Boundary Value Problems
Volume 2008, Article ID 385874,7pages doi:10.1155/2008/385874
Research Article
A Boundary Value Problem for Hermitian
Monogenic Functions
Ricardo Abreu Blaya,1Juan Bory Reyes,2Dixan Pe ˜na Pe ˜na,3and Frank Sommen3
1Facultad de Inform´atica y Matem´atica, Universidad de Holgu´ın,
Holgu´ın 80100, Cuba
2Departamento de Matem´atica, Universidad de Oriente,
Santiago de Cuba 90500, Cuba
3Department of Mathematical Analysis, Ghent University,
Galglaan 2, 9000 Gent, Belgium
Correspondence should be addressed to Dixan Pe ˜na Pe ˜na,[email protected]
Received 14 September 2007; Accepted 7 December 2007
Recommended by Patrick J. Rabier
We study the problem of finding a Hermitian monogenic function with a given jump on a given hypersurface inRm, m2n. Necessary and sufficient conditions for the solvability of this problem
are obtained.
Copyrightq2008 Ricardo Abreu Blaya et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Hermitian Clifford analysis deals with the simultaneous null solutions of the orthogonal Dirac operators∂xand its twisted counterpart∂x|, introduced below. For a thorough treatment of this
higher-dimensional function theory, we refer the reader to, for example,1–5.
Lete1, . . . , e2nbe an orthonormal basis of the Euclidean spaceR2n. Consider the com-plex Clifford algebraC2n constructed overR2n. The noncommutative multiplication inC2n is governed by
ej2−1, j1, . . . ,2n, ejek ekej 0, 1≤j /k≤2n.
1.1
A basis forC2n is obtained by considering for a setA {j1, . . . , jk} ⊂ {1, . . . ,2n}the element
eA ej1 . . . ejk, with j1 < · · · < jk. For the empty set ∅, we put e∅ 1, the latter being the
identity element.
Any Clifford numbera∈C2nmay thus be written as
a
A
and its Hermitian conjugateais defined by
a
A
aAeA, eA −1kk 1/2eA, |A|k. 1.3
The Euclidean space R2n is embedded in the Clifford algebra C
2n by identifying x1,
. . . , x2nwith the real Clifford vectorxgiven by
x
n
j1
e2j−1x2j−1 e2jx2j
. 1.4
The product of two vectors splits up into a scalar part and a so-called bivector part:
xy−x, y x∧y, 1.5
where
x, y
2n
j1
xjyj,
x∧y
2n
j1 2n
kj 1
ejek
xjyk−xkyj
.
1.6
We also introduce for each real Clifford vectorxits twisted counterpart
x|
n
j1
e2j−1x2j−e2jx2j−1
. 1.7
Note that x2 −x, x −|x|2 −|x||2 x|2. Also observe that the Clifford vectors x and
x| are orthogonal with respect to the standard Euclidean scalar product, which implies that
xx|−x|x.
The Fischer dual of the vectorxis the first-order differential operator
∂x n
j1
e2j−1∂x2j−1 e2j∂x2j
1.8
called Dirac operator. Null solutions of this operator are called monogenic functions, which may be regarded as a natural generalization to a higher-dimensional setting of the holomorphic functions of one complex variablesee6,7. A functionf continuously differentiable in an open set Ω of R2n and taking value inC
2n is said to be leftmonogenic in Ω if and only if
∂xf 0 inΩ. In a similar way, a notion of monogenicity can be associated to the Fischer dual of the vectorx|given by
∂x| n
j1
e2j−1∂x2j −e2j∂x2j−1
. 1.9
We notice that the Dirac operators ∂x and ∂x| anticommute and factorize the Laplacian, that
is, −∂2x Δ −∂2x|. Thus, monogenicity with respect to ∂x resp., ∂x|can be regarded as a
Further, a continuously differentiable functionf in an open setΩofR2nwith values in
C2n is called a left Hermitian monogenic or h-monogenic function in Ω if and only if it satisfies inΩthe system
∂xf0∂x|f. 1.10
Throughout the paperΩ will stand for an open-bounded set inR2nwith a boundary compact topological hypersurfaceΓof finite2n−1-dimensional Hausdorffmeasure, andΩ−R2n\Ω . We assume that both open setsΩ±are connected. Finally, suppose thatfbelongs to the H ¨older spaceC0,αΓ, 0< α <1.
The aim of this paper is to the study the following jump problem forh-monogenic func-tions. Under which conditions can we decompose a givenfonΓas
ff −f−, 1.11
wheref±∈C0,αΓare extendable toh-monogenic functionsF±inΩ±withF−∞ 0?
First, it should be noticed that if this jump problem has a solution, then it is unique. This assertion can be easily proved using the Painlev´e and Liouville theorems in the Clifford analysis settingsee6,8.
This work is motivated by the results obtained in 9, 10where a similar problem was studied for two-sided monogenic functions. For the case of harmonic vector fields, we refer the reader to11.
In order to solve problem1.11, we propose two different approaches. The first one uses an integral criterion forh-monogenicitySection 2; and for the second approach, we establish a conservation law forh-monogenic functionsSection 3.
2. An integral criterion forh-monogenicity
Let us denote byH2n−1the2n−1-dimensional Hausdorffmeasuresee12–14. In this sec-tion, we requireΓto be an Ahlfors-David regular hypersurfacesee15, that is, there exists
c >0 such that for allx∈Γand all 0< r ≤diamΓ,
c−1r2n−1≤ H2n−1Γ∩|y−x| ≤r≤cr2n−1. 2.1 The fundamental solutions of the Dirac operators∂xand∂x|introduced in the previous section
are, respectively,
Ex − 1 σ2n
x
|x|2n, E|x − 1
σ2n
x|
|x|2n, 2.2
whereσ2nis the surface area of the unit sphereS2n−1inR2n.
Let us consider the following Cauchy-type integralsCΓf,CΓ|f, and their singular ver-sionsSΓf,SΓ|f, defined as
CΓfx
ΓEy−xνyfydH
2n−1y,
SΓfz 2lim →0
Γ\{|y−z|≤}Ey−zνy
fy−fzdH2n−1y fz,
CΓ|fx
ΓE
y−xνyfydH2n−1y,
SΓ|fz 2lim →0
Γ\{|y−z|≤}E
y−zνy
fy−fzdH2n−1y fz,
2.3
Here and subsequently,νy nj1e2j−1ν2j−1y e2jν2jystands for the unit normal vector onΓat the pointyintroduced by Federersee13.
Note thatCΓfresp.,CΓ|fis monogenic inR2n\Γwith respect to∂xresp.,∂x|and that CΓf∞ CΓ|f∞ 0.
Let us now formulate some important properties of these integral operators. For their proofs, we refer the reader to16,17.
aSΓf,SΓ|f ∈C0,αΓ.
bSokhotski-Plemelj formulae: forz∈Γ,
C±
Γf
z lim
Ω±x→z
CΓfx 1 2
SΓfz±fz,
CΓ|±fz lim Ω±x→z
CΓ|fx 1 2
SΓ|fz±fz.
2.4
Theorem 2.1integral criterion. The functionfhas anh-monogenic extensionF±inΩ±,F−∞
0, if and only ifSΓf ±fSΓ|f.
Proof. Suppose thatf has anh-monogenic extensionF inΩ . By Cauchy’s integral formula for monogenic functionssee6, we have
CΓfx F x CΓ|fx, x∈Ω . 2.5
Propertybnow implies
SΓf f SΓ|f. 2.6
Conversely, assume thatSΓffSΓ|f. From2.6and using again propertyb, we obtain
CΓf fCΓ| f. 2.7
Note thatCΓf−CΓ|fis harmonic inΩ andCΓf−CΓ| f0. The maximum and the minimum principle for harmonic functions now yieldsCΓf CΓ|finΩ , hence thatCΓfish-monogenic inΩ . Therefore by putting
F x
CΓfx, x∈Ω ,
fx, x∈Γ, 2.8
we obtain anh-monogenic extension offinΩ . The caseΩ−is proved similarly.
We are now in the position to give a first solution to1.11. We first claim that iffcan be decomposed as in1.11, thenSΓf SΓ|f. Indeed,Theorem 2.1now leads to
SΓf SΓf −SΓf−SΓf −SΓf−SΓf. 2.9
On the other hand, if SΓf SΓ|f, then an analysis similar to that in the proof of Theorem 2.1 shows thatCΓf CΓ|f, which implies that CΓf is h-monogenic in R2n \Γ. Fi-nally, by aandb, we conclude that f± C±Γf CΓ|±f is a solution of the jump problem 1.11.
Theorem 2.2. The following statements are equivalent: if can be decomposed as in1.11;
iiSΓfSΓ|f; iiiCΓf CΓ|f;
ivCΓf ish-monogenic inR2n\Γ.
Moreover, if the jump problem1.11is solvable, then its unique solution is given by
f±C±Γf 1
2
SΓf±f
CΓ|±f 1 2
SΓ|f±f.
2.10
3. A conservation law forh-monogenic functions
In the remainder of this paper, we assumeΓto be aC1-smooth hypersurface. Then forxsuffi -ciently close toΓ, we may assume that the orthogonal projection ofxontoΓis unique and it is denoted byx⊥. Let us denote byν nj1e2j−1ν2j−1 e2jν2jthe unit normal vector onΓat the pointx⊥.
In a neighborhood ofΓ, we have the decomposition of∂xin the normal and the tangential partssee18
∂x−ν
ν∂x
ν∂ν ∂x, 3.1
where
∂ν
ν, ∂x
, ∂x−ν
ν∧∂x
. 3.2
Similarly,
∂x| −ν|ν|∂x| ν|∂ν ∂x|, 3.3
with
∂x| −ν|
ν| ∧∂x|
. 3.4
The restrictions of the operators∂xand∂x|toΓwill be denoted by∂ωand∂ω|, respectively.
Let us suppose at the outset thatF ∈C1Ω is a monogenic function inΩ with respect to∂xand setg F|Γ. IfFis moreoverh-monogenic inΩ , then from3.1and3.3, we obtain that in a neighbourhood ofΓintersected withΩ
∂νF−ν∂xF 0,
∂νF−ν|∂x|F 0.
3.5
In this way,ν ∂xF ν|∂x|Fin a neighbourhood ofΓintersected withΩ . By continuity, we get
onΓthe relation
On the other hand, ifgsatisfies3.6, then forG∂x|F, we have
Gν|∂νF ∂x|F, 0ν ∂νF ∂xF. 3.7
Therefore in a neighbourhood ofΓintersected withΩ , we obtain
Gν|ν ∂xF ∂x|F. 3.8
It follows immediately thatG|Γ ν|ν ∂ωg ∂ω|g 0. AsG ish-monogenic inΩ and hence harmonic, we conclude that∂x|F G0 inΩ .
Note that this analysis may be also applied to monogenic functions inΩ−with respect to
∂xvanishing at infinity.
We have thus proved the following.
Theorem 3.1conservation law. LetF± ∈ C1Ω±be a monogenic function inΩ±with respect to
∂x,F−∞ 0. Then,F±is anh-monogenic function inΩ±if and only ifgF±|Γsatisfies3.6.
Let us return to the jump problem 1.11. If f can be decomposed as in 1.11, then Theorem 3.1now gives
ν|ν ∂ωf ∂ω|f
ν|ν ∂ωf ∂ω|f
−ν|ν ∂ωf− ∂ω|f−
0. 3.9
Conversely, suppose thatν|ν ∂ωf ∂ω|f 0. Definef± C±Γf. We will prove thatf±is a solution
of1.11. To do this, takeG∂x|CΓf. It follows that
Gν|ν ∂xCΓf ∂x|CΓf. 3.10
Consequently, the limit valuesG±ofGtaken fromΩ±are given by
G±ν|ν ∂ωC±Γf ∂ω|C±Γf. 3.11
Frombwe see thatG −G−ν|ν ∂ωf ∂ω|f0. As the functionGish-monogenic inR2n\Γ
and vanishes at infinity, we haveG≡0 inR2n\Γ, the last equality being a consequence of the Painlev´e and Liouville theorems.
We thus arrive to another characterization for the solvability of the jump problem1.11.
Theorem 3.2. The jump problem1.11is solvable if and only if
ν|ν ∂ωf ∂ω|f 0. 3.12
Acknowledgments
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