Boundary value problems for the quaternionic Hermitian system in R4n

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R E S E A R C H

Open Access

Boundary value problems for the

quaternionic Hermitian system in

R



n

Ricardo Abreu-Blaya

1

_{, Juan Bory-Reyes}

2

_{, Fred Brackx}

3

_{, Hennie De Schepper}

3*

_{and Frank Sommen}

3

*_{Correspondence:} [email protected] 3_{Department of Mathematical} Analysis, Faculty of Engineering, Ghent University, Galglaan 2, 9000 Gent, Belgium

Full list of author information is available at the end of the article

Abstract

In this paper boundary value problems for quaternionic Hermitian monogenic functions are presented using a circulant matrix approach.

MSC: 30G35

Keywords: quaternionic Cliﬀord analysis; Cauchy integral formula

1 Introduction

Euclidean Clifford analysis is a higher dimensional function theory offering a refinement of classical harmonic analysis. The theory is centred around the concept of monogenic functions,i.e.null solutions of a first-order vector-valued rotation invariant differential operator, called Dirac operator, which factorises the Laplacian; monogenic functions may thus also be seen as a generalisation of holomorphic functions in the complex plane. Its roots go back to the s. For more details on this function theory we refer to the standard references [, , –].

More recently Hermitian Cliﬀord analysis emerged as a reﬁnement of the Euclidean set-ting for the case ofRn_{. Here, Hermitian monogenic functions are considered,}_i.e.

func-tions taking values either in a complex Cliﬀord algebra or in complex spinor space, and being simultaneous null solutions of two complex Hermitian Dirac operators, which are invariant under the action of the unitary group. For the systematic development of this function theory we refer to [–].

In the papers [, , , ], the Hermitian Clifford analysis setting was further refined by considering functions onRn_{with values in a quaternionic Clifford algebra, being}

si-multaneous null solutions of four mutually related quaternionic Dirac operators, which are invariant under the action of the symplectic group. In [], Borel-Pompeiu and Cauchy integral formulas are established in this setting, by following a (×) circulant matrix approach, similar in spirit to the circulant (×) matrix approach introduced in [] within the complex Hermitian Clifford case. Subsequently, in [] a quaternionic Hermitian Cauchy integral is introduced, as well as its boundary limit values, leading to the definition of a matrix quaternionic Hermitian Hilbert transform. These operators provide a useful tool for studying boundary value problems for the quaternionic Hermitian system. This is precisely the main objective of the present paper. The main problems that we address are the problem of finding a quaternionic Hermitian monogenic function with a given jump over a given surface ofRn_{as well as problems of Dirichlet type for the quaternionic}

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Hermitian system. Finally, we also prove an equivalence between both-sided quaternionic Hermitian monogenicity and a certain integral conservation law.

2 Preliminaries

Let (e, . . . ,em) be an orthonormal basis of Euclidean spaceRmand consider the real

Clif-ford algebraR,mconstructed overRm. The non-commutative multiplication inRmis

gov-erned by the rules:

e= –, = , . . . ,m,

eek+eke= , =k.

InRmone can consider the following automorphisms: (i) the conjugatione= –eand for anya,b∈Rm,ab=ba (ii) the main involutione˜= –eand for anya,b∈Rm,ab˜ =a˜b˜.

In particular, we consider the skew-ﬁeld of quaternionsHwhose elements will be denoted byq=x+ix+jx+kxwithi=j=k= – andij= –ji=k. Clearly,Hmay be identiﬁed

with the Cliﬀord algebraR,making the identiﬁcationsi↔e,j↔eandk↔ee. The

automorphisms (i) and (ii) then respectively lead to theH-conjugation

q=x–ix–jx–kx

and to the mainH-involution

qγ≡ ˜q=x–ix–jx+kx.

However, it is quite natural to introduce two moreH-involutions deﬁned by

qα₌_x

+ix–jx–kx, qβ=x–ix+jx–kx.

Deﬁnition (see[]) The quaternionic Witt basis ofHm=H⊗RRm,m= n, is given by {f,fα,f

β ,f

γ

},= , . . . ,n, where

f=e+(–)–ie+(–)–je+(–)–ke+(–),

fα=e+(–)–ie+(–)+je+(–)+ke+(–),

fβ=e+(–)+ie+(–)–je+(–)+ke+(–),

fγ =e+(–)+ie+(–)+je+(–)–ke+(–).

We will consider the Cliﬀord vectors

X=X_=

n

=

(e–x–+e–x–+e–x–+ex),

X_=

n

=

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X_=

=

(e–x–+e–x–e–x––ex–),

X_=

n

=

(e–x–e–x–+e–x––ex–)

for whichX_r= –|X_|_{, while}_X

rXs+XsXr= ,r=s,r,s= , . . . , . The corresponding Dirac

operators are denoted by∂X=∂X_,∂X_,∂X_and∂X_. Here we have∂Xr = –n, withn

the Laplacian inRn_{, and}_∂

Xr∂Xs+∂Xs∂Xr= ,r=s,r,s= , . . . , . Next, the quaternionic

Hermitian variables are introduced:

Z=Z_=X_+iX_+jX_+kX_,

Z_=X_+iX_–jX_–kX_,

Z_=X_–iX_+jX_–kX_,

Z_=X_–iX_–jX_+kX_

for whichZ_Z†_+Z_Z†_+Z_Z†_+Z_Z†_= |X|_{, the symbol}†_{denoting Hermitian} quater-nionic conjugation is deﬁned as the composition ofH-conjugation and Cliﬀord conjuga-tion inR,m,i.e.λ†=

AeAλA. The Hermitian Dirac operators are

∂Z=



(∂X+i∂X+j∂X+k∂X),

∂Z=



(∂X+i∂X–j∂X–k∂X),

∂Z=



(∂X–i∂X+j∂X–k∂X),

∂Z=



(∂X–i∂X–j∂X+k∂X)

for whichn= (∂Z∂

†

Z+∂Z∂

†

Z+∂Z∂

†

Z+∂Z∂

†

Z).

Deﬁnition (see []) Let be an open set inRn_{. A continuously diﬀerentiable function}

f : →Hnis said to be (left)q-Hermitian monogenic in (orq-monogenic for short)

iﬀ it satisﬁes in the system∂Zf =∂Zf =∂Zf =∂Zf = , or, equivalently, the system

∂Xf=∂Xf =∂Xf=∂Xf = .

Similarly right q-monogenicity is deﬁned. Left and rightq-monogenic functions are called two-sidedq-monogenic. Aq-monogenic function in is monogenic, and thus har-monic in . Note that Deﬁnition  was proven in [] to be equivalent to the system in-troduced in [] by group invariance considerations.

The fundamental solutions of the Dirac operators∂Xr, r= , . . . , ,i.e.the Euclidean

Cauchy kernels, are respectively given by

Er(X) = –



an X_r

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withanthe area of the unit sphereSn–inRn. Explicitly, this means that∂_{X r}Er(X) =δ(X), r= , . . . , . Next we introduce the Hermitian Cauchy kernels:

Er(Z) =



an Z†_r

|Z|n, r= , . . . , .

Note thatEris not the fundamental solution of∂_{Z r}. However, the following theorem holds,

see [].

Theorem  Introducing the circulant (×) matrices

D= ⎛ ⎜ ⎜ ⎜ ⎝

∂Z_ ∂Z_ ∂Z_ ∂Z_

∂Z_ ∂Z_ ∂Z_ ∂Z_

∂Z_ ∂Z_ ∂Z_ ∂Z_

∂Z_ ∂Z_ ∂Z_ ∂Z_

⎞ ⎟ ⎟ ⎟

⎠, E=

⎛ ⎜ ⎜ ⎜ ⎝

E E E E

E E E E

E E E E

E E E E

⎞ ⎟ ⎟ ⎟ ⎠, δ= ⎛ ⎜ ⎜ ⎜ ⎝

δ   

 δ  

  δ 

   δ

⎞ ⎟ ⎟ ⎟ ⎠

one obtains thatDTE=EDT=δ.

Thus,Eis a fundamental solution ofD, in a matricial interpretation.

We associate, with functionsg,g,gandgdeﬁned in ⊂Rnand taking values in

Hn, the (×) circulant matrix function

G= ⎛ ⎜ ⎜ ⎜ ⎝

g g g g g g g g g g g g g g g g

⎞ ⎟ ⎟ ⎟ ⎠≡circ

⎛ ⎜ ⎜ ⎜ ⎝ g g g g ⎞ ⎟ ⎟ ⎟ ⎠. ()

We say thatGbelongs to some class of functions if all its entries belong to that class. In particular, the spaces ofk-times continuously diﬀerentiable, ofα-Hölder continuous ( <α≤) and ofp-integrable (×) circulant matrix functions on some suitable subset EofRnare respectively denoted byCk(E),C,α_{(E) and}_Lp_{(E). The corresponding spaces}

ofHn-valued functions are denoted byCk(E),C,α(E) andLp(E). Moreover, introducing

the non-negative functionG(X)=maxr=,,,{|gr(X)|}, the classesC,α(E) andLp(E) may

also be deﬁned by means of the respective traditional conditions

Gα=max

X∈E

G(X)+ sup

X,Y∈E,X=Y

G(X) –G(Y)

|X–Y|α < +∞

and

Gp=

E

G(X)p 

p

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Deﬁnition  The (×) circulant matrix functionGis called (left)Q-Hermitian mono-genic in (orQ-monogenic for short) iﬀDTG=Oin , whereOdenotes the matrix with zero entries.

Similarly rightQ-monogenicity is deﬁned by the systemGDT=O. Left and right Q-monogenic matrix functions are called two-sided Q-monogenic. An important special case concerns the diagonal matrix functionG, withg=gandg=g=g= . Indeed,

G is left (respectively right)Q-monogenic iﬀ the functiong is left (respectively right) q-monogenic.

Now, let += be a bounded simply connected domain inRnwith boundary=∂ , and denote by –_{the complementary open domain}_Rn_\₍ _∪_{). We assume}_{to be a}

Liapunov surface. The unit normal vector onatX∈is given by

n_(X) =

n

=

e–n–(X) +e–n–(X) +e–n–(X) +en(X)

and similarly as above, we also introduce the vectorsn_,n_andn_, giving rise in the usual way (up to a constant factor) to their Hermitian counterparts

N=



(n+in+jn+kn)

andN,N,N, as well as to the circulant matrixN. Then, in [], the following Cauchy

integral formulae were proven forQ-monogenic matrix functions and forq-monogenic functions, respectively.

Theorem (Q-Hermitian Cauchy integral formula) If the matrix functionG, (), isQ

-monogenic in then

∂

E(Z–V)NT(Z)G(X)dS(X) = ⎧ ⎨ ⎩

G(Y), Y∈+,

O, Y∈–_.

Theorem (q-Hermitian Cauchy integral formula) If the function g is q-monogenic in then

∂

E(Z–V)NT(Z)G(X)dS(X) =

⎧ ⎨ ⎩

G_(Y), Y∈+,

O, Y∈–,

whereGis the corresponding diagonal matrix.

Next, in [] aQ-Hermitian Cauchy transform was introduced, given by

C[G](Y) =

E(Z–V)NT(Z)G(X)dS(X), Y∈/ ()

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vanishing at inﬁnity; in terms of the Euclidean Cauchy type integrals

Cr,sg(Y) :=

Er(X–Y)ns(X)g(X)dS(X), Y∈/

it reads as

C[G] =  circ ⎛ ⎜ ⎜ ⎜ ⎝

C,+C,+C,+C, C,–C,+j(C,+C,)

C,–C,+C,–C, C,–C,–j(C,+C,)

⎞ ⎟ ⎟ ⎟ ⎠[G].

In particular, for the special case of the matrixG_, the action ofCis reduced to

C[G] =

 circ ⎛ ⎜ ⎜ ⎜ ⎝

C,g+C,g+C,g+C,g C,g–C,g+j(C,g+C,g)

C,g–C,g+C,g–C,g C,g–C,g–j(C,g+C,g)

⎞ ⎟ ⎟ ⎟ ⎠.

In general C[G_] will not be a diagonal matrix, whence its entries will not be left q -monogenic functions. HoweverC[G_] does become diagonal if and only if

C,g=C,g, C,g= –C,g, C,g=C,g+C,g ()

in which case we obtain

C[G] =circ

⎛ ⎜ ⎜ ⎜ ⎝

C,g

   ⎞ ⎟ ⎟ ⎟ ⎠=circ

⎛ ⎜ ⎜ ⎜ ⎝

C,g

   ⎞ ⎟ ⎟ ⎟ ⎠=  circ ⎛ ⎜ ⎜ ⎜ ⎝

C,g+C,g

   ⎞ ⎟ ⎟ ⎟ ⎠.

The following Plemelj-Sokhotski formula, proven in [], then asserts the existence of the continuous boundary limits of theQ-Hermitian Cauchy transform.

Theorem  Let G∈ C,α₍₎ ₍_{ <}_α _≤__{), then the continuous limit values of its} _Q -Hermitian Cauchy transformC[G]exist and are given by

C±_[_G_](_U_{) =}  

H[G](U)±G(U), U∈.

Here we have introduced the matrixQ-Hermitian Hilbert operator

H[G] = 

circ ⎛ ⎜ ⎜ ⎜ ⎝

H,+H,+H,+H, H,–H,+j(H,+H,)

H,–H,+H,–H, H,–H,–j(H,+H,)

⎞ ⎟ ⎟ ⎟ ⎠[G],

where the singular integrals

Hr,sg(U) = 

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are Cauchy principal values.Hshows the following traditional properties, see [].

Theorem  One has

(i) His a bounded linear operator on(C,α₍_),_{• α}₎₍_{ <}_α_{< }₎ (ii) His an involution onC,α₍₎₍_{ <}_α_{< }_).

Similar results may be obtained for right-hand versions of theQ-Hermitian Cauchy and Hilbert transforms by means of the alternative deﬁnitions

[G]C(Y) =

G(X)NT(Z)E(Z–V)dS(X), Y∈/

and

[G]H= [G] circ

⎛ ⎜ ⎜ ⎜ ⎝

H,+H,+H,+H, H,–H,+j(H,+H,)

H,–H,+H,–H, H,–H,–j(H,+H,)

⎞ ⎟ ⎟ ⎟ ⎠,

where

gHr,s(U) = 

g(X)n_s(X)Er(X–U)dS(X), U∈.

3 Boundary value problems forQ-monogenic functions

In this section we study the so-called jump problem (reconstruction problem) for Q-monogenic functions; that is, we will investigate the problem of reconstructing a Q-monogenic matrix function inRn_\ _{vanishing at inﬁnity and having a prescribed}

jumpGacross,i.e.

+(U) ––(U) =G(U), U∈. ()

First, it should be noted that if this problem has a solution, then it necessarily is unique. This assertion can be easily proven using the Painlevé and Liouville theorems in the Clif-ford analysis setting, see []. Next, under the condition thatG∈C,α₍_{), Theorem }

en-sures the solvability of the jump problem (), its unique solution being given by

(Y) =C[G](Y), Y∈Rn\.

Now consider the important special case of the matrix functionG_. The reconstruction problem () then is strongly related to the jump problem for the involvedq-monogenic function, as addressed in the following theorem.

Theorem  For a function g∈C,α₍₎_{, the following statements are equivalent:} (i) the jump problem

ψ+(U) –ψ–(U) =g(U), U∈ ()

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(ii) gsatisﬁes the relations ();

(iii) gsatisﬁes the relationsC,g=C,g=C,g=C,g.

Proof

(i)→(ii)

Associate to the functiongthe diagonal matrix functionG. ThenG∈C,α(), and the

jump problem () forG_has the unique solution

(Y) =C[G](Y), Y∈Rn\.

Letψbe a solution of (), then the circulant matrix

(Y) =circ ⎛ ⎜ ⎜ ⎜ ⎝ ψ

  

⎞ ⎟ ⎟ ⎟ ⎠

is another solution of the jump problem () forG, whence the uniqueness yields

circ ⎛ ⎜ ⎜ ⎜ ⎝ ψ

  

⎞ ⎟ ⎟ ⎟ ⎠=

 circ

⎛ ⎜ ⎜ ⎜ ⎝

C,g+C,g+C,g+C,g C,g–C,g+j(C,g+C,g)

C,g–C,g+C,g–C,g C,g–C,g–j(C,g+C,g)

⎞ ⎟ ⎟ ⎟ ⎠

implying (ii). (ii)→(iii)

From the third relation in (), we have ∂YC,g=∂YC,g+∂YC,g=∂YC,, and

hence

∂YC,g=

∂YE(X–Y)

n_(X)g(X)dS(X)

= –

∂Y_E(X–Y)

n_(X)g(X)dS(X)

= –∂Y_C,g=∂Y_C,g= , Y∈/

the latter following from the second relation in () and the∂Y-monogenicity ofC,g. This

fact means thatC,g–C,gis a∂Y-monogenic function inR

n_\_{. Moreover, it has a null}

jump through, whence it vanishes in the whole ofRn_{. We conclude that}_C

,g=C,g.

Similarly, we arrive atC,g=C,g.

(iii)→(i)

It suﬃces to observe that, under the conditions stated,C,gisq-monogenic, whence it

solves the jump problem ().

For rightq-monogenic functions the following analogue is obtained.

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(i) the jump problem

ψ+(U) –ψ–(U) =g(U), U∈ ()

is solvable in terms of rightq-monogenic functions; (ii) gsatisﬁes the relations

gC,=gC,, gC,= –gC,, gC,=gC,+gC,;

(iii) gsatisﬁes the relationsgC,=gC,=gC,=gC,.

The next result deals with the Dirichlet boundary value problem forQ-monogenic func-tions.

Theorem  LetG∈C,α₍₎_{, then the following statements are equivalent:} (i) The Dirichlet problem

DT_F₌_O _resp._F_DT₌_O_, _in _,

F=G, on

()

has a solution.

(ii) H[G] =G(resp.[G]H=G).

Proof We give the proof for the left-sided version of the theorem, the right-sided one being completely similar.

(i)→(ii)

LetFbe a solution of the Dirichlet problem (). Then, by theQ-Hermitian Cauchy for-mula, we have

C[F](Y) =F(Y), Y∈ .

Taking limits asY→U∈, (ii) follows in view of Theorem . (ii)→(i)

It suﬃces to observe that, under the condition (ii),F=C[G] solves ().

Theorem  Let g∈C,α₍₎_{, then the following statements are equivalent:} (i) The Dirichlet problem

∂Zf=∂Zf =∂Zf=∂Zf= , in ,

f =g, on

()

has a solution. (ii) gsatisﬁes the relations

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(iii) gsatisﬁes the relations

H,g=H,g=H,g=H,g=g.

Proof

(i)→(ii)

From (i) we see that the matrix function

F=circ

⎛ ⎜ ⎜ ⎜ ⎝

f

  

⎞ ⎟ ⎟ ⎟ ⎠

is a solution of the Dirichlet problem

DT_F₌_O_, _in _,

F=G, on

whence by Theorem  we have thatH[G_] =G_. The desired conclusion (ii) then directly follows by comparing the entries in the above equality.

(ii)→(iii)

From the conditionH,g=H,g=git follows thatC,± g=C±,g. Therefore, asC,g– C,gis harmonic in ±andC,± g–C,±g|= , we haveC,g=C,ginRn\. Using

the remaining conditions in (ii) and following a similar reasoning as above, we obtain that

g satisﬁes the relations () and hence by Theorem  we have thatC,g=C,g=C,g= C,g. Consequently, we obtain thatH,g=H,g=H,g=H,g=g, as stated in (iii).

(iii)→(i)

The conditionsH,g=H,g=H,g=H,g=g imply the solvability of the Dirichlet

problems

∂Xrf = , in ,

f =g, on,

()

wherer= , . . . , . Now, letf,f,f,fbe the respective solutions of (), then these

func-tions all are solufunc-tions of the classical Dirichlet problem

nf = , in ,

f =g, on

whence they coincide. The functionf =f=f=f=fthus isq-monogenic and constitutes

a solution of ().

For rightq-monogenic functions the following analogue is obtained.

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(i) The Dirichlet problem

f∂Z=f∂Z=f∂Z=f∂Z= , in ,

f =g, on

()

has a solution. (ii) gsatisﬁes the relations

gH,=gH,=g, gH,= –gH,, gH,+gH,= g.

gH,=gH,=gH,=gH,g=g.

We now turn our attention towards establishing a connection between the two-sided Q-monogenicity of a matrix functionGand the matrix Hilbert transformsH[G|] and [G|]Hof its trace on the boundary.

Theorem  LetG∈C,α₍ _∪₎_{, such that}_DT_G₌_O

in , then the following statements are equivalent:

(i) Gis two-sidedQ-monogenic in . (ii) H[G|] = [G|]H.

Proof Assume that, next to its already assumed leftQ-monogenicity,Galso is right Q-monogenic in . Then by Theorem  it holds that

H[G|] =G|= [G|]H.

Conversely, suppose thatH[G|] = [G|]H. By Theorem  and its right-handed version, we conclude that the corresponding left and rightQ-Hermitian Cauchy transform ofG, C[G] and [G]C, have the same boundary values on. This fact, together with their har-monicity, implies that

C[G] = [G]C.

On the other hand, from the assumed leftQ-monogenicity ofGwe haveG=C[G] and hence

G=C[G] = [G]C

which clearly forcesGto be two-sidedQ-monogenic.

The following result illustrates the utility of the above theorem when considering q -monogenic functions.

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(i) gis two-sidedq-monogenic in . (ii) gsatisﬁes the relations

H,g=gH,, H,g=gH,,

H,g+H,g=gH,+gH,,

H,g+H,g=gH,+gH,.

H,g=gH,, H,g=gH,, H,g=gH,, H,g=gH,.

Proof

(i)↔(ii)

From (i) we see that the matrix function G corresponding to g is two-sided

Q-monogenic in , whence (ii) follows from Theorem (ii) applied toG. Conversely, (ii)

can be rewritten in the matricial formH[G|] = [G|]H, from which (i) follows by

ob-serving that the two-sidedQ-monogenicity ofG_implied by Theorem  is equivalent to theq-monogenicity ofg.

(i)↔(iii)

It follows from (i) thatg is two-sided monogenic w.r.t.∂Xr,r= , . . . , . We may then

invoke [, Theorem .] in order to conclude thatHr,rg=gHr,r,r= , . . . , . Conversely,

suppose that (iii) holds. Each of the conditionsHr,rg=gHr,r,r= , . . . , , implies the

two-sided monogenicity ofgin w.r.t.∂Xr,r= , . . . , , see again [, Theorem .], whenceg

is two-sidedq-monogenic in .

Competing interests

The authors declare that they have no competing interests. Authors’ contributions

All authors have worked jointly on the manuscript, which is the result of an intensive collaboration. All authors read and approved the ﬁnal manuscript.

Author details

1_{Facultad de Informática y Matemática, Universidad de Holguín, Holguín 80100, Cuba.}2_{Departamento de Matemática,} Universidad de Oriente, Santiago de Cuba 90500, Cuba.3_{Department of Mathematical Analysis, Faculty of Engineering,} Ghent University, Galglaan 2, 9000 Gent, Belgium.

Acknowledgement

Ricardo Abreu-Blaya and Juan Bory-Reyes wish to thank all members of the Department of Mathematical Analysis of Ghent University, where the paper was written, for the invitation and hospitality. They were supported respectively by the Research Council of Ghent University and by the Research Foundation - Flanders (FWO, project 31506208).

Received: 5 January 2012 Accepted: 25 May 2012 Published: 12 July 2012 References

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Cite this article as:Abreu-Blaya et al.:Boundary value problems for the quaternionic Hermitian system inR4n_.