. 232], which is based on [19, p. 184]), and it remains to estimate the second order derivatives of u for Ix[ -> r. Using (2.10) we see that [D~Dju(x)] r ( k , j = 1, 2), which gives D k D j u e Lq(l~2\Br) for all 1 < q < oo. Thus u ~ L2'q(G) as asserted and the theorem is proved.

The preceding arguments could be used for the case n > 3 and q > n/2 as well, because, due to ( 2 - 1 / q ) q > n - 1, Sobolev's L e m m a [1] would imply (I) e C(OG) as for n = 2. The case n -> 3 and q < n/2, however, would not be included. Therefore, in the following we develop another approach which works for any q with 1 < q < oo and any n > 3. 3.4 Lemma. Let G _c Nn (n >- 3) be an u n b o u n d e d set as in (1.2) with ~?G of class C 2. For | ~ Lq(SG), 1 < q < oo, set Tq|

=

Dn| - ~En|

(0 < ~ e ~)

with the double layer potential D"| Then the operator Tq: L q(~?G) - , L q(~3G)

is well defined and bounded.

(3.5) and the single layer potential E"|

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Proof. Because G has a c o m p a c t b o u n d a r y ~G o f class C 2 we have [N(x). (x - y)[ -< c[x

-

y]2

(x, y ~ ~G).

Here c is some c o n s t a n t d e p e n d i n g on ~G, N(x) is a unit n o r m a l vector in x ~ ~G, a n d the d o t 9 m e a n s the scalar p r o d u c t in ~n. The above estimate implies [ON(y)En(X

-- y)[ -<

c[x -- y[2-n

(x, y ~ OG)

(3.6)

a n d hence

On

[OU(y)E,,(x--y) --czE,,(x -Y)I <--c=]x -Yl 2-" (x,y tOG). the other h a n d , if 19 e Lq(~3G), by H61der's inequality we

)q

_=

(3.7) obtain

)q

< foa (f~G [x-Y[Z-ndoy)q-l(fo

[x-Y[2-"]19(Y)[qd~176

a n d c o m b i n i n g b o t h estimates (3.7) a n d (3.8), the l e m m a is proved. 3.9 L e m m a . Let G __ N~ (n > 3) be an u n b o u n d e d set as in (1.2) with ~G o f class C 2. Let | e Lq(~G), 1 < q < oo, be given with

- 1/219 + rq19

=

(3.10)

O.

T h e n there is some n u m b e r p with n - 1 < p < oo such t h a t 19 e

LP(OG).

Proof. Because o f (3.10) it suffices to consider the case 1 < q < n - 1. If q = n - 1 we have 19 e L r(c3G)for all 1 < r < n - 1, because OG is b o u n d e d . T h u s let 1 < q < n - 1. A p p l y i n g the H a r d y - L i t t l e w o o d - S o b o l e v inequality [23, p. 119] we o b t a i n Tq19E Ls(~3G) with

IITq19LG cql11911q,

(1/s = 1/q-

1/(n -- 1)).

N o t e t h a t s > q. Because o f (3.10) we find 19 e LS(OG), too, a n d we can repeat this procedure. After a finite n u m b e r o f times, finally, we o b t a i n 19 ~ LP(~G) with some p as asserted, a n d the l e m m a is proved. 3.11 L e m m a . L e t G _~ ~" (n -> 3) be an u n b o u n d e d set as in (1.2) with 0G o f class C 2, a n d let 1 < q < oo. Consider the sets

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Strong solutions of Poisson's equation in Beppo Levi spaces

A = {0 ~ Lq(~G) [ - - ~10 --[-Tq| = 0},

B

I -~O

=

+ D " O - ocE"O = 0}.

Then A = B = {0}. Proof. Because B = {0} by Theorem 2.7 and because B _ A we only have to show A ___B. Let | ~ A. Then, using the estimate (3.7), H61der's inequality, and L e m m a 3.9, we conclude that | is bounded on OG:

I|

=

2]zqO(x)[ < c f

Ix -

y[2-nlO(y)]

doy

de G

Here p > n - 1 is the number from L e m m a 3.9, and the first integral on the right hand side is finite due to ( n - 2 ) p ' < n - 1 iff p > n - 1. It follows from the boundedness of O that Tq| is continuous (compare [10, p. 42] for the case n --3). Thus (3.10) yields the continuity o f | and the temma is proved. 3.12 Lemma. Let G ___~n (n -> 3) be an u n b o u n d e d set as in (1.2) with OG of class C 2, and let 1 < q < ~ . Then the operator zq: Lq(c3G) ~ tq(OG) defined in L e m m a 3.4 is compact. For any (I) ~ Lq(~G) there is one and only one | ~ Zq(G) satisfying on 0G (I)=

1 + Tq| --~|

(3.13)

and the estimate -< Cq 1]- 89 + zqOHq, OG

]]| Proofi

~-

Cq H(l)l]q,OG

(3.14)

Define the cut-off kernel

k~(x, Y)

fl

for x , y ~ G otherwise

with ] x - y I - > e

>0

For e > 0 define the operator Tq'~: Lq(~OG) ~ Lq(OG) by (Tq'eO)(x)

= -- f

doG

( a N ( y ) E n ( X -- y ) -- o~En(x --

y))k~(x, y)|

dog.

Because its kernel is bounded, T q,~ is o f H i l l e - T a m a r k i n type and thus a compact operator for any e > 0 (see [12, p. 169]). Using estimates as in (3.8) we find

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hence Tq'e---, T q as a ~ 0 in the operator norm, which implies that T q is a compact operator, too. Now, applying Fredholm theory and the open mapping theorem [26], the last two assertions follow. This proves the lemma. N o w let us return to the equation (2.1). Because of 9 there are functions Ok ~ C2(OG), k ~ N, such that

[Iok-oll2_i/q,q, aG--,o

as k ~ o o .

~

W2-1/q'q(~G) (3.15)

Let Ok be the solution of the boundary integral equation (2.9) with 9 replaced by Ok, corresponding to Theorem 2.7. Then this implies Ok = --~19k 1 -1- Tq19k 9

Moreover, let 19 e Lq(OG) solve 9 = 119

q_ Tq19 '

according to L e m m a 3.12. Then, using (3.14),

L119-19kllq, G- 0

as k - , 0 .

(3.16)

For x ~ G and k ~ N let us define Uk(X ) -= o n 1 9 k ( X ) -- C~En19k(X),

u(x) = D " 1 9 ( x ) -- ~En19(x).

Then, as shown above, uk ~ C ~ ( G ) n C ( G ) satisfies (2.1) with 9 = Ok, and, in particular, uk ~ C2(t?G~), where G~ = G n B r (see (1.3)). Thus we conclude Uk ~ w2"q(Gr) with the following estimate: N(Uk--Ul)II2,q,

Gr

Cq,r([[Uk -- Uz[12_l/q,q,~G JU [[Uk -- UlN2--1/q,q, aBr)

(3.17)

(see [5, p. 340], which is based on [19, p. 184]). Because u k - ut = O k - O! on OG, the first term on the right hand side of (3.17) tends to zero as k, l ~ ~ . For the second term we find [[Uk -- Ul [12--1/q,q, OBr

-< Cq,r[[ok- o, llq, G

using (1.3) (compare [6, p. 238]). Thus, due to (3.16), uk is a Cauchy sequence in w2'q(Gr). Moreover, H61der's inequality together with (3.16) shows that for any x G we have u k ( x ) ~ u(x) as k --. oo, hence u ~ wZ'q(Gr) with [lu - uk 112,q.Gr ~ 0 as k ~ oo. This implies Ilu - uk [[2- 1/q,q, oa -~ 0 as k ~ ~ , and because uk = Ok on OG, (3.15) yields u = R e WZ-l/q'q(~3G) o n with Au = 0 in G, and because u satisfies the decay ~G. Because u ~ C ~ properties (2.2) for n > 3 , the second order derivatives D k D j u ( x ) ( k , j = 1 . . . . , n) for all x with Ix] -> r can be estimated as in the case n = 2

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(see the p r o o f of T h e o r e m 3.3). Thus u ~ LZq(G) and the following theorem is proved. 3.18 Theorem. Let G _ N" (n >/2) be an u n b o u n d e d set as in (1.2) with b o u n d a r y c3G of class C 2, and let rb ~ W 2 - 1/q,q(~G) be given. In addition, if n = 2 , let a e N be given. Then there is one and only one function u ~ L2'q(G)~ C~176 satisfying (2.1) and the decay conditions (2.2).

4. The Poisson equation The first theorem ensures the solvability of Poisson's equation (1.1) in the Beppo Levi spaces L2"q(G), defined by (1.4). 4.1 Theorem. Let G ___N" (n -> 2) be an u n b o u n d e d set as in (1.2) with b o u n d a r y c3G of class C 2, and let 1 < q < ~ . Then for e v e r y f e Lq(G) and (~) E W 2 - 1/q'q(OG) there exists some u ~ L2'q(G) satisfying the Poisson equation (1.1) in G. Proof. Setting f = 0 in N"/G we obtain some function f e Lq(~ n) with fig = f i n G. L e t ~ ~ C ~ ( R n) denote a sequence such that f~---,fin z q ( ~ n) a s i ~ oo. Consider n o w for fixed i the equation - A t ~ i =./7//in Nn. We can solve it by convolution with En (see (2.3)), obtaining in x e Rn the representation (x) =

9 f, )(x) =

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

Moreover, by the theorem of C a l d e r o n - Z y g m u n d [3], for the second order derivatives we obtain the estimate NV2gi lie-< c][f/[[q with some constant c independent of i e ~, which implies [IV2(~e- ~Tk)[Iq~ 0 as i,k ~ o o . Following [22, p. 6] we consider a sequence of open balls (Bj)j with Bj _ Bj +t and U~=1Bj = ~ . Let us define the space P = {P: x ~ P ( x ) = a + b . x [ b, x ~ ~", a ~ ~}

(4.2)

of linear functions P: ~" -+ ~. Then by the generalized Poincar6 inequality (compare [14, p. 22] or [16, p. 112]) we obtain for every v ~ L2"q([~n) the estimate

IIv]lLq(,,,/p:= i n f ]Iv + PllLq(,j, <- cj IIV21Qt[Lq(Bj)n 2

(4.3)

with some constants cj > 0. Because ~7~e L2"q([~n) we conclude that (~Tz)iis a Cauchy sequence with respect to the n o r m tI'][Lq/Pon the left h a n d side of (4.3) for fixed j = 1. This implies the existence of linear functions P~ s P such that (~Ts+ P~)i is a Cauchy sequence in Lq(B1). Repeating this argument now for j = 2, there exist linear functions Qe ~ [P such that (zT~+ Qi)i is a

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Cauchy sequence in Lq(B2), hence in Lq(B1), because B~ ___B 2. Thus the difference (P~ - Qi)i is a Cauchy sequence in Zq(B1), and using the representation Pi(x) = ~i + ,6i " x,

Qi(x) = 7i + 5i. x,

we obtain that (ai - 7i)i and (/?i - 6~)i are Cauchy sequences in ~ and in ~', respectively. F r o m this we find that ( P ; - Q~)i is a Cauchy sequence in Lq(B2), and thus also (5~ + P~),- = (tT~+ Qi)i + (P~ - Q~)i. Repeating this procedure it follows that (tTz+ P;), is a Cauchy sequence in Lq(Bj) for all j = 1, 2 . . . . . Thus we can find some fie Lzq([~ ") such that (~i -}- Pi) ~ 5

in Lfo~(~"), as

Moreover, ~ satisfies - A5 = f in R n together with the estimate Ilv=~Tllq -< c ][fll q. Since 5 ~ W2,;q(~ ") we find from the usual trace theorem [1, p. 217] that ~tle6 ~ W 2 - 1/q'q(~a). Following T h e o r e m 3.18 there is a function w ~ L2'q(G) satisfying the equations

-Aw=0

in G,

wl0a=Ulaa-~,

where r e W2-1/q'q(OG) is the prescribed b o u n d a r y value. N o w setting u = Z~l~- w we obtain the desired solution and the theorem is proved. Because functions u ~ L2'q(G) have no suitable decay properties at infinity, in general we cannot expect uniqueness for the solution of (1.1) constructed in T h e o r e m 4.1. Thus we consider in G the h o m o g e n e o u s equations and define the nullspace of (1.1) by Nq(G) = {u ~ L2'q(G)]- Au = 0 in G, ul~ = 0}.

(4.4)

4.5 Theorem. Let G _ ~" (n -> 2) be an u n b o u n d e d set as in (1.2) with b o u n d a r y 0G of class C 2, and let 1 < q < oe. Then for the dimension dim Nq(G) of the nullspace defined in (4.4) we have dim N q ( G ) = n + 1 independent of q. The following explicit example is due to Simader, Sohr (compare [21, p. 1]). Let G = R"\/~I denote the complement of the closed unit ball in ~" (n > 2). In this case a basis B = {e0, e l , . . . , en} of Nq(G) is explicitly known: In Ixl eo(X) = [ l x l 2 _ , , _ 1

(n --- 2) (n >- 3)'

ex(X) = xk(1 - I x [ - ' )

(k = 1, 2 , . . . ,

n).

Vol. 43, 1992

Strong solutions of Poisson's equation in Beppo Levi spaces

1035

P r o o f of Theorem 4.5. Consider the space P of linear functions defined in (4.2). Because for every P e P we have P ( x ) = a + b 9 x with some a e and some vector b e E ", we find dim P = n + 1. Let u P denote the uniquely determined solution of the equation

-Au=0,

UI~G=--PleG

with P e P, according to Theorem 3.18. Here in the case n = 2 we require u(x)-a

lnlx I = O ( 1 )

as ]xl--+or

(4.6)

where the constant a is chosen from P ( x ) = a + b 9 x. Setting

Me(C) =

{.P + elsie

p)

we obtain M q ( G ) ~ Nv(G), obviously. Consider the surjective mapping s q : P --+M q ( G ) ,

s q ( e ) = lgP-}- P!~.

Then from the uniqueness mentioned above it follows immediately that S q is linear. Furthermore, S q is injective, which can be shown as follows: Let P ( x ) = a + b . x and let u*' + PF~ = 0 in G. Then from the decay properties o f u p and Vu P established in Theorem 3.18 we find a = 0 and b = 0, hence P = 0. Here in the case n = 2 we obtain a = 0 due to the special choice of the number a in (4.6). Thus S q is linear and bijective, hence dim M q ( G ) = dim P = n + 1, and it remains to show (4.7)

Nq(G) <_ Mq(G).

To do so, let us first determine the null space Nq(N ~) = {u[u ~ L2'q(N n) with - k u

= 0 in ~n}.

F r o m k u = 0 , hence Ag2u = 0 with D2ku e L q ( N ") ( j , k = 1 , . . . , n ) we obtain V2u = 0 in ~ , which implies u = P for some P e P. Thus we have shown that N q ( ~ ' ) = P.

(4.8)

N o w let u e Nq(G). We extend u on the whole space obtaining a function ff e L2,q([~ n) with ~lc = u [1, p. 83]. Moreover, this function satisfies on Nn the identity - k ~ = f e L q ( N " ) , where the function f has a compact support in the b o u n d e d set Rn\~. Consider the equations -kw

=f

in ~ .

(4.9)

Again, it can be solved by convolution with the fundamental solution E, of the Laplacian: We obtain w = En ,27 in ~ and the C a l d e r o n - Z y g m u n d 2 ~ Lr([R ~) for all 1 < r < q (j, k = 1 , . . . , n). Here we theorem implies Djew used 27e L r(~n)" for all 1 < r < q due to its compact support. N o w by a well k n o w n estimate o f H a r d y - L i t t l e w o o d - S o b o l e v - t y p e [23, p. 119] we

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Paul Deuring and Werner Varnhorn

ZAMP

find w ~ L S ( ~ ") for s o m e s > q, hence w ~ L ~ o c ( ~ " ) ~ L ~ o c ( ~ n ) . T h u s we have constructed s o m e s o l u t i o n w o f (4.9) such that w ~ L2'q(R"). Setting W = ~ - w we o b t a i n W ~ Nq([~"), and (4.8) leads to fi = w + P for s o m e P E P. Because UleG = 0 and since ulG = u we find u E Mq(G), w h i c h proves (4.7) and thus the theorem.

Acknowledgement The authors w o u l d like to t h a n k Professor Simader and Professor y o n W a h l for valuable c o m m e n t s .

References

[1] Adams, R. A., Sobolev Spaces. Academic Press, New York 1975. [2] Brakhage, H. and Werner, P., Ober das Diriehlet'sche Auflenraumproblem fiir die Helmholtz'sche Schwingungsgleichung. Arch. Math. 16, 325 329 (1965). [3] Calderon, A. P. and Zygmund, A., On singular integrals. Amer. J. Math. 78, 289-309 (1956). [4] Deny, J. and Lions, J.-L., Les espaees du type de Beppo Levi. Ann. Inst. Fourier 5, 305-370 (1953-1954). [5] Deuring, P., The resolvent problem for the Stokes system in exterior domains: An elementary approach. Math. Meth. Appl. Sci. 13, 335-349 (1990). [6] Deuring, P., Wahl, W. von and Weidemaier, P., Das lineare Stokes-System im R 3 (1. Vorlesungen iiber das lnnenraumproblem). Bayreuther Math. Schriften 27, 1-252 (1988). [7] Dieudonnr, J., Elements d'Analyse 1. Gauthier-Villars, Paris 1972. [8] Fortunato, D., On the index of elliptic partial differential operators in Nn. Ann. Mat. Pura Appl. 119, 317-331 (1979). [9] Giroire, J., [~tude de quelques problOmes aux limites exterieurs et rdsolution par equations integrales. Th~se de doctorat d'~tat +s sci. math. Universit~ Pierre et Marie Curie, Paris 6, 1987. [10] G/inter, N. M., Die Potentialtheorie und ihre Anwendungen auf Grundaufgaben der mathematischen Physik. Verlagsgesellschaft, Leipzig 1957. [ 11] Hsiao, G. C. and Kress, R., On an integral equation for the two-dimensional exterior Stokes problem. Appl. Numer. Math. 1, 77-93 (1985). [12] J6rgens, K., Lineare Integraloperatoren. Teubner, Stuttgart 1970. [13] Leis, R., Zur Eindeutigkeit der Randwertaufgaben der Helmholtz'schen Schwingungsgleichung. Math. Z. 85, 141-153 (1964). [14] Maz'ja, V. G., Sobolev Spaces. Springer, Berlin 1985. [15] McOwen, R., The behaviour of the Laplacian on weighted Sobolev spaces. Comm. Pure Appl. Math. 32, 783-795 (1979). [16] Neras, J., Les mdthodes direetes en thOorie des kquations elliptiques. Academia, Prague 1967. [17] Saranen, J. and Witsch, K. J., Exterior boundary value problems for elliptic equations. Ann. Acad. Sci. Fennicae Ser. A.I. Mathematica 8, 3-42 (1983). [18] Smirnow, W. I., Lehrgang der hdheren Mathematik 4. Deutscher Verlag der Wissenschaften, Berlin 1979. [19] Simader, C. G., On Dirichlet's Boundary Value Problem. Springer, Berlin 1972. [20] Simader, C. G., The weak Dirichlet and Neumann problem for the Laplacian in L q for bounded and exterior domains. Applications. In: Krbec, M., Kufner, A., Opic, B. and Rfikosnic, J. (eds.): Nonlinear Analysis, Function Spaces and Applications Vol. 4, 180-223. Teubner, Leipzig 1990. [21] Simader, C. G. and Sohr, H., The weak Dirichlet problem for A in L q in bounded and exterior domains. Preprint Univ. Bayreuth 1991. [22] Simader, C. G. and Sohr, H., A new approach for the Helmholtz decomposition and the Neumann 1991 problem in Lq-spaces for bounded and exterior domains. Preprint Univ. Bayreuth 1991. [23] Stein, E. M., Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton 1970. [24] Varnhorn, W., The Poisson equation with weights in exterior domains ofN n. Applic. Anal. 43, 135-145 (1992).

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Strong solutions of Poisson's equation in Beppo Levi spaces

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[25] Wahl, W. von, Vorlesung iiber das Auflenraumproblem for die instationdren Gleichungen Navier-Stokes. Rudolph-Lipschitz-Vorlesung, SFB 256, Univ. Bonn 1989. [26] Yosida, K., Functional Analysis. Springer, Berlin 1965.

yon

Abstract Let G ~_ ~n (n-> 2) be an unbounded open set having a compact complement and a smooth boundary 3G of class C 2. In G we consider the equations - A u = f , ulna = ~ and prove the existence of a solution u ~ Lz'q(G) proyided f E Lq(G) and g# ~ W 2 - l/q,q(3G) (1 <'q < oo). Here L2'q(G) is the space of all functions u ~ Lfoc(G) having all second order distributional derivatives in Lq(G). Concerning the uniqueness of this solution we show that the corresponding nullspace has dimension n + 1 (n -> 2).

Zusammenfassung Sei G _~ R" (n -> 2) eine unbeschr~inkte offene Menge mit kompaktem Komplement und mit glattem Rand dG der Klasse C 2. In G betrachten wir das Randwertproblem - A u = f , u,oo = dp und beweisen die Existenz einer L6sung u ~ L2'q(G) f/ir beliebige f ~ Lq(G) und Randwerte (I) e I3j2 - ~/e,a(BG) (1 < q < m). Dabei ist L2,q(G) der Raum aller Funktionen u ~ Lfoc(G), die Distributionsableitungen zweiter Ordnung in Lq(G) besitzen. Bezfiglich der Eindeutigkeit solcher L6sungen zeigen wir, daJ3 der entsprechende Nullraum die Dimension n + 1 (n -> 2) besitzt. (Received: April 2, 1991; revised: September 17, 1991)