MATHEMATICAL METHODS IN THE APPLIED SCIENCES Math. Meth. Appl. Sci. 2008; 31:1427–1439 Published online 4 February 2008 in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/mma.980 MOS subject classification: 30 G 35; 42 B 30 On the solutions of the Moisil–Th´ eodoresco system Juan Bory Reyes 1, ∗, † and Richard Delanghe 2 1 Departamento de Matem´ aticas, Universidad de Oriente, Santiago de Cuba 90500, Cuba 2 Department of Mathematical Analysis, Ghent University, Galglaan 2, B-9000 Ghent, Belgium Communicated by K. Guerlebeck SUMMARY A structure theorem is proved for the solutions to the Moisil–Th´ eodoresco system in open subsets of R 3 . Furthermore, it is shown that the Cauchy transform maps L 2 (R 2 , R + 0,2 ) isomorphically onto H 2 (R 3 + , R + 0,3 ), thus proving an elegant generalization to R 2 of the classical notion of an analytic signal on the real line. Copyright 2008 John Wiley & Sons, Ltd. KEY WORDS: Clifford analysis; Moisil–Th´ eodoresco system; Cauchy transform; analytic signal 1. INTRODUCTION: THE MOISIL–TH ´ EODORESCO SYSTEM The aim of this paper is to show how the study of solutions to the Moisil–Th´ eodoresco (M–T) system in open subsets of R 3 may benefit from an approach within the framework of Clifford analysis. In this section, we list all possible approaches known to us and describe various ways of representing the solutions to this system. Let ⊂ R 3 be open and f i , i = 0, 1, 2, 3, be real-valued C 1 -functions in . For reasons that will become clear later on, we denote, henceforth, an arbitrary element x ∈ R 3 by x = (x 0 , x 1 , x 2 ) = (x 0 , x ), where x = (x 1 , x 2 ) ∈ R 2 . Substituting ˜ f = ( f 0 , f) where f = ( f 1 , f 2 , f 3 ), then ˜ f is said to ∗ Correspondence to: Juan Bory Reyes, Departamento de Matem´ aticas, Universidad de Oriente, Santiago de Cuba 90500, Cuba. † E-mail: [email protected], [email protected]Contract/grant sponsor: Clifford Research Group in Ghent and the Cuban Research Group in Clifford analysis; contract/grant number: 01T13804 Copyright 2008 John Wiley & Sons, Ltd. Received 5 December 2006
Transcript
MATHEMATICAL METHODS IN THE APPLIED SCIENCESMath. Meth. Appl. Sci. 2008; 31:1427–1439Published online 4 February 2008 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/mma.980MOS subject classification: 30G 35; 42B 30
On the solutions of the Moisil–Theodoresco system
Juan Bory Reyes1,∗,† and Richard Delanghe2
1Departamento de Matematicas, Universidad de Oriente, Santiago de Cuba 90500, Cuba2Department of Mathematical Analysis, Ghent University, Galglaan 2, B-9000 Ghent, Belgium
Communicated by K. Guerlebeck
SUMMARY
A structure theorem is proved for the solutions to the Moisil–Theodoresco system in open subsets of R3.Furthermore, it is shown that the Cauchy transform maps L2(R
2,R+0,2) isomorphically onto H2(R3+,R+
0,3),
thus proving an elegant generalization to R2 of the classical notion of an analytic signal on the real line.Copyright q 2008 John Wiley & Sons, Ltd.
KEY WORDS: Clifford analysis; Moisil–Theodoresco system; Cauchy transform; analytic signal
1. INTRODUCTION: THE MOISIL–THEODORESCO SYSTEM
The aim of this paper is to show how the study of solutions to the Moisil–Theodoresco (M–T)system in open subsets of R3 may benefit from an approach within the framework of Cliffordanalysis.
In this section, we list all possible approaches known to us and describe various ways ofrepresenting the solutions to this system.
Let �⊂R3 be open and fi , i=0,1,2,3, be real-valued C1-functions in �. For reasons thatwill become clear later on, we denote, henceforth, an arbitrary element x ∈R3 by x=(x0, x1, x2)=(x0, x), where x=(x1, x2)∈R2. Substituting f =( f0, f) where f=( f1, f2, f3), then f is said to
∗Correspondence to: Juan Bory Reyes, Departamento de Matematicas, Universidad de Oriente, Santiago de Cuba90500, Cuba.
Contract/grant sponsor: Clifford Research Group in Ghent and the Cuban Research Group in Clifford analysis;contract/grant number: 01T13804
Copyright q 2008 John Wiley & Sons, Ltd. Received 5 December 2006
1428 J. B. REYES AND R. DELANGHE
satisfy the M–T-system in � if {div f=0
grad f0+curl f=0(1)
Writing the system (M–T) explicitly, f thus has to satisfy the following system of partial differentialequations introduced in [1] (see also [2]):⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
0+ � f1�x0
+ � f2�x1
+ � f3�x2
=0
� f0�x0
+0− � f2�x2
+ � f3�x1
=0
� f0�x1
+ � f1�x2
+0− � f3�x0
=0
� f0�x2
− � f1�x1
+ � f2�x0
+0=0
(2)
As observed by Stein in [3], the system (M–T) may also be obtained by considering two complex-valued functions u=u1+ iu2 and v=v1+ iv2 defined in � and satisfying the system of equations{
�u+�x2v=0
�v−�x2u=0(3)
where �=�x0 + i�x1 and �=�x0 − i�x1 .The identification of (2) and (3) is then made through substituting f0=v2 and f=(u1,u2,v1).The system (M–T) may also be obtained by making use of the algebra H of real quaternions
by considering in R3 the so-called Moisil–Theodoresco operator
D3= i�x0+j�x1+k�x2
where (i, j,k) is the standard set of imaginary units in H.Let us recall that these units satisfy the basic multiplication rules{
i2= j2=k2=−1
ij=k, jk= i, ki= j
If f =( f0, f) associates with the H-valued function
F= f0+ i f1+j f2+k f3 (4)
then f satisfying system (1) is equivalent to F satisfying the equation (see e.g. [2, 4])D3F=0 (5)
As is well known, H is isomorphic to the even subalgebra R+0,3 of the universal Clifford algebra
R0,3 constructed over R0,3. Hereby, R0,3 is the space R3 equipped with a quadratic form of
ON THE SOLUTIONS OF THE MOISIL–THEODORESCO SYSTEM 1429
signature (0,3). Taking an orthogonal basis e=(e0,e1,e2) of R0,3, the basic multiplication rulesin R0,3 are governed by {
e2i =−1, i=0,1,2
ei e j +e j ei =0, i �= j, i, j =0,1,2
The even subalgebra R+0,3 then consists of all linear combinations of scalars and bivectors, i.e. (see
e.g. [5])R+0,3={a0+a12e12+a20e20+a01e01}
where ei j =ei e j and a0,ai j ∈R.Identifying e12,e20 and e01, respectively, with i, j and k, it is readily seen that R+
0,3 is isomorphicto H.
Furthermore, defining the conjugation on R0,3 as the anti-involution a→a,a∈R0,3, such thatfor the basic elements
ei =−ei , i=0,1,2
then the conjugation on R0,3 restricted to R+0,3 coincides with the conjugation on H.
Introducing the Dirac operator �x and the Cauchy–Riemann operator Dx in R3, where
�x =e0�x0 +e1�x1 +e2�x2
and
Dx =e0�x =�x0 +e0e1�x1 +e0e2�x2
and identifying f =( f0, f) with the R+0,3-valued C1-function W in �, where
W = f0+e12 f1+e20 f2+e01 f3 (6)
then saying that f satisfies the system (M–T) is equivalent to saying that W satisfies the equation
�x W =0⇐⇒Dx W =0 (7)
In the terminology of Clifford analysis (see e.g. [5]), any f ∈C1(�;R0,3), satisfying �x f =0 in �is called a left monogenic function in �. The space of these functions is denoted by M(�;R0,3).Note that each f ∈M(�;R0,3) is smooth in �; even more, it is real-analytic in �.
Note also that, by applying the natural linear isomorphism
� :E(�;R0,3)→E(�;�R3)
between the space of smooth R0,3-valued functions and the space of smooth differential formsin � (see [6]), then by virtue of (7), saying that f =( f0, f) is a solution to the (M–T)-system isequivalent to saying that the form
Hereby d and d∗ are, respectively, the exterior derivative and the Hodge co-derivative operatorsacting on E(�;�R3).
Indeed, as has been indicated in [6], the action of �x on E(�;R0,3) corresponds to the actionof d+d∗ on E(�;�R3).
For the interpretation of the (M–T)-system in terms of differential forms, see also [7, 8].Identifying
R2={x=(x1, x2); x j ∈R}with the subspace
R0,2=spanR(e1,e2)
of R0,3, it is clear that R0,2 generates inside R0,3 the Clifford algebra R0,2. The even subalgebraR+0,2 of R0,2 is then given by
R+0,2={a0+a12e12;a0,a12∈R}
whereas its odd subspace R−0,2 is determined by
R−0,2=R0,2=spanR{e1,e2}
It thus means that the function W in (6) may also be expressed as
W =U+e0V (9)
where
U = f0+e12 f1, V =−e1 f3+e2 f2 (10)
As
R0,3=R0,2⊕e0R0,2 (11)
it follows in (9) that W has been decomposed along (11).Note that by means of splitting (11), any function W :�→R0,3 may be decomposed into
W =U+e0V (12)
where U and V are R0,2-valued in �.We then have that for W ∈C1(�,R0,3), in � (see [9, 10])
�xW =0 ⇐⇒ DxW =0 ⇐⇒{
�x0U+�x V =0
�xU+�x0V =0(13)
As mentioned before, Dx =e0�x =�x0 +e0�x is the Cauchy–Riemann operator in R3, where �x =e1�x1 +e2�x2 is the Dirac operator in R2.
Note that �2x =−�x , whereas Dx Dx =Dx Dx =�x , �x being the Laplace operator in R3.Consequently, the components (U,V ) of a function W satisfying (13) are R0,2 valued and
ON THE SOLUTIONS OF THE MOISIL–THEODORESCO SYSTEM 1431
Following the terminology from [9, 10], any pair (U,V ) of harmonic R0,2- valued functions in� satisfying (13) is called a conjugate harmonic pair in �.
Summarizing we thus have the following:
(i) The space of solutions of the (M–T)-system in �⊂R3 open may be identified with thespace of R+
0,3-valued left monogenic functions in �, denoted, henceforth, by MT(�,R+0,3).
The identification is made through (6).(ii) Each W ∈MT(�,R+
0,3) may be decomposed into W =U+e0V . Hereby, (U,V ) is a conju-
gate harmonic pair in �, where U and V are, respectively, R+0,2 and R−
0,2 valued.
In Section 2, some properties of the Clifford algebra R0,3 are recalled.In Section 3, by fully exploiting general results from conjugate harmonicity in Euclidean space
and by making use of the decomposition of R+0,3=R+
0,2⊕e0R−0,2 induced by (11), we arrive at
proving structure theorems for the solutions of the (M–T)-system.In Section 4, it is proved that the Cauchy transform C maps the space L2(R
2,R+0,2) isomorphi-
cally onto H2(R3+,R+0,3). In such a way, a direct generalization is obtained to R2 of the classical
notion of an analytic signal on the real line.
2. THE CLIFFORD ALGEBRA R0,3
Let e=(e0,e1,e2) be an orthogonal basis of R0,3, R0,3 being the vector space R3 provided witha quadratic form of signature (0,3). Then, the basis of the Clifford algebra R0,3 constructed overR0,3 is given by the elements
1; e0,e1,e2; e1e2; e2e0; e0e1; e0e1e2
Hereby, 1 is the identity element in R0,3, and R is identified with R1 in R0,3, R1 being the so-calledset of scalars.
Let us recall that the basic multiplication rules in R0,3 are governed by{e2i =−1, i=0,1,2
ei e j +e j ei =0, i �= j, i, j =0,1,2
Substituting R0,2=spanR{e1,e2}, then R0,2 generates inside R0,3 the Clifford algebra R0,2 withbasis
1; e1,e2; e1e2
The conjugation a→a in R0,3 is defined by ei →ei =−ei , i=0,1,2, and ab=ba for a,b∈R0,3.A norm | | may be introduced on R0,3 by substituting |a|2=(aa), a∈R0,3, where for b∈R0,3,
b denotes the scalar part of b.Clearly, R0,3 splits into
R0,3=R0,2⊕e0R0,2 (14)
Inside R0,3, the even subalgebras R+0,3 of R0,3 and R+
∼=C and that the conjugation in R0,3 is carried over to the conjugationin H, respectively, in C.
The odd subspaces of R0,3 and R0,2 are defined by
R−0,3 = spanR(e0,e1,e2)⊕spanR(e0e1e2)
∼= R0,3⊕e0e1e2R
and
R−0,2 = spanR(e1,e2)
∼= R0,2
Clearly,
R0,3 = R+0,3⊕R−
0,3
R0,2 = R+0,2⊕R−
0,2
(15)
It thus turns out that in terms of decomposition (14),
H∼=R+0,3=R+
0,2⊕e0R−0,2 (16)
R3 is usually imbedded in the following way into subspaces of R0,3.Let x=(x0, x1, x2)=(x0, x)∈R3. Then
(i) x→ x=∑2i=0 xi ei , i.e. R3∼=R0,3 or
(ii) x→ x= x0+e0x , i.e. R3∼=R⊕e0R0,2.
In the latter case, x=(x1, x2)∈R2 has thus been identified with
x=2∑
i=1x j e j ∈R0,2
Following (15), any a∈R0,3, respectively, any q∈R0,2, splits into
a=a++a−, a± ∈R±0,3
and
q=q++q−, q± ∈R±0,2
For x ∈R0,3, respectively, x ∈R0,2, fixed, it is easily seen that the associated left multiplicationoperators x :a→ xa,a∈R0,3, respectively x :q→ xq,q∈R0,2, obey the following rules:
ON THE SOLUTIONS OF THE MOISIL–THEODORESCO SYSTEM 1433
3. STRUCTURE THEOREMS
Let for �⊂R3 open, H(�,R+0,2) denote the space of harmonic R+
0,2-valued functions in �.
Writing H ∈H(�,R+0,2) as H =H0+e1e2H12, we have by virtue of Dx Dx =�x that Dx H ∈
MT(�,R+0,3).
Note hereby that
Dx H = �x0H−e0�x H
=U+e0V
where
U =�x0H+e1e2�x0H12
and
−V =e1(�x1H0+�x2H12)+e2(�x2H0−�x1H12)
We now prove that under certain geometric conditions upon �, the converse also holds, i.e. ifW ∈MT(�,R+
0,3), then there exists H ∈H(�,R+0,2) such that W =Dx H .
Denote by � the orthogonal projection of � on R2.Henceforth, we suppose that � satisfies the following conditions:
(i) � is normal w.r.t. the e0-direction, i.e. there exists x∗0 ∈R such that for all x ∈ �, �∩{x+ te0 :
t ∈R} is connected and contains the element (x∗0 , x).
(ii) � is contractible to a point.
An example of such a domain � is �=]a,b[×�, �⊂R2 being contractible to a point.Condition (i) is sufficient for ensuring the existence of a conjugate harmonic function to a given
U ∈H(�,R+0,2).
Condition (ii) is sufficient to ensure the validity of the inverse Poincare lemma in �.Let W =U+e0V ∈MT(�,R+
0,3), U and V being, respectively, R+0,2 and R−
0,2 valued.Adapting where necessary the reasonings made in [9, 10], we proceed as follows.Let
H(x0, x)=∫ x0
x∗0
U (t, x)dt− h(x) (18)
where h is an R+0,2-valued C∞-function in � satisfying
�x h(x)=�x0U (x∗0 , x)
Then H ∈H(�,R+0,2), where W =Dx H =U+e0V ∈MT(�,R+
0,3) with U =�x0 H =U and V =−�x H .
It thus follows that W −W =e0(V −V ), where V −V is R−0,2 valued and left monogenic in �,
i.e. Dx (e0(V −V ))=0 in �. Consequently, V −V is independent of x0 and �x (V −V )=0 in �.
Denote by H(�,R) and M(�,R−0,2) the spaces of R-valued harmonic and R−
0,2-valued left
monogenic functions in �. Then, as
�x :H(�,R)→M(�,R−0,2)
is surjective, we may find h∈H(�,R) such that V −V =�xh.Substituting H = H+h, then by construction H ∈H(�,R+
0,2) and W =Dx H .We have thus proved.
Theorem 3.1W ∈MT(�,R+
0,3) if and only if there exists H ∈H(�,R+0,2) such that W =Dx H .
Remarks
(i) The mapping a→ae0e1e2,a∈R+0,3, is an isomorphism of R+
0,3 onto R−0,3, its inverse being
given by b→be0e1e2,b∈R−0,3.
As right multiplication with e0e1e2 does not affect left monogenicity of functions in someopen subset � of R3, it means that the solutions to the (M–T)-system may also be consideredas elements of the spaceMT(�,R−
0,3) consisting of all R−0,3-valued left monogenic functions
in �.To be more precise, let W ∈MT(�,R+
0,3) with
W =(W0+e12W12)+e0(W1e1+W2e2)
Then W ∗ =We0e1e2∈MT(�,R−0,3), where
W ∗ = W0e0e1e2−(e0W12+e1W2+e2W1)
= (e1W2+e2W1)+e0(W12−W0e12)
(ii) Notice also that, by using the linear isomorphism � between E(�;R0,3) and E(�;�R3)
(see [6]), right multiplication with the pseudo-scalar e0e1e2 on E(�;R0,3) corresponds totaking the Hodge ∗-operator on E(�;�R3). This leads to an interpretation of the spaceMT(�,R−
0,3) in terms of differential forms (see also [8]).
Theorem 3.1 thus implies
Theorem 3.2W ∗ ∈MT(�,R−
0,3) if and only if there exists an R−0,2-valued harmonic function H∗ in � such that
W ∗ =Dx H∗.
Now let F ∈M(�;R0,3) be decomposed following (15), i.e.
F=F++F−
where F± is R±0,3 valued.
By virtue of (17), �x F=0 implies �x F+ =0 and �x F− =0, where F± ∈MT(�,R±0,3). Moreover,
by means of the isomorphism
MT(�,R+0,3)→MT(�,R+
0,3)e0e1e2=MT(�,R−0,3)
there ought to exist F+ ∈MT(�,R+0,3) such that F− = F+e0e1e2.
ON THE SOLUTIONS OF THE MOISIL–THEODORESCO SYSTEM 1435
Consequently, we obtain
Theorem 3.3Let F ∈C1(�,R0,3). Then the following properties are equivalent:
(i) F ∈M(�,R0,3)
(ii) There exists a pair (F+, F+) in MT(�,R+0,3) such that
F=F++ F+e0e1e2(iii) There exists a pair (H+, H+) in H(�,R+
0,2) such that
F=Dx (H++ H+e0e1e2)
RemarkTheorem 3.3, in fact, indicates that M(�,R0,3) is completely determined by MT(�,R+
0,3).
4. THE CAUCHY TRANSFORM ON L2(R2,R+
0,2)
The aim of this section is to apply results from the interplay between harmonic analysis andClifford analysis in Euclidean space to the three-dimensional case we are dealing with. The mostrelevant result obtained is that the Hardy space H2(R3+,R+
0,3) may be recovered by taking the
Cauchy transform on L2(R2,R+
0,2), thus generalizing the notion of analytic signal on R in an
elegant way to the case R2.For the sake of completeness, we first recall some general properties of the Cauchy transform
and the associated Hilbert transform, the study of which has been the subject of intensive researchever since the mid-1980s (see e.g. [11]).
We formulate these properties for the case R2, the latter being considered as the boundary ofR3+ ={(x0, x) : x0>0}, the upper half space in R3.
The fundamental solution E of �x in R3 is given by
E(x)= 1
4�
x
|x |3 , x ∈R3\{0}
It gives rise to the definition of the Cauchy kernel C(x, y) in R3+ with
C(x, y) = e0E(y, x)
= 1
4�e0
x− y
|x− y|3 , x ∈R3+, y∈R2
e0 being the outward pointing unit normal on R2.The corresponding Cauchy transform C on L2(R
It thus means that any f ∈H2(R2,R+0,3) may be expressed as
f =[ f ]++e0H([ f ]+) (24)
with [ f ]+ ∈L2(R2,R+
0,2).
The elements of H2(R2,R+0,3) are called analytic signals in R2.
Finally, take f ∈L2(R2,R+
0,2) and consider its Cauchy transform C[ f ]. Then from (19) weobtain that
C[ f ](x) = 12 (P[ f ](x)+e0Q[ f ](x))
=U (x)+e0V (x)
where U (x)= 12P[ f ](x) is R+
0,2 valued and V (x)= 12Q[ f ](x)= 1
2P[H[ f ]](x) is R−0,2 valued, i.e.
C[ f ] splits directly in terms of decomposition (15).Consequently, (U,V ) is a conjugate harmonic pair in R3+.Moreover,
C+[ f ]= 12 ( f +e0H[ f ])
i.e. for f ∈L2(R2,R+
0,2), C+[ f ] is an analytic signal.
Summarizing, we have
Theorem 4.1
(i) C : L2(R2,R+
0,2)→H2(R3+,R+0,3) is an isomorphism.
(ii) H2(R2,R+0,3)={ f +e0H[ f ] : f ∈L2(R
2,R+0,2)}
Remarks
(i) Analytic signals in Rm (m�2) were first introduced by J. Gilbert and M. Murray byusing the decomposition R0,m+1=R0,m⊕e0R0,m . They proved that C maps L2(R
m,R0,m)
isomorphically onto H2(Rm+1+ ,R0,m+1), even more generally L p(Rm,R0,m)} onto
H p(Rm+1+ ,R0,m+1), 1�p<+∞(see [11, Theorem 5.33]).In the case R2 we are dealing with, the notion of analytic signal given above is a refinementof theirs. This is mainly a consequence of the fact that, by virtue of the splitting R0,3=R+0,3⊕R−
0,3 and the isomorphism of R+0,3 and R−
0,3 through right multiplication with the
pseudo-scalar e0e1e2, it suffices to consider the Hardy space H2(R3+,R+0,3).
(ii) Note also that if complex analysis in an open subset � of R2 is interpreted as being the studyof R+
0,2-valued null solutions in � of the Cauchy–Riemann operator Dx =�x0 +e0e1�x1 in
R2, then Theorem 4.1 is a direct generalization of the classical one on the real line (see[11, Theorem 4.28]). It is hereby understood that R0,2 is generated by the orthogonal basise=(e0,e1) of R0,2.
ON THE SOLUTIONS OF THE MOISIL–THEODORESCO SYSTEM 1439
(iii) As we have already mentioned in Section 1, the (M–T)-system may also be obtainedwithin the framework of quaternionic analysis in R3, where Theorem 4.1 should have itscounterpart within this theory.For a much elaborated quaternionic analysis approach to the (M–T)-system we refer to[2]. In fact, Section 1 of this study is even dealing with the more general theory of �-holomorphic functions, which is related to the factorization of the Helmholtz operator�3+� in R3 by making use of the algebra H(C) of complex quaternions.
ACKNOWLEDGEMENTS
This paper was written while the first author was visiting the Department of Mathematical Analysis ofGhent University. He was supported by the Special Research Fund No. 01T13804 obtained for collaborationbetween the Clifford Research Group in Ghent and the Cuban Research Group in Clifford analysis, onthe subject Boundary value theory in Clifford analysis. Juan Bory Reyes wishes to thank the members ofthis Department for their hospitality.
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