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Clifford Analysis over Unbounded Domains
ADVANCES IN APPLIED MATHEMATICS ARTICLE NO.
19, 216]239 Ž1997.
AM970541
Clifford Analysis over Unbounded Domains Klaus Gurlebeck and Uwe Kahler ¨ ¨ Fakultat ¨ fur ¨ Mathematik, Technische Uni¨ erstitat ¨ Chemnitz-Zwickau, D-09107 Chemnitz, Germany
John Ryan Department of Mathematics, Uni¨ ersity of Arkansas, Fayette¨ ille, Arkansas 72701
and Wolfgang Sproßig ¨ T U Bergakademie Freiberg, Fakultat ¨ fur ¨ Mathematik und Informatik, D-09596 Freiberg, Germany Received July 14, 1996; accepted December 30, 1996
A modified Cauchy kernel is introduced over unbounded domains whose complement contains nonempty open sets. Basic results on Clifford analysis over bounded domains are now carried over to this more general context and to functions that are no longer assumed to be bounded. In particular Plemelj formulae are explicitly computed. Basic properties of the Cauchy transform over unbounded domains lying in a half space are investigated, and an orthogonal decomposition of the L2 space for such a domain is set up. At the end a boundary value problem will be studied in the case of an unbounded domain without using weighted Sobolev spaces. Q 1997 Academic Press
1. INTRODUCTION One of the most important parts of Clifford analysis is the possibility to study boundary value problems of mathematical physics in a self-contained form. In w5x the authors studied the application of Clifford analysis, namely, quaternionic analysis, to boundary value problems for various kinds of partial differential equations. However, the authors were only 216 0196-8858r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
217
CLIFFORD ANALYSIS
able to apply their theory to boundary value problems over bounded domains. However, in physics there are many problems stated over unbounded domains. Clifford analysis over bounded domains was developed and applied by a number of authors using different techniques; see for instance w1, 13, 9, 10, 18, and 19x. Moreover, in w3x it was shown that there is an idea to study monogenic functions on unbounded domains, which can be effectively used to develop a similar theory to w5x. In w3x the authors have proved the existence of a Cauchy transform as well as a Cauchy integral formula in the case of Hardy spaces and Holder spaces under some restrictive ¨ conditions. The main problem in w5x is that the Cauchy kernel does not have ‘‘good enough’’ behaviour near infinity; for instance, the T-operator, which is based on the Cauchy kernel, is unbounded in the usual function spaces. The above-mentioned idea from w3x solves this problem by adding an extra term to the Cauchy kernel. That means the authors introduce a ‘‘generalized’’ Cauchy kernel of the form kŽ j , z. s
jyz jqz , n y < j q z
ž
/
where v is the surface area of the unit sphere in R n. This kernel works on unbounded domains V lying in a half space with a sufficiently smooth boundary. Using this kernel they prove a Cauchy integral formula for monogenic functions over unbounded domains and they introduce an analog to the usual FG-operator F˜G f Ž z . s
HGk Ž j , z . a Ž j . f Ž j . dG,
z g V.
We start by introducing a modification l Ž j , x . of this kernel. The advantage of this kernel over the kernel k Ž j , x . is that it can be used over arbitrary unbounded domains whose complement contains a nonempty open set. We show that many basic results from w3x also hold for this kernel, and we also set up analogs of the Plemelj formulae and a Borel]Pompeiu formula for a wide class of C 1 functions defined on unbounded domains. Other basic results from Clifford analysis are shown to hold in the context described here using the kernel l Ž j , x .. Again using the kernel k Ž j , z ., we also introduce analogs to the usual operators from w5x as well as show similar properties. In Section 4 the domains V have to satisfy the condition dŽ V, yV . G d1 ) 0, where dŽ?, ? . is the usual distance between sets. We will prove an orthogonal decomposition theorem for domains V satisfying the preceding conditions. Later on
218
GURLEBECK ET AL. ¨
we show how easy it is to generalize the results from w5x for boundary value problems over bounded domains to boundary value problems over unbounded domains lying in a half space. As a first example, we use the Dirichlet problem for the Laplace equation. This example shows that after the proof of some mapping properties of the integral operators described in the first part of the paper, we are able to use the same tools as known for the case of bounded domains. It is not necessary to introduce weighted Sobolev spaces Žas for instance in w1x.. Consequently, all results connected with the inner product and orthoprojections remain true. Let us remark that for the kernel k Ž j , z . the whole theory from the bounded case can be carried over to the unbounded case. We have to pay for this with a loss of generality in the possible class of unbounded domains. On the other hand, the new kernel l Ž j , z . allows more general domains for investigations in Holder spaces. However, at the moment it is ¨ not clear how we can prove the corresponding properties for this kernel in Sobolev spaces because some of our results from the theory of singular integral operators are applicable only to operators based on k Ž j , z ..
2. PRELIMINARIES Here we outline some basic background material. From R n with orthonormal basis e1 , . . . , e n , we may construct the real 2 n-dimensional Clifford algebra A n with basis 1, e1 , . . . , e n , . . . , e j1 . . . e j r, . . . , e1 . . . e n , where 1 F r F n and j1 - ??? - jr . The algebra A n is set up so that for each x g R n we have that x 2 s y5 x 5 2 . It follows that each nonzero vector x g R n is invertible with multiplicative inverse yxr5 x 5 2 . Up to the minus sign this corresponds to the Kelvin inverse of the vector x. For each A s a0 q ??? qa1 ? ? ? n e1 ??? e n the norm of A is defined to be 5 A 5 s Ž a20 q ??? qa12 ? ? ? n .1r2 . DEFINITION 2.1. For U a domain in R n a differentiable function f : U ª A n is called left monogenic if Df s 0, where D s Ý njs1 e j r x j . A similar definition can be given for right monogenic functions. Basic facts on monogenic function theory can be found in w2x.
3. CAUCHY KERNELS ON UNBOUNDED DOMAINS We shall first give a modification of some of the results described in w3x. In w3x attention is restricted to unbounded domains lying in a half space. This restriction can easily be lifted. Here the type of domain that we shall consider is an unbounded domain U with a sufficiently smooth boundary
219
CLIFFORD ANALYSIS
U. Usually we can consider U to be Lipschitz continuous, but sometimes we will restrict to having a Liapunov domain. For such a domain we may choose a point y lying in the complement of the closure of U. Now we may introduce the Cauchy kernel lŽ j , x. s
jyx jyy . n y 5 j y y5n v 5 j y x5 1
ž
/
It is straightforward to deduce that for some constant C g Rq, lŽ j , x. F C5 x y y5
ny1
Ý 5 j y x 5yj 5 j y y 5 jyn .
js1
Using this inequality one can easily adapt arguments given in w3x to deduce the following form of the Cauchy integral formula over U. THEOREM 3.1. Suppose that f Ž x . is a bounded, left monogenic function on U and extends continuously in the L` sense to the boundary of U. Then for each x g U, f Ž x. s
H Ul Ž j , x . a Ž x . f Ž j . d s Ž j . .
The previous theorem is also true if we replace f by a function defined on U j U such that g is left monogenic on U and satisfies 5 g Ž x .5 F C 5 x 5 s on U for some C g Rq and s g Ž0, 1.. Moreover, on U the function g belongs to the weighted L p space L`Ž5 x 5ys . and g extends continuously in the L`Ž5 x 5ys .5 topology on U from U to U. This continuation can be achieved via homotopy deformations of U within U. We shall denote by AŽU, s . the right Clifford module of functions which are left monogenic on U, satisfy 5 f Ž x .5 F C 5 x 5 s on U for some C g Rq and s g w0, 1., and extend continuously to U in the L`Ž5 x 5ys . topology. It follows from Theorem 3.1 and the remarks following Theorem 3.1 that the space AŽU, s . is a Banach space. For each closed subset K of U we denote the Banach space of left monogenic functions defined in a neighbourhood of K and satisfying 5 f Ž x .5 F C 5 x 5 s by AŽ K, s .. It now follows that one can easily adapt arguments given in w12x and w2x to obtain the following approximation theorem. THEOREM 3.2. Suppose that K is a closed subset of U and that the set U _ K has only one component. Then AŽU, s . is dense in AŽ K, s . with respect to the supremum norm sup x g K 5 f Ž x .5 5 x 5ys .
GURLEBECK ET AL. ¨
220
Proof. The function l Ž j , x . belongs to both AŽU, s . and AŽ K, s . for each x g R n _clŽU . and for j in U, or, respectively, in K. On substituting the function l Ž j , x . for the standard Clifford]Cauchy kernel in the proof of Theorem 18.4 in w2x the result follows. Q.E.D. We now turn to look at the analogs of the Plemelj formulae. We begin with: PROPOSITION 3.1. Suppose that c : U ª A n is both bounded and Holder ¨ continuous with exponent s g Ž0, 1.. Then the integral
H Ul Ž j , x . a Ž j . c Ž j . d s Ž j .
P.V.
is well defined for each x g U. Proof. In the case where U is compact we first place l Ž j , x . s GŽ j y x . y GŽ j y y . , where GŽ x . s Ž1rv . x 5 x 5yn . The integral
H UG Ž j y x . a Ž j . c Ž j . d s Ž j .
P.V.
is handled the same way as in w6x, while the integral
H UG Ž j y y . a Ž j . c Ž j . d s Ž j . defines a constant. Consequently, when U is compact, the integral
H Ul Ž j , x . a Ž j . c Ž j . d s Ž j .
P.V.
is well defined on U. When U is not compact we may note that
H U _UŽ x .l Ž j , x . a Ž j . c Ž j . d s Ž j .
P.V.
F C5 x y y5
yn
H U _UŽ x .5 j 5
c Ž j . ds Ž j . ,
where UŽ x . is a suitable neighbourhood in U of x. Because c is bounded, it follows that the right-hand integral in the previous expression is bounded by C 5 x y y 5 sup
jg U
cŽj .
yn
H U _UŽ x .5 j 5
ds Ž j . ,
221
CLIFFORD ANALYSIS
and this integral is bounded. By integrating over UŽ x . the result now follows by the same argument as presented for the case where U is bounded. Q.E.D. The proof of the previous proposition may readily be adapted to establish the analogous result if we assume that c g L`Ž5 x y y 5.ys for some s g Ž0, 1.. By substituting the kernel l Ž j , x . for the kernel GŽ j y x . we may easily adapt arguments in w6x to deduce Suppose that a g A n is a constant. Then for each x g U,
LEMMA 3.1.
H Ul Ž j , x . a Ž x . a d s Ž j . s
P.V.
1 2
a.
Using the previous results we may deduce continuous with THEOREM 3.3. Suppose that c : U ª A n is Holder ¨ exponent s g Ž0, 1. and that c also belongs to L`Ž5 x y y 5ys . on U. Then
H Ul Ž j , x . a Ž j . c Ž j . d s Ž j .
C Ž x . s P.V.
is also Holder ¨ continuous with exponent s and it also belongs to L`Ž5 x y y 5ys .. Proof. We shall first show that C Ž x . belongs to L`Ž5 x y y 5ys .. First we rewrite C Ž x . as
H Ul Ž j , x . a Ž j . Ž c Ž j . y c Ž x . . d s Ž j .
P.V.
H Ul Ž j , x . a Ž j . c Ž x . d s Ž j . .
q
By the previous lemma this evaluates to 1 2
c Ž x . q P.V.
H Ul Ž j , x . a Ž j . Ž c Ž j . y c Ž x . . d s Ž j . .
Let B U Ž x, 12 5 x y y 5. denote the restriction to U of the ball centered at x and of radius 21 5 x y y 5. Now
H U _ B
U Ž x ,
F
5 xyy 5r2 .
H U _ B
l Ž j , x . a Ž j . Ž c Ž j . y c Ž x . . ds Ž j .
U Žw , 35
xyy 5r2 .
l Ž j , x . a Ž j . Ž c Ž j . y c Ž x . . ds Ž j . ,
GURLEBECK ET AL. ¨
222
where w is the point in U such that 5 w y y 5 s d is the distance from y to U, and B U Ž w, 32 5 x y y 5. is the restriction to U of the ball centered at w and of radius 32 5 x y y 5. The previous integral is dominated by C
H U _ B
U Žw , 35
q
xyy 5r2 .
H U _ B
U Žw , 35
l Ž j , x . a Ž j . c Ž j . ds Ž j .
xyy 5r2 .
l Ž j , x . a Ž j . c Ž x . ds Ž j . .
Moreover,
H U _ B
U Žw , 35
xyy 5r2 .
l Ž j , x . a Ž x . c Ž j . ds Ž j .
F C 5 x y y 5 5 c 5 L`Žys.
`
y2 qs
H3 5 xyy 5r2r
dr ,
where 5 c 5 L`Žys. is the L`Ž5 x y y 5ys . norm of c . Also,
H U _ B
U Žw , 35
xyy 5r2 .
l Ž j , x . a Ž j . c Ž x . ds Ž j .
F C 5 x y y 5 1q s 5 c 5 L`Žys.
`
y2
H3 5 xyy 5r2r
dr.
Furthermore, P.V.
HB
U Ž x ,
5
5 xyy 5r2 5
H0 xyy r2r
FC
l Ž j , x . a Ž j . Ž c Ž j . y c Ž x . . ds Ž j . sy 1
dr.
This last inequality follows because one can find a constant C g Rq, independent of the choice of x and such that 5 j y y 5 - C 5 x y y 5 whenever x g B U Ž x, 12 5 x y y 5.. Now let us consider the restriction, A U Ž w, 21 5 x y y 5, 23 5 x y y 5., of the spherical shell AŽ w, 21 5 x y y 5, 23 5 x y y 5. to U. Let us consider the set K s A U Ž w, 21 5 x y y 5, 23 5 x y y 5._ B U Ž x, 21 5 x y y 5.. On the set K we find that 5 j y x 5 ) 12 5 j y y 5 and for some constant C g Rq we have that 5 y y x 5 - C 5 j y y 5. Consequently, the integral
HKl Ž j , x . a Ž j . Ž c Ž j . y c Ž x . . d s Ž x .
223
CLIFFORD ANALYSIS
is dominated by C 5 x y y 5 1q s 5 x y y 5yn
5
5
H5 3xyyxyy5r2r2r
ny2
dr.
It remains to consider the integral
HB
U Žw ,
5 xyy 5r2 .
l Ž j , x . a Ž j . Ž c Ž j . y c Ž x . . ds Ž j . .
We begin by replacing c Ž j . y c Ž x . by Ž c Ž j . y c Ž w .. q Ž c Ž w . y c Ž x ... It follows that
HB
U Žw ,
F
5 xyy 5r2 .
HB
U Žw ,
q
HB
l Ž j , x . a Ž j . Ž c Ž j . y c Ž x . . ds Ž j .
5 xyy 5r2 .
U Žw ,
l Ž j , x . a Ž j . Ž c Ž j . y c Ž w . . ds Ž j .
5 xyy 5r2 .
G Ž j y y . a Ž j . ds Ž j .
c Ž x. y c Ž w. .
On B U Ž w, 12 5 x y y 5. we have that 5 j y x 5 ) 12 5 x y y 5, 5 j y w 5 F 5 x y y 5, and 5 j y y 5 ) 5 w y j 5. Consequently, the expression
HB
U Žw ,
5 xyy 5r2 .
l Ž j , x . a Ž j . Ž c Ž j . y c Ž w . . ds Ž j .
is dominated by 5 xyy 5r2 sy1
žH
C
r
0
ny2
dr q
5 xyy 5r2 jy1
Ý 5 x y y 5 syj H
r
0
js1
/
dr .
The term
HB
U Žw ,
5 xyy 5r2 .
G Ž j y x . a Ž j . ds Ž j .
c Ž x. y c Ž w.
is dominated by C 5 x y y 5 sq 1yn
5
5
H0 xyy r2r
ny2
dr.
GURLEBECK ET AL. ¨
224
Using Cauchy’s integral formula it may be observed that the expression
HB
U Žw ,
5 xyy 5r2 .
GŽ j y y . a Ž j . ds Ž j .
is dominated by 1q
HS
ny1 Ž
y, r.
GŽ j y y . a Ž j . ds Ž j . ,
where S ny 1 Ž y, r . is the sphere in R n centered at y and of radius r, for any r g Rq. It now follows that the function c satisfies the inequality 5 C Ž x .5 F 5 C x y y 5 s. To show that C is Holder continuous with exponent s one first ¨ notes that
H U Ž G Ž j y x . y G Ž j y u . . a Ž j . c Ž j . d s Ž j . .
C Ž x . y C Ž u . s P.V.
The proof now follows the same lines as the proof of the analogous result in w6x. Q.E.D. It is straightforward to note that if c satisfies the conditions outlined in Theorem 3.3, then the integral
H Ul Ž j , u . a Ž j . c Ž j . d s Ž j . defines a left monogenic function on U. We shall denote the domain, or domains, complementary to U in R n by V. Similarly the integral
H Ul Ž j , ¨ . a Ž j . c Ž j . d s Ž j . defines a left monogenic function on V which takes the value zero at y. Following a similar arguments to those presented in w6x and elsewhere, we can now present analogs of the Plemelj formulae. THEOREM 3.4. Suppose that c satisfies the conditions described in Theorem 3.3. Suppose also that f : w0, 1. ª U is a C 1 function and that lim t ª 1 f Ž t . s x g U, and that lim t ª 1 Ž d frdt .Ž t . is nontangential to U. Then lim
H
tª1 U
l Ž j , f Ž t . . a Ž j . c Ž j . ds Ž j .
s 12 c Ž x . q P.V.
H Ul Ž j , x . a Ž j . c Ž j . d s Ž j . .
225
CLIFFORD ANALYSIS
Similarly if we assume that f : w0, 1. ª V, with lim t ª 1 f Ž t . s x g U and lim t ª 1 Ž d frdt .Ž t . is nontangential to U, then lim
H
tª1 U
l Ž j , f Ž t . . a Ž j . c Ž j . ds Ž x .
s y 12 c Ž x . q P.V.
H Ul Ž j , x . a Ž j . c Ž j . d s Ž j . .
We now proceed to establish: THEOREM 3.5. Suppose that c Ž x . is a Holder-continuous function de¨ fined on U with exponent s g Ž0, 1. and that 5 c Ž x .5 F C 5 x y y 5 s for some C g Rq. Then the left monogenic function F Ž u. s
H Ul Ž j , u . a Ž j . c Ž j . d s Ž j .
defined on U satisfies 5 F Ž u.5 F C 5 x y y 5 s. Proof. Consider the set B U Ž y, 5 5 y y u 5 . s U l B Ž y, 5 y y u 5 . . The integral
H U _ B
U Ž y , 55
yyu 5.
l Ž j , u. a Ž j . c Ž j . ds Ž j .
is dominated by C 5 u y y 5 1q s
`
H5 5 uyy 5r
y2
dr.
On the other hand,
HB
U Ž y , 5 5 uyy 5.
s
l Ž j , u. a Ž j . c Ž j . ds Ž j .
HB
U Ž y , 5 5 uyy 5.
y
HB
GŽ j y y . a Ž j . c Ž j . ds Ž j .
U Ž y , 5 5 uyy 5.
G Ž j y u. a Ž j . c Ž j . ds Ž j . .
GURLEBECK ET AL. ¨
226 The integral
HB
U Ž y , 5 5 uyy 5.
G Ž j y y . a Ž j . c Ž j . ds Ž j .
is dominated by
HB
U Ž y , 5 5 uyy 5.
q
GŽ j y y . a Ž j . Ž c Ž j . y c Ž w . . ds Ž j .
HB
U Ž y , 5 5 uyy 5.
G Ž j y y . a Ž j . ds Ž j .
c Ž w. .
The second term here is part of the Cauchy integral formula for a constant, and 5 c Ž w .5 F C 5 w y y 5 s F CX 5 u y y 5. Moreover, the first term is dominated by
HB
U Ž y , 5 5 uyy 5.
G Ž j y w . a Ž j . Ž c Ž j . y c Ž w . . ds Ž j . .
By similar reasoning to that used in the proof of Theorem 3.3 it may be observed that this term satisfies the desired inequality. The term
HB
U Ž y , 5 5 uyy 5.
G Ž j y u. a Ž j . c Ž j . ds Ž j .
is dominated by
HB
U Ž y , 5 5 uyy 5.
q
G Ž j y u. a Ž j . Ž c Ž j . y c Ž ¨ . . ds Ž j .
HB
U Ž y , 5 5 uyy 5.
G Ž j y u. a Ž j . ds Ž j .
c Ž¨. ,
where ¨ is the point on U that is closest to u. The second term contains the Cauchy integral formula for a constant, and 5 c Ž ¨ .5 F C 5 ¨ y y 5 s. Moreover, 5 ¨ y y 5 - 5 5 u y y 5. On the other hand, the first term in the previous expression is dominated by
HB
U Ž ¨ , 10 5 uyy 5.
GŽ j y ¨ . a Ž j . Ž c Ž j . y c Ž ¨ . . ds Ž j . .
It therefore follows by similar reasoning to that given in the proof of Theorem 3.3, that this integral is dominated by C 5 u y y 5 s. The result follows. Q.E.D.
227
CLIFFORD ANALYSIS
On combining this last result with the Cauchy integral formula described at the beginning of this section and Theorem 3.3 we arrive at THEOREM 3.6. Suppose that c is a Holder-continuous function defined on ¨ U and has exponent s g Ž0, 1.. Suppose also that 5 c Ž x .5 F C 5 x y y 5 s for some C g Rq. Then
H Ul Ž n , x . n Ž n . P.V.H Ul Ž j , n . a Ž j . c Ž j . d s Ž j . d s Ž n . s
P.V.
1 4
c Ž x. .
The previous result shows that the Plemelj formulae, seen as operators acting over the appropriate function space, are projection operators. We now turn to look at the other results that can be obtained using the modified kernel l Ž j , x . instead of the Cauchy kernel GŽ j y x .. We begin with an analog of the Borel]Pompeiu formula. THEOREM 3.7. Suppose that f is a C 1 function defined on the domain U and with a continuous extension to the boundary of U. Suppose also that 5 f Ž x .5 F C1 5 x 5 s and 5 Df Ž x .5 F C2 5 x 5 sy1 , for some constants C1 , C2 g Rq and s g Ž0, 1.. Then for each u g U, f Ž u. s
H Ul Ž j , u . a Ž j . f Ž j . d s Ž j . y HUl Ž j , u . Df Ž j . d j
n
.
The proof of this result follows similar lines to proofs given in w3x. We conclude this section by noting that in w14x it is shown that one can set up a Cauchy integral formula and Dirac operator on the hyperbola Hn . This is done via a Mobius transformation. One can similarly introduce the ¨ Cauchy kernel l H nŽ j , x . s GŽ j y x . y GŽ j y y ., where now j , x, and y are all points on Hn and G is the Cauchy kernel on Hn . It is now a relatively easy exercise to show that all the results obtained so far in this section now may be rederived over the hyperbola.
4. SOME NEW OPERATORS ON UNBOUNDED DOMAINS In this section we develop a new operator theory, which will allow us to study boundary value problems of mathematical physics in a self-contained form. Using the kernel k Ž j , z . s GŽ j y z . y GŽ j q z . we introduce analogs to the usual operators from w5x as well as show similar properties. For the sake of brevity, first we cite a theorem from w11, Chap. XI, Theorem 9.1x, which we will use more than once later.
GURLEBECK ET AL. ¨
228 LEMMA 4.1.
Let the symbol FAŽ Q . of a singular integral operator
Ž Au . Ž z . s au Ž z . q H nrynk Ž Q . u Ž j . , dR n q Ku Ž z . , R
where a is a constant, r s < j y z <, Q s Ž j y z .r< j y z <, k Ž Q . is the characteristic, and K is a compact operator, be essentially bounded. Then we ha¨ e that this operator is bounded in the spaces W 2l ŽR n . for each natural l or l s 0 and the estimate 5 A 5 W 2l ŽR n . F C sup ess FA Ž Q . with C is a space constant Ž in case of L 2 : C s 1.. Preparing the following investigations in Sobolev spaces we start with some auxiliary statements in Holder spaces. In the following we shorten ¨ 5 ? 5 L r Ž V , C l 0, n . to 5 ? 5 r . LEMMA 4.2 w7x. Let a1 , a2 , . . . , a n be real positi¨ e numbers. Furthermore, let s G 1. Then the following inequality is ¨ alid: s
n
n
F n sy1
žÝ / ai
Ý ais .
is1
is1
LEMMA 4.3. Let V be a bounded domain in R n. For any real numbers a and b with 0 F a , b - n, a q b ) n and two different points x 1 , x 2 g V a constant C exists which depends on the domain V with Is
HV < x
dy a
1
y y < < x2 y y <
F
b
C < x 1 y x 2 < aq byn
.
Proof. Let d be the diameter of the domain V. We enclose V in a ball of radius d centered at the point x 1. Now we perform the coordinate transform zX s x 1 y y and obtain
HB Ž x . < z <
X a
d
1
dzX < x 1 y x 2 y zX < b
.
Now setting < x 1 y x 2 < s r, Ž x 1 y x 2 .rr s u , and zX s rz, clearly, we have dzX s r n dz. This leads to the estimates dz
< I < F r ny ay b
HB
s r ny ay b
žH
dr r Ž x 1 .
< z
a
< z <
< z <
b
q
dz
H2-< z <-drr < u y z <
b
< z
/
.
229
CLIFFORD ANALYSIS
The first integral is bounded by a constant A because < u < s 1. Therefore, the order of the singularity of the kernel is less than n. For the second integral, which we denote by I2 , we calculate as < I2 < F
H2-< z <-drr < z <
dz a
b
dz
F 2a
H1-< z <-drr < z <
aq b
.
In this estimation the inequality < z y u < G < z < y < u < s < z < y 1 G 12 < z <
Ž < z < ) 2! .
was used. The change to polar coordinates leads to drr
< I2 < F 2 av
H1
r ny1y ay b d r F 2 av
1
aqbyn
.
This proves our assertion with the constant C s A q 2 av Ž a q b y n.y1 . Note that v again denotes the surface area of the n-dimensional unit sphere. Remark 4.1. The reader can find statements of this type in w17x and w11x, for instance. We begin now with the investigation of the usual T-operator defined by
Ž TV f . Ž x . s H G Ž x y y . f Ž y . dy, V
where GŽ x . s Ž1rv . y xr5 x 5 n. THEOREM 4.1. Let V be a domain Ž bounded or unbounded. in R n, u g L r Ž V, Cl 0, n . Ž r ) n.. Then we get
Ž Tu . Ž x 1 . y Ž Tu . Ž x 2 . F C Ž V , r . 5 u 5 r < x 1 y x 2 < s for x 1 / x 2 and s s 1 y nrr. Proof. At first let V be a bounded domain. From Holder’s inequality ¨ we obtain s
Ž Tu . Ž x 1 . y Ž Tu . Ž x 2 . F
1
v
HV
< u 1 < x 2 y y < ny1 y u 2 < x 1 y y < ny1 < x 1 y y < Ž ny1. s < x 2 y y < Ž ny1. s
s
dy 5 u 5 rs ,
GURLEBECK ET AL. ¨
230
where u i s Ž x i y y .r< x i y y < Ž i s 1, 2.. Applying Lemma 4.2 we get s
Ž Tu . Ž x 1 . y Ž Tu . Ž x 2 . F
2 sy1
v
Ž I1 q I2 . 5 u 5 rs .
Here we use the notation I1 s
HV
s
< x 2 y y < ny1 y < x 1 y y < ny1 < x 1 y y < Ž ny1. s < x 2 y y < Ž ny1. s
dy and
I2 s
HV < x
2
y y < Ž ny1. s
dy.
At first we will consider the integral I1. Without any difficulties we obtain by using Lemmas 4.2 and 4.3 the estimate I1 F Ž n y 2 .
sy1
HV
Ý aq bsny2 < x 2 y y < a s < x 1 y y < b s < x 1 y y < Ž ny1. s < x 2 y y < Ž ny1. s
dy < x 1 y x 2 < s
F C Ž V , r . < x 1 y x 2 < ny n sqs . Introducing the abbreviation z i s x i y y Ž i s 1, 2., the second integral I2 can be estimated in the manner I2 s
HV < z
F2
2
Ž sy1.
F 2 2 sy1
< Ž ny1. s
HV
dy F
HV
z1r< z1 < y z 2r< z1 < q z 2r< z1 < y z 2r< z 2 < < z 2 < Ž ny1. s
< z1 y z 2 < sr< z1 < s q < z 2 < s
HV < z
Ž < z 2 < y < z1 < . r Ž < z1 < < z 2 < .
< z 2 < Ž ny1. s 1 < 2
Ž ny1. s
< z1 < s
s
dy
s
dy
dy < x 1 y x 2 < s .
Note that always ns ) n. Lemma 4.3 now yields the result I2 F C Ž V , r . < x 1 y x 2 < ny Ž ny1. sqsys s C Ž V , r . < x 1 y x 2 < ny n sqs . For 1rs q 1rr s 1 we conclude
Ž Tu . Ž x 1 . y Ž Tu . Ž x 2 . F C Ž V , r . < x 1 y x 2 < 1y n r r 5 u 5 r , where C Ž V, r . are different positive constants which only depend on V and r. If the domain V is unbounded, then we decompose V into a bounded part which contains the points x 1 , x 2 and an exterior domain. Showing the Holder continuity for ŽTu.Ž x . we only have to consider points x 1 , x 2 whose ¨
231
CLIFFORD ANALYSIS
mutual distance is bounded. The T-operator now is split into corresponding parts. For the integral over the bounded part of the domain we have already shown our statement. The kernel function of the integral over the unbounded part satisfies a Lipschitz condition, because it does not contain any singularities. Therefore, this second part also yields an inequality of the previously mentioned type. Remark 4.2. For bounded domains the result of Theorem 4.1 is proved in w5x. The case n s 2 is treated in w16x. COROLLARY 4.1. operator
Let V be a domain with V l y V s B. For the
˜ . Ž x . s H k Ž x, y . u Ž y . dy Ž Tu V
with k Ž x, y . s GŽ x y y . y GŽ x q y . the statement of Theorem 4.1 remains true. Proof. The position of the domain in a half space of R n leads to the fact that GŽ x q y . has no singularities in V. In this way the T-operator is disturbed by an integral operator with a smooth kernel, which realizes such an estimate in a natural way. Q.E.D. Let us remark here that the previously proved results on the boundedness of FG and TV are necessary for a classical theory of boundary value problems stated in Holder spaces. However, our goal is the generalization ¨ of Hilbert space methods to the case of boundary value problems in unbounded domains. Therefore, we only use the previously obtained results for extension procedures from Holder spaces to Sobolev spaces. ¨ These extensions are impossible using TV and FG without corrections of the kernel function or the introduction of weighted Sobolev spaces. For all that follows let V be an unbounded domain lying in a half space and satisfying the condition dŽ V, yV . G d1 ) 0, where dŽ?, ? . is the usual distance between sets. We assume the boundary to be a Liapunov boundary. THEOREM 4.2. Suppose that V is an unbounded domain satisfying our preceding conditions. Then T˜ defined by
˜s Tf
HVk Ž j , z . f Ž j . dV
j
is a continuous mapping from L 2 Ž V . to W 21 Ž V .. Proof. From the results of w3x and Corollary 4.1 the weakly singular integral operator T˜ can easily be extended as a bounded operator to the space L 2 Ž V .. Applying the fact that T˜ is a weakly singular integral
GURLEBECK ET AL. ¨
232
˜ is a differentiable L 2 operator we get from w11, Chap. IX, Sect. 7x that Tf ˜ For function for f g L 2 Ž V .. Now let us consider the operator Ž r z i .T. this operator we have zi
˜s Tf
1
e i y n Ž j i y z i . Ž j y z . r< j y z < 2
HV
v y
< j y z
e i y n Ž j i q z i . Ž j q z . r< j q z < 2 < j q z<
n
f Ž z . dV y
ei n
f Ž z. .
If we ask for the boundedness of this operator as an operator from the space L 2 Ž V . to the space L 2 Ž V ., then we have only to look at the operator
zi
˜s Tf
1
HR
v y
e i y n Ž j i y z i . Ž j y z . r< j y z < 2 < j y z
n
e i y n Ž j i q z i . Ž j q z . r< j q z < 2 < j q z<
n
f Ž z . dR n y
ei n
f Ž z.
as an operator over the whole space L 2 ŽR n .. Obviously, we get our operator if we extend all functions f g L 2 Ž V . by zero to the space R n and then restrict this operator to the domain V. Now all we have to do is look at the symbol FŽ r z i .T˜Ž Q . of the operator
zi
˜s Tf
1
v y
HR
e i y n Ž j i y z i . Ž j y z . r< j y z < 2 n
< j y z
e i y n Ž j i q z i . Ž j q z . r< j q z < 2 < j q z<
n
f Ž z . dR n y
ei n
f Ž z. .
For this symbol the formula FŽ r x i .T˜ Ž Q . s 2 Ž e i y nQXi QX . ln
HS
1
q
ip 2
sign cos g dSQX y
holds. Using Lemma 4.1 we get Ž r z i .T˜: L 2 Ž V . ¬ L 2 Ž V ..
ei n
Q.E.D.
LEMMA 4.4. Assume that V is a domain like in the pre¨ ious theorem. Then for all functions f g L 2 Ž V . the equation
˜ sf DTf holds.
CLIFFORD ANALYSIS
233
This was proved for the case of Holder-continuous functions in w3x and ¨ can easily be extended to L 2 functions by the help of Theorem 4.2 and the continuity of D: W 21 ¬ L 2 . THEOREM 4.3. Let V be a domain like in Theorem 4.2. If f g W 21 Ž V ., then we ha¨ e the Borel]Pompeiu formula
˜ . F˜G f s f y TDf ˜ Applying partial integration Proof. Let us consider the expression TDf. ˜ s f, we get our formula. Q.E.D. with respect to D and the fact that DTf For the F˜G-operator we ha¨ e
PROPOSITION 4.1.
F˜G : W 21r2 Ž G . ¬ W 21 Ž V . l ker D. Proof. Let ¨ g W 21r2 Ž G .. Then there exists a function u g W 21 Ž V . with tr u s ¨ . Using the foregoing Borel]Pompeiu formula and Theorem 4.2 ˜ g W 21 Ž V .. we get FG ¨ s u y TDu Q.E.D. THEOREM 4.4. Suppose that the domain V is as in the pre¨ ious theorem, f g C a Ž G ., 0 - a - 1. Then we ha¨ e for all ˜ z g G, lim F˜G f Ž z . s 12 f Ž ˜ z . q 12 S˜G f Ž ˜ z. ,
zªz˜ zgV
where S˜G f Ž ˜ z . s 2HG k Ž j , ˜ z . a Ž j . f Ž j . dG. Proof. Obviously, we have S˜G : C a Ž G . ¬ C a Ž G . or S˜G : L 2 Ž G . ¬ L 2 Ž G ., respectively, Lemma 4.1, because we have for the symbol of the singular integral part of S˜G FS˜GŽ Q . s iQ. Consider ˜ z g G, « ) 0. Also, consider a point z g V and the sphere SŽ0, r Ž z .. centered at 0 and of radius r Ž z .. We denote by DŽ0, r Ž z .. the open disc centered at 0 and of radius r Ž z .. For r Ž z . sufficiently large z will lie in the domain V l DŽ0, r Ž z ... Let f g C a Ž G .. Then there exists a function u g C a Ž V . with tr u s f. For this function u we have F˜G u Ž z . s
HGlD Ž0, r Ž z ..k Ž j , z . a Ž j . u Ž j . dG HVlSŽ0, r Ž z ..k Ž j , z . a Ž j . u Ž j . dG
q
HG _ D Ž0, r Ž z ..k Ž j , z . a Ž j . u Ž j . dG
q
HVlSŽ0, r Ž z ..k Ž j , z . Ž ya Ž j . . u Ž j . dG,
q
GURLEBECK ET AL. ¨
234 S˜G u Ž ˜ z. s
HGlD Ž0, r Ž z ..k Ž j , ˜z . a Ž j . u Ž j . dG HVlSŽ0, r Ž z ..k Ž j , ˜z . a Ž j . u Ž j . dG
q
HG _ D Ž0, r Ž z ..k Ž j , ˜z . a Ž j . u Ž j . dG
q
HVlSŽ0, r Ž z ..k Ž j , ˜z . Ž ya Ž j . . u Ž j . dG.
q
Because r Ž z . is sufficiently large, we have
HG _ D Ž0, r Ž z ..k Ž j , z . a Ž j . u Ž j . dG
-
HVlSŽ0, r Ž z ..k Ž j , z . Ž ya Ž j . . u Ž j . dG
-
HG _ D Ž0, r Ž z ..k Ž j , ˜z . a Ž j . u Ž j . dG
-
HVlSŽ0, r Ž z ..k Ž j , ˜z . Ž ya Ž j . . u Ž j . dG
-
« 8
« 8
« 8
« 8
, , , ,
because < k Ž j , z .< - C Ž< z
HG _ D Ž0, r Ž z ..k Ž j , z . a Ž j . u Ž j . dG HVlSŽ0, r Ž z ..k Ž j , z . Ž ya Ž j . . u Ž j . dG
q
HG _ D Ž0, r Ž z ..k Ž j , ˜z . a Ž j . u Ž j . dG
y
HVlSŽ0, r Ž z ..k Ž j , ˜z . Ž ya Ž j . . dG
y
-
« 2
.
Ž 1.
Moreover, from w5x it follows that there exists a d ) 0 so that for all z with
HGlD Ž0, r Ž z ..k Ž j , z . a Ž j . u Ž j . dG 1
HVlSŽ0, r Ž z ..k Ž j , z . a Ž j . u Ž j . dG y 2 u Ž ˜z .
q
235
CLIFFORD ANALYSIS
y y
1 2
HGlD Ž0, r Ž z ..k Ž j , ˜z . a Ž j . u Ž j . dG
1
kŽ j , ˜ z . a Ž j . u Ž j . dG H 2 VlS Ž0, r Ž z ..
-
« 2
.
Ž 2.
Consequently, we can conclude from Ž1. and Ž2. that ;« ) 0 'd :
F˜G u Ž z . y 12 u Ž ˜ z . y 12 S˜G u Ž ˜ z. - «
;z: < z y ˜ z< - d.
Because ˜ z g G is arbitrary and f s tr u we have our statement.
Q.E.D.
PROPOSITION 4.2. Using the continuity of the trace operator tr from the domain V to the boundary G Ž see w15x. we can extend our pre¨ ious Plemelj]Sokhotzki formula to Sobole¨ spaces W 2l Ž G ., 0 - l - 1, in the form tr F˜G f s 12 f q 12 S˜G f for f g W 2l Ž G ., 0 - l F 1. Proof. Suppose u g W 2l Ž G ., 0 - l - 1. Then there exists a sequence u n of Holder-continuous functions with u n ª u in W 2l Ž G ., 0 - l - 1. For ¨ these functions u n we have from our previous theorem the formula
Ž tr F˜G . u n s Ž I q S˜G . u n . 1 2
Note that the operator I q S˜G is a continuous mapping from W 2l Ž G ., l ) 0, to W 2l Ž G ., l ) 0 ŽLemma 4.1.. If we now let u n ª u, then we get our result. Q.E.D. PROPOSITION 4.3. Let f g C a Ž G ., 0 - a - 1. Then we can pro¨ e in the same way as in Theorem 4.4 that for all ˜ z g G, lim
zªz˜ zgR n _ Ž VjyV .
F˜G f Ž z . s y 12 f Ž ˜ z . q 12 S˜G f Ž ˜ z. .
From these propositions we obtain that S˜G f Ž ˜ z. s f Ž ˜ z . is necessary and Ž . sufficient for the fact that f ˜ z are boundary values of an A n-valued function which is left monogenic in V. Let us now introduce the operators PG s
1 2
Ž I q S˜G .
and
QG s
1 2
Ž I y S˜G . .
Consequently, we have for PG and Q G the algebraic properties PG2 s PG , Q G2 s Q G , and PG Q G s Q G PG s 0, so we get that PG and Q G are projec-
GURLEBECK ET AL. ¨
236
tions. It may be observed that PG is the projection onto the space of all ŽHolder-continuous or W 2l , l ) 0. A n-valued functions which are left ¨ monogenic extendable into the domain V. THEOREM 4.5. The right Hilbert module L 2 Ž V . allows the orthogonal decomposition L 2 Ž V . s ker D Ž V . l L 2 Ž V . [ D Ž W˚21 Ž V . .
Ž 3.
with respect to the inner product
Ž u, ¨ . s H u¨ dV . V
For the idea of the proof we refer to a proof of a similar theorem in w4x. Proof. The right linear sets X1 s L 2 Ž V . l ker D Ž V .
X 2 s L2 Ž V . ] X1
and
˜ g W 21 Ž V .. From are subspaces of L 2 Ž V .. For any u g L 2 Ž V . we have Tu 1Ž . this it follows that there exists a function ¨ g W 2 V with u s D¨ . Let u g X 2 . Then, we have for all g g X1 ,
HVD¨ g dV s 0, and, in particular, for any l g N,
HVD¨ g
l
dV s 0,
Ž 4.
with gl Ž x . s
Ž x y yl .
Ž x q yl .
y
< x y yl < n
< x q yl < n
l g N, y l g R n _ Ž V j y V . .
,
Obviously, g l g ker DŽ V . l L 2 Ž V .. We assume that the set y l , l g N4 is dense in R n _Ž V j yV.. Then we get for any y l g R n _Ž V j yV.,
n
HVD¨ g
l
dV x s
Ý i , js0
HVe
i
xi
¨ j e j g l dV x
n
sy
Ý i , js0
sy
HVe e ¨
HV¨ Dg
j
l
i j
xi
dV x q
n
g l dV x q
HG¨ a g
l
dGx
Ý
He e ¨ a g
i , js0 G
j
i j
i
l
dGx
CLIFFORD ANALYSIS
s
HGg a ¨ dG
s
HG
l
ž
237
x
yl y x < x y yl < n
q
x q yl < x q yl < n
/
a ¨ dGx
s yv Ž F˜G Ž tr ¨ . . Ž y l . , where tr ¨ denotes the trace of ¨ . Using Ž4. we get F˜G Žtr ¨ . s 0 in R n _Ž V j yV.. Hence, it follows that tr ¨ g im PG l W 21r2 Ž G .. Consequently, there exists a function h g W 21 Ž V . l ker DŽ V . with the property that tr h s tr ¨ . Taking the function w s ¨ y h g W˚21 Ž V . we get that Dw g DŽ W˚21 Ž V ... The result now follows from u s D¨ s Dw. Q.E.D. Remark 4.3. From our decomposition of the space L 2 Ž V . we get the orthoprojections P: L 2 Ž V . ¬ L 2 Ž V . l ker D, Q: L 2 Ž V . ¬ D Ž W˚21 Ž V . . . At this point we have built up an operator theory similar to w5x. In the next section we will take a look at investigating boundary value problems of elliptic partial differential equations. We will show how easy it is to solve questions like existence, and uniqueness with the help of our theory. Moreover, we will be able to give convenient representations of the solutions. These integral representation formulas are adapted for a possible numerical evaluation of the solutions.
5. THE DIRICHLET BOUNDARY VALUE PROBLEM OF THE LAPLACE EQUATION IN UNBOUNDED DOMAINS As an example we study the Laplace equation. This is not only because the Laplace equation provides the simplest example of an elliptic boundary value problem, but also because we can easily demonstrate how powerful our operator theory works in investigating such problems. THEOREM 5.1. problem
Suppose f g L 2 Ž V . and g g W 23r2 Ž G .. Then Dirichlet’s
has a solution u g
W 22, loc Ž V .
yDu s f in V , u s g on G l
W 21
of the form
˜, u s F˜G g q T˜PDh q T˜QTf where h is a
W 22
extension of g.
Ž 5.
238
GURLEBECK ET AL. ¨
Proof. First we investigate the boundary value problem yDu1 s f in V , u1 s 0 on G.
Ž 6.
Now let us mention that the orthoprojection Q maps W2k into W2k, loc . To prove this we consider Q f s f y P f and we use P f g ker DŽ V . ; C`Ž V .. ˜ Using Now let us apply the operator yD s DD to the function ¨ s T˜QTf. the fact that ¨ is at least a W21 function we can apply the D-operator and ˜ Obviously, D¨ is also a W21, loc function, that means D is get D¨ s QTf. applicable again and we obtain that yD¨ s f g L 2 Ž V .. Using known properties of elliptic operators Žsee e.g., w8x. we get that ¨ g W 22, loc Ž V .. Because T˜Q f has vanishing boundary values for each L 2 function f we have that ¨ is a solution of Ž6.. Moreover, we can use this result to solve the boundary value problem yDu 2 s 0 in V , u 2 s g on G.
Ž 7.
Whereas g g W 23r2 Ž G . there exists a W 22 extension h with tr h s g. If we set u 2 s ¨ q h, then we have transformed the last boundary value problem into yD¨ s D h in V , ¨ s 0 on G. Applying our solution of the boundary value problem Ž6. one may determine that ¨ s T˜QT˜ D h. Using the Borel]Pompeiu formula, P s I y Q, and DD s yD we find ¨ s yT˜Q Dh q T˜Q F˜G Dh s yh q T˜PDh q F˜G h or we get u 2 s ¨ q h s F˜G g q T˜PDh. On noting that if u1 , u 2 are solutions of the boundary value problems Ž6. and Ž7. then u s u1 q u 2 is a solution of the boundary value problem Ž5., it can be seen that
˜. u s F˜G g q T˜PDh q T˜QTf
Q.E.D.
˜ is the only soluWe note that the solution u s F˜G g q T˜PDh q T˜QTf tion of the boundary value problem Ž5.. ACKNOWLEDGMENT Part of the work covered in this paper was done while the third author was the recipient of a von Humboldt Research Fellowship visiting the Bergakademie in Freiberg in Sachsen.
CLIFFORD ANALYSIS
239
REFERENCES 1. S. Bernstein, Elliptic Boundary Value Problems in Unbounded Domains, in ‘‘Clifford Algebras and their Applications in Mathematical Physics,’’ ŽF. Brackx, H. Serras, and R. Delanghe, Eds.., pp. 45]53, Kluwer, Dordrecht, 1993. 2. F. Brackx, R. Delanghe, and F. Sommen, ‘‘Clifford Analysis,’’ Research Notes in Mathematics, Vol. 76, Pitman, London, 1982. 3. E. Franks and J. Ryan, Bounded monogenic functions on unbounded domains, Contemp. Math. to appear. 4. K. Gurlebeck and F. Kippig, Complex Clifford analysis and elliptic boundary value ¨ problems, Ad¨ ances in Applied Clifford Algebra 5 Ž1995., 51]62. 5. K. Gurlebeck and W. Sprossig, ‘‘Quaternionic Analysis and Elliptic Boundary Value ¨ ¨ Problems,’’ International Series of Numerical Mathematics, Vol. 89, Birkhauser, Basel, ¨ 1990. 6. V. Iftimie, Fonctions hypercomplexes, Bull. Math Soc. Sci. Math. R.S. Roumanie Ž N.S.. 9 Ž1965. 279]332. 7. P. P. Korowkin, ‘‘Ungleichungen,’’ Deutscher Verlag der Wissenschaften, Berlin, 1973. 8. J.-L. Lions and E. Magenes, ‘‘Problemes aux Limites Non Homogenes et Applications,’’ ` ` Vol. 1, Dunod, Paris, 1968. 9. C. Li, A. McIntosh, and S. Semmes, Convolution singular integrals on Lipschitz surfaces, J. Amer. Math. Soc., 5 Ž1992. 455]481. 10. A. McIntosh, Clifford algebras, Fourier theory, singular integral operators, and partial differential equations on Lipschitz domains, in ‘‘Clifford Algebras in Analysis and Related Topics’’ ŽJ. Ryan Eds.., CRC Press, Boca Raton, FL, 1995. 11. S. G. Michlin and S. Prossdorf, ‘‘Singulare ¨ ¨ Integraloperatoren,’’ Akademie-Verlag, Berlin, 1980. 12. W. Rudin, ‘‘Real and Complex Analysis,’’ 3rd ed., McGraw-Hill, New York, 1987. 13. J. Ryan, Iterated Dirac operators and conformal transformations in R n , in Proceedings of the Eighteenth International Conference on Differential Geometric Methods in Theoretical Physics,’’ pp. 390]399, World Scientific, Singapore, 1987. 14. J. Ryan, Dirac operators over spheres and hyperbolae, to appear. 15. J. Wloka, ‘‘Partielle Differentialgleichungen,’’ Verlag B. G. Teubner, Stuttgart, 1982. 16. I. N. Vekua, ‘‘Verallgemeinerte Analytische Funktionen,’’ Akademie-Verlag, Berlin, 1963. 17. W. S. Vladimirov, ‘‘Gleichungen der Mathematischen Physik,’’ Nauka, Moscow, 1967 Žin Russian.. 18. Z. Xu and C. Zhou, On boundary value problems of Riemann-Hilbert type for monogenic functions in a half space of R n Ž n G 2., Complex Variables Theory Appl. 3–4, Ž1993., 181]193. 19. C. Zhou, On boundary value problems of Neumann type for the Dirac operator in a half space of R n Ž n G 2., Complex Variables Theory Appl. 1–2, Ž1993., 1]16 .
19, 216]239 Ž1997.
AM970541
Clifford Analysis over Unbounded Domains Klaus Gurlebeck and Uwe Kahler ¨ ¨ Fakultat ¨ fur ¨ Mathematik, Technische Uni¨ erstitat ¨ Chemnitz-Zwickau, D-09107 Chemnitz, Germany
John Ryan Department of Mathematics, Uni¨ ersity of Arkansas, Fayette¨ ille, Arkansas 72701
and Wolfgang Sproßig ¨ T U Bergakademie Freiberg, Fakultat ¨ fur ¨ Mathematik und Informatik, D-09596 Freiberg, Germany Received July 14, 1996; accepted December 30, 1996
A modified Cauchy kernel is introduced over unbounded domains whose complement contains nonempty open sets. Basic results on Clifford analysis over bounded domains are now carried over to this more general context and to functions that are no longer assumed to be bounded. In particular Plemelj formulae are explicitly computed. Basic properties of the Cauchy transform over unbounded domains lying in a half space are investigated, and an orthogonal decomposition of the L2 space for such a domain is set up. At the end a boundary value problem will be studied in the case of an unbounded domain without using weighted Sobolev spaces. Q 1997 Academic Press
1. INTRODUCTION One of the most important parts of Clifford analysis is the possibility to study boundary value problems of mathematical physics in a self-contained form. In w5x the authors studied the application of Clifford analysis, namely, quaternionic analysis, to boundary value problems for various kinds of partial differential equations. However, the authors were only 216 0196-8858r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
217
CLIFFORD ANALYSIS
able to apply their theory to boundary value problems over bounded domains. However, in physics there are many problems stated over unbounded domains. Clifford analysis over bounded domains was developed and applied by a number of authors using different techniques; see for instance w1, 13, 9, 10, 18, and 19x. Moreover, in w3x it was shown that there is an idea to study monogenic functions on unbounded domains, which can be effectively used to develop a similar theory to w5x. In w3x the authors have proved the existence of a Cauchy transform as well as a Cauchy integral formula in the case of Hardy spaces and Holder spaces under some restrictive ¨ conditions. The main problem in w5x is that the Cauchy kernel does not have ‘‘good enough’’ behaviour near infinity; for instance, the T-operator, which is based on the Cauchy kernel, is unbounded in the usual function spaces. The above-mentioned idea from w3x solves this problem by adding an extra term to the Cauchy kernel. That means the authors introduce a ‘‘generalized’’ Cauchy kernel of the form kŽ j , z. s
jyz jqz , n y < j q z
ž
/
where v is the surface area of the unit sphere in R n. This kernel works on unbounded domains V lying in a half space with a sufficiently smooth boundary. Using this kernel they prove a Cauchy integral formula for monogenic functions over unbounded domains and they introduce an analog to the usual FG-operator F˜G f Ž z . s
HGk Ž j , z . a Ž j . f Ž j . dG,
z g V.
We start by introducing a modification l Ž j , x . of this kernel. The advantage of this kernel over the kernel k Ž j , x . is that it can be used over arbitrary unbounded domains whose complement contains a nonempty open set. We show that many basic results from w3x also hold for this kernel, and we also set up analogs of the Plemelj formulae and a Borel]Pompeiu formula for a wide class of C 1 functions defined on unbounded domains. Other basic results from Clifford analysis are shown to hold in the context described here using the kernel l Ž j , x .. Again using the kernel k Ž j , z ., we also introduce analogs to the usual operators from w5x as well as show similar properties. In Section 4 the domains V have to satisfy the condition dŽ V, yV . G d1 ) 0, where dŽ?, ? . is the usual distance between sets. We will prove an orthogonal decomposition theorem for domains V satisfying the preceding conditions. Later on
218
GURLEBECK ET AL. ¨
we show how easy it is to generalize the results from w5x for boundary value problems over bounded domains to boundary value problems over unbounded domains lying in a half space. As a first example, we use the Dirichlet problem for the Laplace equation. This example shows that after the proof of some mapping properties of the integral operators described in the first part of the paper, we are able to use the same tools as known for the case of bounded domains. It is not necessary to introduce weighted Sobolev spaces Žas for instance in w1x.. Consequently, all results connected with the inner product and orthoprojections remain true. Let us remark that for the kernel k Ž j , z . the whole theory from the bounded case can be carried over to the unbounded case. We have to pay for this with a loss of generality in the possible class of unbounded domains. On the other hand, the new kernel l Ž j , z . allows more general domains for investigations in Holder spaces. However, at the moment it is ¨ not clear how we can prove the corresponding properties for this kernel in Sobolev spaces because some of our results from the theory of singular integral operators are applicable only to operators based on k Ž j , z ..
2. PRELIMINARIES Here we outline some basic background material. From R n with orthonormal basis e1 , . . . , e n , we may construct the real 2 n-dimensional Clifford algebra A n with basis 1, e1 , . . . , e n , . . . , e j1 . . . e j r, . . . , e1 . . . e n , where 1 F r F n and j1 - ??? - jr . The algebra A n is set up so that for each x g R n we have that x 2 s y5 x 5 2 . It follows that each nonzero vector x g R n is invertible with multiplicative inverse yxr5 x 5 2 . Up to the minus sign this corresponds to the Kelvin inverse of the vector x. For each A s a0 q ??? qa1 ? ? ? n e1 ??? e n the norm of A is defined to be 5 A 5 s Ž a20 q ??? qa12 ? ? ? n .1r2 . DEFINITION 2.1. For U a domain in R n a differentiable function f : U ª A n is called left monogenic if Df s 0, where D s Ý njs1 e j r x j . A similar definition can be given for right monogenic functions. Basic facts on monogenic function theory can be found in w2x.
3. CAUCHY KERNELS ON UNBOUNDED DOMAINS We shall first give a modification of some of the results described in w3x. In w3x attention is restricted to unbounded domains lying in a half space. This restriction can easily be lifted. Here the type of domain that we shall consider is an unbounded domain U with a sufficiently smooth boundary
219
CLIFFORD ANALYSIS
U. Usually we can consider U to be Lipschitz continuous, but sometimes we will restrict to having a Liapunov domain. For such a domain we may choose a point y lying in the complement of the closure of U. Now we may introduce the Cauchy kernel lŽ j , x. s
jyx jyy . n y 5 j y y5n v 5 j y x5 1
ž
/
It is straightforward to deduce that for some constant C g Rq, lŽ j , x. F C5 x y y5
ny1
Ý 5 j y x 5yj 5 j y y 5 jyn .
js1
Using this inequality one can easily adapt arguments given in w3x to deduce the following form of the Cauchy integral formula over U. THEOREM 3.1. Suppose that f Ž x . is a bounded, left monogenic function on U and extends continuously in the L` sense to the boundary of U. Then for each x g U, f Ž x. s
H Ul Ž j , x . a Ž x . f Ž j . d s Ž j . .
The previous theorem is also true if we replace f by a function defined on U j U such that g is left monogenic on U and satisfies 5 g Ž x .5 F C 5 x 5 s on U for some C g Rq and s g Ž0, 1.. Moreover, on U the function g belongs to the weighted L p space L`Ž5 x 5ys . and g extends continuously in the L`Ž5 x 5ys .5 topology on U from U to U. This continuation can be achieved via homotopy deformations of U within U. We shall denote by AŽU, s . the right Clifford module of functions which are left monogenic on U, satisfy 5 f Ž x .5 F C 5 x 5 s on U for some C g Rq and s g w0, 1., and extend continuously to U in the L`Ž5 x 5ys . topology. It follows from Theorem 3.1 and the remarks following Theorem 3.1 that the space AŽU, s . is a Banach space. For each closed subset K of U we denote the Banach space of left monogenic functions defined in a neighbourhood of K and satisfying 5 f Ž x .5 F C 5 x 5 s by AŽ K, s .. It now follows that one can easily adapt arguments given in w12x and w2x to obtain the following approximation theorem. THEOREM 3.2. Suppose that K is a closed subset of U and that the set U _ K has only one component. Then AŽU, s . is dense in AŽ K, s . with respect to the supremum norm sup x g K 5 f Ž x .5 5 x 5ys .
GURLEBECK ET AL. ¨
220
Proof. The function l Ž j , x . belongs to both AŽU, s . and AŽ K, s . for each x g R n _clŽU . and for j in U, or, respectively, in K. On substituting the function l Ž j , x . for the standard Clifford]Cauchy kernel in the proof of Theorem 18.4 in w2x the result follows. Q.E.D. We now turn to look at the analogs of the Plemelj formulae. We begin with: PROPOSITION 3.1. Suppose that c : U ª A n is both bounded and Holder ¨ continuous with exponent s g Ž0, 1.. Then the integral
H Ul Ž j , x . a Ž j . c Ž j . d s Ž j .
P.V.
is well defined for each x g U. Proof. In the case where U is compact we first place l Ž j , x . s GŽ j y x . y GŽ j y y . , where GŽ x . s Ž1rv . x 5 x 5yn . The integral
H UG Ž j y x . a Ž j . c Ž j . d s Ž j .
P.V.
is handled the same way as in w6x, while the integral
H UG Ž j y y . a Ž j . c Ž j . d s Ž j . defines a constant. Consequently, when U is compact, the integral
H Ul Ž j , x . a Ž j . c Ž j . d s Ž j .
P.V.
is well defined on U. When U is not compact we may note that
H U _UŽ x .l Ž j , x . a Ž j . c Ž j . d s Ž j .
P.V.
F C5 x y y5
yn
H U _UŽ x .5 j 5
c Ž j . ds Ž j . ,
where UŽ x . is a suitable neighbourhood in U of x. Because c is bounded, it follows that the right-hand integral in the previous expression is bounded by C 5 x y y 5 sup
jg U
cŽj .
yn
H U _UŽ x .5 j 5
ds Ž j . ,
221
CLIFFORD ANALYSIS
and this integral is bounded. By integrating over UŽ x . the result now follows by the same argument as presented for the case where U is bounded. Q.E.D. The proof of the previous proposition may readily be adapted to establish the analogous result if we assume that c g L`Ž5 x y y 5.ys for some s g Ž0, 1.. By substituting the kernel l Ž j , x . for the kernel GŽ j y x . we may easily adapt arguments in w6x to deduce Suppose that a g A n is a constant. Then for each x g U,
LEMMA 3.1.
H Ul Ž j , x . a Ž x . a d s Ž j . s
P.V.
1 2
a.
Using the previous results we may deduce continuous with THEOREM 3.3. Suppose that c : U ª A n is Holder ¨ exponent s g Ž0, 1. and that c also belongs to L`Ž5 x y y 5ys . on U. Then
H Ul Ž j , x . a Ž j . c Ž j . d s Ž j .
C Ž x . s P.V.
is also Holder ¨ continuous with exponent s and it also belongs to L`Ž5 x y y 5ys .. Proof. We shall first show that C Ž x . belongs to L`Ž5 x y y 5ys .. First we rewrite C Ž x . as
H Ul Ž j , x . a Ž j . Ž c Ž j . y c Ž x . . d s Ž j .
P.V.
H Ul Ž j , x . a Ž j . c Ž x . d s Ž j . .
q
By the previous lemma this evaluates to 1 2
c Ž x . q P.V.
H Ul Ž j , x . a Ž j . Ž c Ž j . y c Ž x . . d s Ž j . .
Let B U Ž x, 12 5 x y y 5. denote the restriction to U of the ball centered at x and of radius 21 5 x y y 5. Now
H U _ B
U Ž x ,
F
5 xyy 5r2 .
H U _ B
l Ž j , x . a Ž j . Ž c Ž j . y c Ž x . . ds Ž j .
U Žw , 35
xyy 5r2 .
l Ž j , x . a Ž j . Ž c Ž j . y c Ž x . . ds Ž j . ,
GURLEBECK ET AL. ¨
222
where w is the point in U such that 5 w y y 5 s d is the distance from y to U, and B U Ž w, 32 5 x y y 5. is the restriction to U of the ball centered at w and of radius 32 5 x y y 5. The previous integral is dominated by C
H U _ B
U Žw , 35
q
xyy 5r2 .
H U _ B
U Žw , 35
l Ž j , x . a Ž j . c Ž j . ds Ž j .
xyy 5r2 .
l Ž j , x . a Ž j . c Ž x . ds Ž j . .
Moreover,
H U _ B
U Žw , 35
xyy 5r2 .
l Ž j , x . a Ž x . c Ž j . ds Ž j .
F C 5 x y y 5 5 c 5 L`Žys.
`
y2 qs
H3 5 xyy 5r2r
dr ,
where 5 c 5 L`Žys. is the L`Ž5 x y y 5ys . norm of c . Also,
H U _ B
U Žw , 35
xyy 5r2 .
l Ž j , x . a Ž j . c Ž x . ds Ž j .
F C 5 x y y 5 1q s 5 c 5 L`Žys.
`
y2
H3 5 xyy 5r2r
dr.
Furthermore, P.V.
HB
U Ž x ,
5
5 xyy 5r2 5
H0 xyy r2r
FC
l Ž j , x . a Ž j . Ž c Ž j . y c Ž x . . ds Ž j . sy 1
dr.
This last inequality follows because one can find a constant C g Rq, independent of the choice of x and such that 5 j y y 5 - C 5 x y y 5 whenever x g B U Ž x, 12 5 x y y 5.. Now let us consider the restriction, A U Ž w, 21 5 x y y 5, 23 5 x y y 5., of the spherical shell AŽ w, 21 5 x y y 5, 23 5 x y y 5. to U. Let us consider the set K s A U Ž w, 21 5 x y y 5, 23 5 x y y 5._ B U Ž x, 21 5 x y y 5.. On the set K we find that 5 j y x 5 ) 12 5 j y y 5 and for some constant C g Rq we have that 5 y y x 5 - C 5 j y y 5. Consequently, the integral
HKl Ž j , x . a Ž j . Ž c Ž j . y c Ž x . . d s Ž x .
223
CLIFFORD ANALYSIS
is dominated by C 5 x y y 5 1q s 5 x y y 5yn
5
5
H5 3xyyxyy5r2r2r
ny2
dr.
It remains to consider the integral
HB
U Žw ,
5 xyy 5r2 .
l Ž j , x . a Ž j . Ž c Ž j . y c Ž x . . ds Ž j . .
We begin by replacing c Ž j . y c Ž x . by Ž c Ž j . y c Ž w .. q Ž c Ž w . y c Ž x ... It follows that
HB
U Žw ,
F
5 xyy 5r2 .
HB
U Žw ,
q
HB
l Ž j , x . a Ž j . Ž c Ž j . y c Ž x . . ds Ž j .
5 xyy 5r2 .
U Žw ,
l Ž j , x . a Ž j . Ž c Ž j . y c Ž w . . ds Ž j .
5 xyy 5r2 .
G Ž j y y . a Ž j . ds Ž j .
c Ž x. y c Ž w. .
On B U Ž w, 12 5 x y y 5. we have that 5 j y x 5 ) 12 5 x y y 5, 5 j y w 5 F 5 x y y 5, and 5 j y y 5 ) 5 w y j 5. Consequently, the expression
HB
U Žw ,
5 xyy 5r2 .
l Ž j , x . a Ž j . Ž c Ž j . y c Ž w . . ds Ž j .
is dominated by 5 xyy 5r2 sy1
žH
C
r
0
ny2
dr q
5 xyy 5r2 jy1
Ý 5 x y y 5 syj H
r
0
js1
/
dr .
The term
HB
U Žw ,
5 xyy 5r2 .
G Ž j y x . a Ž j . ds Ž j .
c Ž x. y c Ž w.
is dominated by C 5 x y y 5 sq 1yn
5
5
H0 xyy r2r
ny2
dr.
GURLEBECK ET AL. ¨
224
Using Cauchy’s integral formula it may be observed that the expression
HB
U Žw ,
5 xyy 5r2 .
GŽ j y y . a Ž j . ds Ž j .
is dominated by 1q
HS
ny1 Ž
y, r.
GŽ j y y . a Ž j . ds Ž j . ,
where S ny 1 Ž y, r . is the sphere in R n centered at y and of radius r, for any r g Rq. It now follows that the function c satisfies the inequality 5 C Ž x .5 F 5 C x y y 5 s. To show that C is Holder continuous with exponent s one first ¨ notes that
H U Ž G Ž j y x . y G Ž j y u . . a Ž j . c Ž j . d s Ž j . .
C Ž x . y C Ž u . s P.V.
The proof now follows the same lines as the proof of the analogous result in w6x. Q.E.D. It is straightforward to note that if c satisfies the conditions outlined in Theorem 3.3, then the integral
H Ul Ž j , u . a Ž j . c Ž j . d s Ž j . defines a left monogenic function on U. We shall denote the domain, or domains, complementary to U in R n by V. Similarly the integral
H Ul Ž j , ¨ . a Ž j . c Ž j . d s Ž j . defines a left monogenic function on V which takes the value zero at y. Following a similar arguments to those presented in w6x and elsewhere, we can now present analogs of the Plemelj formulae. THEOREM 3.4. Suppose that c satisfies the conditions described in Theorem 3.3. Suppose also that f : w0, 1. ª U is a C 1 function and that lim t ª 1 f Ž t . s x g U, and that lim t ª 1 Ž d frdt .Ž t . is nontangential to U. Then lim
H
tª1 U
l Ž j , f Ž t . . a Ž j . c Ž j . ds Ž j .
s 12 c Ž x . q P.V.
H Ul Ž j , x . a Ž j . c Ž j . d s Ž j . .
225
CLIFFORD ANALYSIS
Similarly if we assume that f : w0, 1. ª V, with lim t ª 1 f Ž t . s x g U and lim t ª 1 Ž d frdt .Ž t . is nontangential to U, then lim
H
tª1 U
l Ž j , f Ž t . . a Ž j . c Ž j . ds Ž x .
s y 12 c Ž x . q P.V.
H Ul Ž j , x . a Ž j . c Ž j . d s Ž j . .
We now proceed to establish: THEOREM 3.5. Suppose that c Ž x . is a Holder-continuous function de¨ fined on U with exponent s g Ž0, 1. and that 5 c Ž x .5 F C 5 x y y 5 s for some C g Rq. Then the left monogenic function F Ž u. s
H Ul Ž j , u . a Ž j . c Ž j . d s Ž j .
defined on U satisfies 5 F Ž u.5 F C 5 x y y 5 s. Proof. Consider the set B U Ž y, 5 5 y y u 5 . s U l B Ž y, 5 y y u 5 . . The integral
H U _ B
U Ž y , 55
yyu 5.
l Ž j , u. a Ž j . c Ž j . ds Ž j .
is dominated by C 5 u y y 5 1q s
`
H5 5 uyy 5r
y2
dr.
On the other hand,
HB
U Ž y , 5 5 uyy 5.
s
l Ž j , u. a Ž j . c Ž j . ds Ž j .
HB
U Ž y , 5 5 uyy 5.
y
HB
GŽ j y y . a Ž j . c Ž j . ds Ž j .
U Ž y , 5 5 uyy 5.
G Ž j y u. a Ž j . c Ž j . ds Ž j . .
GURLEBECK ET AL. ¨
226 The integral
HB
U Ž y , 5 5 uyy 5.
G Ž j y y . a Ž j . c Ž j . ds Ž j .
is dominated by
HB
U Ž y , 5 5 uyy 5.
q
GŽ j y y . a Ž j . Ž c Ž j . y c Ž w . . ds Ž j .
HB
U Ž y , 5 5 uyy 5.
G Ž j y y . a Ž j . ds Ž j .
c Ž w. .
The second term here is part of the Cauchy integral formula for a constant, and 5 c Ž w .5 F C 5 w y y 5 s F CX 5 u y y 5. Moreover, the first term is dominated by
HB
U Ž y , 5 5 uyy 5.
G Ž j y w . a Ž j . Ž c Ž j . y c Ž w . . ds Ž j . .
By similar reasoning to that used in the proof of Theorem 3.3 it may be observed that this term satisfies the desired inequality. The term
HB
U Ž y , 5 5 uyy 5.
G Ž j y u. a Ž j . c Ž j . ds Ž j .
is dominated by
HB
U Ž y , 5 5 uyy 5.
q
G Ž j y u. a Ž j . Ž c Ž j . y c Ž ¨ . . ds Ž j .
HB
U Ž y , 5 5 uyy 5.
G Ž j y u. a Ž j . ds Ž j .
c Ž¨. ,
where ¨ is the point on U that is closest to u. The second term contains the Cauchy integral formula for a constant, and 5 c Ž ¨ .5 F C 5 ¨ y y 5 s. Moreover, 5 ¨ y y 5 - 5 5 u y y 5. On the other hand, the first term in the previous expression is dominated by
HB
U Ž ¨ , 10 5 uyy 5.
GŽ j y ¨ . a Ž j . Ž c Ž j . y c Ž ¨ . . ds Ž j . .
It therefore follows by similar reasoning to that given in the proof of Theorem 3.3, that this integral is dominated by C 5 u y y 5 s. The result follows. Q.E.D.
227
CLIFFORD ANALYSIS
On combining this last result with the Cauchy integral formula described at the beginning of this section and Theorem 3.3 we arrive at THEOREM 3.6. Suppose that c is a Holder-continuous function defined on ¨ U and has exponent s g Ž0, 1.. Suppose also that 5 c Ž x .5 F C 5 x y y 5 s for some C g Rq. Then
H Ul Ž n , x . n Ž n . P.V.H Ul Ž j , n . a Ž j . c Ž j . d s Ž j . d s Ž n . s
P.V.
1 4
c Ž x. .
The previous result shows that the Plemelj formulae, seen as operators acting over the appropriate function space, are projection operators. We now turn to look at the other results that can be obtained using the modified kernel l Ž j , x . instead of the Cauchy kernel GŽ j y x .. We begin with an analog of the Borel]Pompeiu formula. THEOREM 3.7. Suppose that f is a C 1 function defined on the domain U and with a continuous extension to the boundary of U. Suppose also that 5 f Ž x .5 F C1 5 x 5 s and 5 Df Ž x .5 F C2 5 x 5 sy1 , for some constants C1 , C2 g Rq and s g Ž0, 1.. Then for each u g U, f Ž u. s
H Ul Ž j , u . a Ž j . f Ž j . d s Ž j . y HUl Ž j , u . Df Ž j . d j
n
.
The proof of this result follows similar lines to proofs given in w3x. We conclude this section by noting that in w14x it is shown that one can set up a Cauchy integral formula and Dirac operator on the hyperbola Hn . This is done via a Mobius transformation. One can similarly introduce the ¨ Cauchy kernel l H nŽ j , x . s GŽ j y x . y GŽ j y y ., where now j , x, and y are all points on Hn and G is the Cauchy kernel on Hn . It is now a relatively easy exercise to show that all the results obtained so far in this section now may be rederived over the hyperbola.
4. SOME NEW OPERATORS ON UNBOUNDED DOMAINS In this section we develop a new operator theory, which will allow us to study boundary value problems of mathematical physics in a self-contained form. Using the kernel k Ž j , z . s GŽ j y z . y GŽ j q z . we introduce analogs to the usual operators from w5x as well as show similar properties. For the sake of brevity, first we cite a theorem from w11, Chap. XI, Theorem 9.1x, which we will use more than once later.
GURLEBECK ET AL. ¨
228 LEMMA 4.1.
Let the symbol FAŽ Q . of a singular integral operator
Ž Au . Ž z . s au Ž z . q H nrynk Ž Q . u Ž j . , dR n q Ku Ž z . , R
where a is a constant, r s < j y z <, Q s Ž j y z .r< j y z <, k Ž Q . is the characteristic, and K is a compact operator, be essentially bounded. Then we ha¨ e that this operator is bounded in the spaces W 2l ŽR n . for each natural l or l s 0 and the estimate 5 A 5 W 2l ŽR n . F C sup ess FA Ž Q . with C is a space constant Ž in case of L 2 : C s 1.. Preparing the following investigations in Sobolev spaces we start with some auxiliary statements in Holder spaces. In the following we shorten ¨ 5 ? 5 L r Ž V , C l 0, n . to 5 ? 5 r . LEMMA 4.2 w7x. Let a1 , a2 , . . . , a n be real positi¨ e numbers. Furthermore, let s G 1. Then the following inequality is ¨ alid: s
n
n
F n sy1
žÝ / ai
Ý ais .
is1
is1
LEMMA 4.3. Let V be a bounded domain in R n. For any real numbers a and b with 0 F a , b - n, a q b ) n and two different points x 1 , x 2 g V a constant C exists which depends on the domain V with Is
HV < x
dy a
1
y y < < x2 y y <
F
b
C < x 1 y x 2 < aq byn
.
Proof. Let d be the diameter of the domain V. We enclose V in a ball of radius d centered at the point x 1. Now we perform the coordinate transform zX s x 1 y y and obtain
HB Ž x . < z <
X a
d
1
dzX < x 1 y x 2 y zX < b
.
Now setting < x 1 y x 2 < s r, Ž x 1 y x 2 .rr s u , and zX s rz, clearly, we have dzX s r n dz. This leads to the estimates dz
< I < F r ny ay b
HB
s r ny ay b
žH
dr r Ž x 1 .
< z
a
< z <
< z <
b
q
dz
H2-< z <-drr < u y z <
b
< z
/
.
229
CLIFFORD ANALYSIS
The first integral is bounded by a constant A because < u < s 1. Therefore, the order of the singularity of the kernel is less than n. For the second integral, which we denote by I2 , we calculate as < I2 < F
H2-< z <-drr < z <
dz a
b
dz
F 2a
H1-< z <-drr < z <
aq b
.
In this estimation the inequality < z y u < G < z < y < u < s < z < y 1 G 12 < z <
Ž < z < ) 2! .
was used. The change to polar coordinates leads to drr
< I2 < F 2 av
H1
r ny1y ay b d r F 2 av
1
aqbyn
.
This proves our assertion with the constant C s A q 2 av Ž a q b y n.y1 . Note that v again denotes the surface area of the n-dimensional unit sphere. Remark 4.1. The reader can find statements of this type in w17x and w11x, for instance. We begin now with the investigation of the usual T-operator defined by
Ž TV f . Ž x . s H G Ž x y y . f Ž y . dy, V
where GŽ x . s Ž1rv . y xr5 x 5 n. THEOREM 4.1. Let V be a domain Ž bounded or unbounded. in R n, u g L r Ž V, Cl 0, n . Ž r ) n.. Then we get
Ž Tu . Ž x 1 . y Ž Tu . Ž x 2 . F C Ž V , r . 5 u 5 r < x 1 y x 2 < s for x 1 / x 2 and s s 1 y nrr. Proof. At first let V be a bounded domain. From Holder’s inequality ¨ we obtain s
Ž Tu . Ž x 1 . y Ž Tu . Ž x 2 . F
1
v
HV
< u 1 < x 2 y y < ny1 y u 2 < x 1 y y < ny1 < x 1 y y < Ž ny1. s < x 2 y y < Ž ny1. s
s
dy 5 u 5 rs ,
GURLEBECK ET AL. ¨
230
where u i s Ž x i y y .r< x i y y < Ž i s 1, 2.. Applying Lemma 4.2 we get s
Ž Tu . Ž x 1 . y Ž Tu . Ž x 2 . F
2 sy1
v
Ž I1 q I2 . 5 u 5 rs .
Here we use the notation I1 s
HV
s
< x 2 y y < ny1 y < x 1 y y < ny1 < x 1 y y < Ž ny1. s < x 2 y y < Ž ny1. s
dy and
I2 s
HV < x
2
y y < Ž ny1. s
dy.
At first we will consider the integral I1. Without any difficulties we obtain by using Lemmas 4.2 and 4.3 the estimate I1 F Ž n y 2 .
sy1
HV
Ý aq bsny2 < x 2 y y < a s < x 1 y y < b s < x 1 y y < Ž ny1. s < x 2 y y < Ž ny1. s
dy < x 1 y x 2 < s
F C Ž V , r . < x 1 y x 2 < ny n sqs . Introducing the abbreviation z i s x i y y Ž i s 1, 2., the second integral I2 can be estimated in the manner I2 s
HV < z
F2
2
Ž sy1.
F 2 2 sy1
< Ž ny1. s
HV
dy F
HV
z1r< z1 < y z 2r< z1 < q z 2r< z1 < y z 2r< z 2 < < z 2 < Ž ny1. s
< z1 y z 2 < sr< z1 < s q < z 2 < s
HV < z
Ž < z 2 < y < z1 < . r Ž < z1 < < z 2 < .
< z 2 < Ž ny1. s 1 < 2
Ž ny1. s
< z1 < s
s
dy
s
dy
dy < x 1 y x 2 < s .
Note that always ns ) n. Lemma 4.3 now yields the result I2 F C Ž V , r . < x 1 y x 2 < ny Ž ny1. sqsys s C Ž V , r . < x 1 y x 2 < ny n sqs . For 1rs q 1rr s 1 we conclude
Ž Tu . Ž x 1 . y Ž Tu . Ž x 2 . F C Ž V , r . < x 1 y x 2 < 1y n r r 5 u 5 r , where C Ž V, r . are different positive constants which only depend on V and r. If the domain V is unbounded, then we decompose V into a bounded part which contains the points x 1 , x 2 and an exterior domain. Showing the Holder continuity for ŽTu.Ž x . we only have to consider points x 1 , x 2 whose ¨
231
CLIFFORD ANALYSIS
mutual distance is bounded. The T-operator now is split into corresponding parts. For the integral over the bounded part of the domain we have already shown our statement. The kernel function of the integral over the unbounded part satisfies a Lipschitz condition, because it does not contain any singularities. Therefore, this second part also yields an inequality of the previously mentioned type. Remark 4.2. For bounded domains the result of Theorem 4.1 is proved in w5x. The case n s 2 is treated in w16x. COROLLARY 4.1. operator
Let V be a domain with V l y V s B. For the
˜ . Ž x . s H k Ž x, y . u Ž y . dy Ž Tu V
with k Ž x, y . s GŽ x y y . y GŽ x q y . the statement of Theorem 4.1 remains true. Proof. The position of the domain in a half space of R n leads to the fact that GŽ x q y . has no singularities in V. In this way the T-operator is disturbed by an integral operator with a smooth kernel, which realizes such an estimate in a natural way. Q.E.D. Let us remark here that the previously proved results on the boundedness of FG and TV are necessary for a classical theory of boundary value problems stated in Holder spaces. However, our goal is the generalization ¨ of Hilbert space methods to the case of boundary value problems in unbounded domains. Therefore, we only use the previously obtained results for extension procedures from Holder spaces to Sobolev spaces. ¨ These extensions are impossible using TV and FG without corrections of the kernel function or the introduction of weighted Sobolev spaces. For all that follows let V be an unbounded domain lying in a half space and satisfying the condition dŽ V, yV . G d1 ) 0, where dŽ?, ? . is the usual distance between sets. We assume the boundary to be a Liapunov boundary. THEOREM 4.2. Suppose that V is an unbounded domain satisfying our preceding conditions. Then T˜ defined by
˜s Tf
HVk Ž j , z . f Ž j . dV
j
is a continuous mapping from L 2 Ž V . to W 21 Ž V .. Proof. From the results of w3x and Corollary 4.1 the weakly singular integral operator T˜ can easily be extended as a bounded operator to the space L 2 Ž V .. Applying the fact that T˜ is a weakly singular integral
GURLEBECK ET AL. ¨
232
˜ is a differentiable L 2 operator we get from w11, Chap. IX, Sect. 7x that Tf ˜ For function for f g L 2 Ž V .. Now let us consider the operator Ž r z i .T. this operator we have zi
˜s Tf
1
e i y n Ž j i y z i . Ž j y z . r< j y z < 2
HV
v y
< j y z
e i y n Ž j i q z i . Ž j q z . r< j q z < 2 < j q z<
n
f Ž z . dV y
ei n
f Ž z. .
If we ask for the boundedness of this operator as an operator from the space L 2 Ž V . to the space L 2 Ž V ., then we have only to look at the operator
zi
˜s Tf
1
HR
v y
e i y n Ž j i y z i . Ž j y z . r< j y z < 2 < j y z
n
e i y n Ž j i q z i . Ž j q z . r< j q z < 2 < j q z<
n
f Ž z . dR n y
ei n
f Ž z.
as an operator over the whole space L 2 ŽR n .. Obviously, we get our operator if we extend all functions f g L 2 Ž V . by zero to the space R n and then restrict this operator to the domain V. Now all we have to do is look at the symbol FŽ r z i .T˜Ž Q . of the operator
zi
˜s Tf
1
v y
HR
e i y n Ž j i y z i . Ž j y z . r< j y z < 2 n
< j y z
e i y n Ž j i q z i . Ž j q z . r< j q z < 2 < j q z<
n
f Ž z . dR n y
ei n
f Ž z. .
For this symbol the formula FŽ r x i .T˜ Ž Q . s 2 Ž e i y nQXi QX . ln
HS
1
q
ip 2
sign cos g dSQX y
holds. Using Lemma 4.1 we get Ž r z i .T˜: L 2 Ž V . ¬ L 2 Ž V ..
ei n
Q.E.D.
LEMMA 4.4. Assume that V is a domain like in the pre¨ ious theorem. Then for all functions f g L 2 Ž V . the equation
˜ sf DTf holds.
CLIFFORD ANALYSIS
233
This was proved for the case of Holder-continuous functions in w3x and ¨ can easily be extended to L 2 functions by the help of Theorem 4.2 and the continuity of D: W 21 ¬ L 2 . THEOREM 4.3. Let V be a domain like in Theorem 4.2. If f g W 21 Ž V ., then we ha¨ e the Borel]Pompeiu formula
˜ . F˜G f s f y TDf ˜ Applying partial integration Proof. Let us consider the expression TDf. ˜ s f, we get our formula. Q.E.D. with respect to D and the fact that DTf For the F˜G-operator we ha¨ e
PROPOSITION 4.1.
F˜G : W 21r2 Ž G . ¬ W 21 Ž V . l ker D. Proof. Let ¨ g W 21r2 Ž G .. Then there exists a function u g W 21 Ž V . with tr u s ¨ . Using the foregoing Borel]Pompeiu formula and Theorem 4.2 ˜ g W 21 Ž V .. we get FG ¨ s u y TDu Q.E.D. THEOREM 4.4. Suppose that the domain V is as in the pre¨ ious theorem, f g C a Ž G ., 0 - a - 1. Then we ha¨ e for all ˜ z g G, lim F˜G f Ž z . s 12 f Ž ˜ z . q 12 S˜G f Ž ˜ z. ,
zªz˜ zgV
where S˜G f Ž ˜ z . s 2HG k Ž j , ˜ z . a Ž j . f Ž j . dG. Proof. Obviously, we have S˜G : C a Ž G . ¬ C a Ž G . or S˜G : L 2 Ž G . ¬ L 2 Ž G ., respectively, Lemma 4.1, because we have for the symbol of the singular integral part of S˜G FS˜GŽ Q . s iQ. Consider ˜ z g G, « ) 0. Also, consider a point z g V and the sphere SŽ0, r Ž z .. centered at 0 and of radius r Ž z .. We denote by DŽ0, r Ž z .. the open disc centered at 0 and of radius r Ž z .. For r Ž z . sufficiently large z will lie in the domain V l DŽ0, r Ž z ... Let f g C a Ž G .. Then there exists a function u g C a Ž V . with tr u s f. For this function u we have F˜G u Ž z . s
HGlD Ž0, r Ž z ..k Ž j , z . a Ž j . u Ž j . dG HVlSŽ0, r Ž z ..k Ž j , z . a Ž j . u Ž j . dG
q
HG _ D Ž0, r Ž z ..k Ž j , z . a Ž j . u Ž j . dG
q
HVlSŽ0, r Ž z ..k Ž j , z . Ž ya Ž j . . u Ž j . dG,
q
GURLEBECK ET AL. ¨
234 S˜G u Ž ˜ z. s
HGlD Ž0, r Ž z ..k Ž j , ˜z . a Ž j . u Ž j . dG HVlSŽ0, r Ž z ..k Ž j , ˜z . a Ž j . u Ž j . dG
q
HG _ D Ž0, r Ž z ..k Ž j , ˜z . a Ž j . u Ž j . dG
q
HVlSŽ0, r Ž z ..k Ž j , ˜z . Ž ya Ž j . . u Ž j . dG.
q
Because r Ž z . is sufficiently large, we have
HG _ D Ž0, r Ž z ..k Ž j , z . a Ž j . u Ž j . dG
-
HVlSŽ0, r Ž z ..k Ž j , z . Ž ya Ž j . . u Ž j . dG
-
HG _ D Ž0, r Ž z ..k Ž j , ˜z . a Ž j . u Ž j . dG
-
HVlSŽ0, r Ž z ..k Ž j , ˜z . Ž ya Ž j . . u Ž j . dG
-
« 8
« 8
« 8
« 8
, , , ,
because < k Ž j , z .< - C Ž< z
HG _ D Ž0, r Ž z ..k Ž j , z . a Ž j . u Ž j . dG HVlSŽ0, r Ž z ..k Ž j , z . Ž ya Ž j . . u Ž j . dG
q
HG _ D Ž0, r Ž z ..k Ž j , ˜z . a Ž j . u Ž j . dG
y
HVlSŽ0, r Ž z ..k Ž j , ˜z . Ž ya Ž j . . dG
y
-
« 2
.
Ž 1.
Moreover, from w5x it follows that there exists a d ) 0 so that for all z with
HGlD Ž0, r Ž z ..k Ž j , z . a Ž j . u Ž j . dG 1
HVlSŽ0, r Ž z ..k Ž j , z . a Ž j . u Ž j . dG y 2 u Ž ˜z .
q
235
CLIFFORD ANALYSIS
y y
1 2
HGlD Ž0, r Ž z ..k Ž j , ˜z . a Ž j . u Ž j . dG
1
kŽ j , ˜ z . a Ž j . u Ž j . dG H 2 VlS Ž0, r Ž z ..
-
« 2
.
Ž 2.
Consequently, we can conclude from Ž1. and Ž2. that ;« ) 0 'd :
F˜G u Ž z . y 12 u Ž ˜ z . y 12 S˜G u Ž ˜ z. - «
;z: < z y ˜ z< - d.
Because ˜ z g G is arbitrary and f s tr u we have our statement.
Q.E.D.
PROPOSITION 4.2. Using the continuity of the trace operator tr from the domain V to the boundary G Ž see w15x. we can extend our pre¨ ious Plemelj]Sokhotzki formula to Sobole¨ spaces W 2l Ž G ., 0 - l - 1, in the form tr F˜G f s 12 f q 12 S˜G f for f g W 2l Ž G ., 0 - l F 1. Proof. Suppose u g W 2l Ž G ., 0 - l - 1. Then there exists a sequence u n of Holder-continuous functions with u n ª u in W 2l Ž G ., 0 - l - 1. For ¨ these functions u n we have from our previous theorem the formula
Ž tr F˜G . u n s Ž I q S˜G . u n . 1 2
Note that the operator I q S˜G is a continuous mapping from W 2l Ž G ., l ) 0, to W 2l Ž G ., l ) 0 ŽLemma 4.1.. If we now let u n ª u, then we get our result. Q.E.D. PROPOSITION 4.3. Let f g C a Ž G ., 0 - a - 1. Then we can pro¨ e in the same way as in Theorem 4.4 that for all ˜ z g G, lim
zªz˜ zgR n _ Ž VjyV .
F˜G f Ž z . s y 12 f Ž ˜ z . q 12 S˜G f Ž ˜ z. .
From these propositions we obtain that S˜G f Ž ˜ z. s f Ž ˜ z . is necessary and Ž . sufficient for the fact that f ˜ z are boundary values of an A n-valued function which is left monogenic in V. Let us now introduce the operators PG s
1 2
Ž I q S˜G .
and
QG s
1 2
Ž I y S˜G . .
Consequently, we have for PG and Q G the algebraic properties PG2 s PG , Q G2 s Q G , and PG Q G s Q G PG s 0, so we get that PG and Q G are projec-
GURLEBECK ET AL. ¨
236
tions. It may be observed that PG is the projection onto the space of all ŽHolder-continuous or W 2l , l ) 0. A n-valued functions which are left ¨ monogenic extendable into the domain V. THEOREM 4.5. The right Hilbert module L 2 Ž V . allows the orthogonal decomposition L 2 Ž V . s ker D Ž V . l L 2 Ž V . [ D Ž W˚21 Ž V . .
Ž 3.
with respect to the inner product
Ž u, ¨ . s H u¨ dV . V
For the idea of the proof we refer to a proof of a similar theorem in w4x. Proof. The right linear sets X1 s L 2 Ž V . l ker D Ž V .
X 2 s L2 Ž V . ] X1
and
˜ g W 21 Ž V .. From are subspaces of L 2 Ž V .. For any u g L 2 Ž V . we have Tu 1Ž . this it follows that there exists a function ¨ g W 2 V with u s D¨ . Let u g X 2 . Then, we have for all g g X1 ,
HVD¨ g dV s 0, and, in particular, for any l g N,
HVD¨ g
l
dV s 0,
Ž 4.
with gl Ž x . s
Ž x y yl .
Ž x q yl .
y
< x y yl < n
< x q yl < n
l g N, y l g R n _ Ž V j y V . .
,
Obviously, g l g ker DŽ V . l L 2 Ž V .. We assume that the set y l , l g N4 is dense in R n _Ž V j yV.. Then we get for any y l g R n _Ž V j yV.,
n
HVD¨ g
l
dV x s
Ý i , js0
HVe
i
xi
¨ j e j g l dV x
n
sy
Ý i , js0
sy
HVe e ¨
HV¨ Dg
j
l
i j
xi
dV x q
n
g l dV x q
HG¨ a g
l
dGx
Ý
He e ¨ a g
i , js0 G
j
i j
i
l
dGx
CLIFFORD ANALYSIS
s
HGg a ¨ dG
s
HG
l
ž
237
x
yl y x < x y yl < n
q
x q yl < x q yl < n
/
a ¨ dGx
s yv Ž F˜G Ž tr ¨ . . Ž y l . , where tr ¨ denotes the trace of ¨ . Using Ž4. we get F˜G Žtr ¨ . s 0 in R n _Ž V j yV.. Hence, it follows that tr ¨ g im PG l W 21r2 Ž G .. Consequently, there exists a function h g W 21 Ž V . l ker DŽ V . with the property that tr h s tr ¨ . Taking the function w s ¨ y h g W˚21 Ž V . we get that Dw g DŽ W˚21 Ž V ... The result now follows from u s D¨ s Dw. Q.E.D. Remark 4.3. From our decomposition of the space L 2 Ž V . we get the orthoprojections P: L 2 Ž V . ¬ L 2 Ž V . l ker D, Q: L 2 Ž V . ¬ D Ž W˚21 Ž V . . . At this point we have built up an operator theory similar to w5x. In the next section we will take a look at investigating boundary value problems of elliptic partial differential equations. We will show how easy it is to solve questions like existence, and uniqueness with the help of our theory. Moreover, we will be able to give convenient representations of the solutions. These integral representation formulas are adapted for a possible numerical evaluation of the solutions.
5. THE DIRICHLET BOUNDARY VALUE PROBLEM OF THE LAPLACE EQUATION IN UNBOUNDED DOMAINS As an example we study the Laplace equation. This is not only because the Laplace equation provides the simplest example of an elliptic boundary value problem, but also because we can easily demonstrate how powerful our operator theory works in investigating such problems. THEOREM 5.1. problem
Suppose f g L 2 Ž V . and g g W 23r2 Ž G .. Then Dirichlet’s
has a solution u g
W 22, loc Ž V .
yDu s f in V , u s g on G l
W 21
of the form
˜, u s F˜G g q T˜PDh q T˜QTf where h is a
W 22
extension of g.
Ž 5.
238
GURLEBECK ET AL. ¨
Proof. First we investigate the boundary value problem yDu1 s f in V , u1 s 0 on G.
Ž 6.
Now let us mention that the orthoprojection Q maps W2k into W2k, loc . To prove this we consider Q f s f y P f and we use P f g ker DŽ V . ; C`Ž V .. ˜ Using Now let us apply the operator yD s DD to the function ¨ s T˜QTf. the fact that ¨ is at least a W21 function we can apply the D-operator and ˜ Obviously, D¨ is also a W21, loc function, that means D is get D¨ s QTf. applicable again and we obtain that yD¨ s f g L 2 Ž V .. Using known properties of elliptic operators Žsee e.g., w8x. we get that ¨ g W 22, loc Ž V .. Because T˜Q f has vanishing boundary values for each L 2 function f we have that ¨ is a solution of Ž6.. Moreover, we can use this result to solve the boundary value problem yDu 2 s 0 in V , u 2 s g on G.
Ž 7.
Whereas g g W 23r2 Ž G . there exists a W 22 extension h with tr h s g. If we set u 2 s ¨ q h, then we have transformed the last boundary value problem into yD¨ s D h in V , ¨ s 0 on G. Applying our solution of the boundary value problem Ž6. one may determine that ¨ s T˜QT˜ D h. Using the Borel]Pompeiu formula, P s I y Q, and DD s yD we find ¨ s yT˜Q Dh q T˜Q F˜G Dh s yh q T˜PDh q F˜G h or we get u 2 s ¨ q h s F˜G g q T˜PDh. On noting that if u1 , u 2 are solutions of the boundary value problems Ž6. and Ž7. then u s u1 q u 2 is a solution of the boundary value problem Ž5., it can be seen that
˜. u s F˜G g q T˜PDh q T˜QTf
Q.E.D.
˜ is the only soluWe note that the solution u s F˜G g q T˜PDh q T˜QTf tion of the boundary value problem Ž5.. ACKNOWLEDGMENT Part of the work covered in this paper was done while the third author was the recipient of a von Humboldt Research Fellowship visiting the Bergakademie in Freiberg in Sachsen.
CLIFFORD ANALYSIS
239
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