Boundary value problems for the quaternionic Hermitian system in R 4 n

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

MSC: 30G35.

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 1930s. For more details on this function theory we refer to the standard references [5,12,14-16].

More recently Hermitian Clifford analysis emerged as a refinement of the Euclidean setting for the case of . Here, Hermitian monogenic functions are considered, i.e. functions 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 [6-8].

In the papers [10,11,13,17], the Hermitian Clifford analysis setting was further refined by considering functions on with values in a quaternionic Clifford algebra, being simultaneous null solutions of four mutually related quaternionic Dirac operators, which are invariant under the action of the symplectic group. In [3], 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 [9] within the complex Hermitian Clifford case. Subsequently, in [4] 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 of as well as problems of Dirichlet type for the quaternionic Hermitian system. Finally, we also prove an equivalence between both-sided quaternionic Hermitian monogenicity and a certain integral conservation law.

Let be an orthonormal basis of Euclidean space and consider the real Clifford algebra constructed over . The non-commutative multiplication in is governed by the rules:

In one can consider the following automorphisms:

(i) the conjugation and for any ,

(ii) the main involution and for any , .

In particular, we consider the skew-field of quaternions whose elements will be denoted by with and . Clearly, may be identified with the Clifford algebra making the identifications , and . The automorphisms (i) and (ii) then respectively lead to the -conjugation

and to the main -involution

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

Definition 1 (see[17])

The quaternionic Witt basis of , , is given by , , where

We will consider the Clifford vectors

for which , while , , . The corresponding Dirac operators are denoted by , , and . Here we have , with the Laplacian in , and , , . Next, the quaternionic Hermitian variables are introduced:

for which , the symbol ^† denoting Hermitian quaternionic conjugation is defined as the composition of -conjugation and Clifford conjugation in , i.e.. The Hermitian Dirac operators are

for which .

Definition 2 (see [17])

Let Ω be an open set in . A continuously differentiable function is said to be (left) q-Hermitian monogenic in Ω (or q-monogenic for short) iff it satisfies in Ω the system , or, equivalently, the system .

Similarly right q-monogenicity is defined. Left and right q-monogenic functions are called two-sided q-monogenic. A q-monogenic function in Ω is monogenic, and thus harmonic in Ω. Note that Definition 2 was proven in [10] to be equivalent to the system introduced in [13] by group invariance considerations.

The fundamental solutions of the Dirac operators , , i.e. the Euclidean Cauchy kernels, are respectively given by

with the area of the unit sphere in . Explicitly, this means that , . Next we introduce the Hermitian Cauchy kernels:

Note that is not the fundamental solution of . However, the following theorem holds, see [3].

Theorem 1Introducing the circulant () matrices

one obtains that.

Thus, is a fundamental solution of , in a matricial interpretation.

We associate, with functions , , and defined in and taking values in , the () circulant matrix function

We say that G belongs to some class of functions if all its entries belong to that class. In particular, the spaces of k-times continuously differentiable, of α-Hölder continuous () and of p-integrable () circulant matrix functions on some suitable subset E of are respectively denoted by , and . The corresponding spaces of -valued functions are denoted by , and . Moreover, introducing the non-negative function , the classes and may also be defined by means of the respective traditional conditions

and

Definition 3 The () circulant matrix function G is called (left) Q-Hermitian monogenic in Ω (or Q-monogenic for short) iff in Ω, where O denotes the matrix with zero entries.

Similarly right Q-monogenicity is defined by the system . Left and right Q-monogenic matrix functions are called two-sided Q-monogenic. An important special case concerns the diagonal matrix function , with and . Indeed, is left (respectively right) Q-monogenic iff the function g is left (respectively right) q-monogenic.

Now, let be a bounded simply connected domain in with boundary , and denote by the complementary open domain . We assume Γ to be a Liapunov surface. The unit normal vector on Γ at is given by

and similarly as above, we also introduce the vectors , and , giving rise in the usual way (up to a constant factor) to their Hermitian counterparts

and , , , as well as to the circulant matrix . Then, in [3], the following Cauchy integral formulae were proven for Q-monogenic matrix functions and for q-monogenic functions, respectively.

Theorem 2 (Q-Hermitian Cauchy integral formula)

If the matrix functionG, (1), isQ-monogenic in Ω then

Theorem 3 (q-Hermitian Cauchy integral formula)

If the functiongisq-monogenic in Ω then

whereis the corresponding diagonal matrix.

Next, in [4] a Q-Hermitian Cauchy transform was introduced, given by

for a matrix function , where and denote the Hermitian versions of the Clifford vectors and , respectively. is a left Q-monogenic matrix function in , vanishing at infinity; in terms of the Euclidean Cauchy type integrals

it reads as

In particular, for the special case of the matrix , the action of is reduced to

In general will not be a diagonal matrix, whence its entries will not be left q-monogenic functions. However does become diagonal if and only if

in which case we obtain

The following Plemelj-Sokhotski formula, proven in [4], then asserts the existence of the continuous boundary limits of the Q-Hermitian Cauchy transform.

Theorem 4Let (), then the continuous limit values of itsQ-Hermitian Cauchy transformexist and are given by

Here we have introduced the matrix Q-Hermitian Hilbert operator

where the singular integrals

are Cauchy principal values. shows the following traditional properties, see [4].

Theorem 5One has

(i) is a bounded linear operator on ()

(ii) is an involution on ().

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

and

where

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 Ψ in vanishing at infinity and having a prescribed jump G across Γ, i.e.

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 Clifford analysis setting, see [1]. Next, under the condition that , Theorem 4 ensures the solvability of the jump problem (4), its unique solution being given by

Now consider the important special case of the matrix function . The reconstruction problem (4) then is strongly related to the jump problem for the involved q-monogenic function, as addressed in the following theorem.

Theorem 6For a function, the following statements are equivalent:

(i) the jump problem

is solvable in terms ofq-monogenic functions;

(ii) gsatisfies the relations (3);

(iii) gsatisfies the relations.

Proof (i) → (ii)

Associate to the function g the diagonal matrix function . Then , and the jump problem (4) for has the unique solution

Let ψ be a solution of (5), then the circulant matrix

is another solution of the jump problem (4) for , whence the uniqueness yields

implying (ii).

(ii) → (iii)

From the third relation in (3), we have , and hence

the latter following from the second relation in (3) and the -monogenicity of . This fact means that is a -monogenic function in . Moreover, it has a null jump through Γ, whence it vanishes in the whole of . We conclude that . Similarly, we arrive at .

(iii) → (i)

It suffices to observe that, under the conditions stated, is q-monogenic, whence it solves the jump problem (5). □

For right q-monogenic functions the following analogue is obtained.

Theorem 7For a function, the following statements are equivalent:

(i) the jump problem

is solvable in terms of rightq-monogenic functions;

(ii) gsatisfies the relations

(iii) gsatisfies the relations.

The next result deals with the Dirichlet boundary value problem for Q-monogenic functions.

Theorem 8Let, then the following statements are equivalent:

(i) The Dirichlet problem

has a solution.

(ii) (resp. ).

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

(i) → (ii)

Let F be a solution of the Dirichlet problem (7). Then, by the Q-Hermitian Cauchy formula, we have

Taking limits as , (ii) follows in view of Theorem 4.

(ii) → (i)

It suffices to observe that, under the condition (ii), solves (7). □

Theorem 9Let, then the following statements are equivalent:

(i) The Dirichlet problem

has a solution.

(ii) gsatisfies the relations

(iii) gsatisfies the relations

Proof (i) → (ii)

From (i) we see that the matrix function

is a solution of the Dirichlet problem

whence by Theorem 8 we have that . The desired conclusion (ii) then directly follows by comparing the entries in the above equality.

(ii) → (iii)

From the condition it follows that . Therefore, as is harmonic in and , we have in . Using the remaining conditions in (ii) and following a similar reasoning as above, we obtain that g satisfies the relations (3) and hence by Theorem 6 we have that . Consequently, we obtain that , as stated in (iii).

(iii) → (i)

The conditions imply the solvability of the Dirichlet problems

where . Now, let , , , be the respective solutions of (9), then these functions all are solutions of the classical Dirichlet problem

whence they coincide. The function thus is q-monogenic and constitutes a solution of (8). □

For right q-monogenic functions the following analogue is obtained.

Theorem 10Let, then the following statements are equivalent:

(i) The Dirichlet problem

has a solution.

(ii) gsatisfies the relations

(iii) gsatisfies the relations

We now turn our attention towards establishing a connection between the two-sided Q-monogenicity of a matrix function G and the matrix Hilbert transforms and of its trace on the boundary Γ.

Theorem 11Let, such thatin Ω, then the following statements are equivalent:

(i) Gis two-sidedQ-monogenic in Ω.

(ii) .

Proof Assume that, next to its already assumed left Q-monogenicity, G also is right Q-monogenic in Ω. Then by Theorem 8 it holds that

Conversely, suppose that . By Theorem 4 and its right-handed version, we conclude that the corresponding left and right Q-Hermitian Cauchy transform of G, and , have the same boundary values on Γ. This fact, together with their harmonicity, implies that

On the other hand, from the assumed left Q-monogenicity of G we have and hence

which clearly forces G to be two-sided Q-monogenic. □

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

Theorem 12Letbe leftq-monogenic in Ω, then the following statements are equivalent:

(i) gis two-sidedq-monogenic in Ω.

(ii) gsatisfies the relations

(iii) gsatisfies the relations

Proof (i) ↔ (ii)

From (i) we see that the matrix function corresponding to g is two-sided Q-monogenic in Ω, whence (ii) follows from Theorem 11(ii) applied to . Conversely, (ii) can be rewritten in the matricial form , from which (i) follows by observing that the two-sided Q-monogenicity of implied by Theorem 11 is equivalent to the q-monogenicity of g.

(i) ↔ (iii)

It follows from (i) that g is two-sided monogenic w.r.t. , . We may then invoke [[2], Theorem 3.2] in order to conclude that , . Conversely, suppose that (iii) holds. Each of the conditions , , implies the two-sided monogenicity of g in Ω w.r.t. , , see again [[2], Theorem 3.2], whence g is two-sided q-monogenic in Ω. □

The authors declare that they have no competing interests.

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

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).

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