Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 18 (1999), No. 4, 839–848

On a Direct Decomposition of the Space L_{p}(Ω)

U. Kähler

Abstract. A direct decomposition of the space L_{p}(Ω) is obtained as a generalization of the orthogonal decomposition of the space L_{2}(Ω), where one of the subspaces is the space of all monogenic L_{p}-functions. Basic results about the orthogonal decomposition are carried over to this more general context. In the end a boundary value problem of the Stokes equations will be studied by a method based on this direct decomposition.

Keywords: Clifford analysis, L_{p}-decomposition, Stokes equations AMS subject classification: Primary 30 G 35, secondary 31 B 10, 35 Q 30, 46 B 25

1. Introduction

One of the most interesting facts of complex and hypercomplex function theory is the orthogonal decomposition of the space L_{2}(Ω):

L_{2}(Ω) = ker D(Ω) ∩ L_{2}(Ω) ⊕ D(W̊_{2}^{1}(Ω)),  (1)

where ker D(Ω) denotes the set of all holomorphic resp. monogenic functions over Ω. This decomposition has a lot of applications, especially to the theory of partial differential equations (e.g. [3] for the complex case and [6] for the hypercomplex case). In the second paper the authors use this orthogonal decomposition to study boundary value problems of mathematical physics over bounded domains in the scale of Hilbert spaces W_{2}^{k}(Ω) in a self-contained theory. This was extended in [1, 5, 7] to the case of unbounded domains. But, a lot of applications, like investigations of Navier-Stokes equations in higher dimensions, require the usage of the scales of Banach spaces W_{p}^{k}(Ω), where we do not have an orthogonal decomposition.

The aim of this paper is to extend the orthogonal decomposition (1) to the spaces L_{p}(Ω) (1 < p < ∞) in form of a direct decomposition in the hypercomplex case, c.f. in the case of Clifford analysis. In the latter part, we show the easy applicability of this direct decomposition with the example of a Stokes system.

U. Kähler: Universidade de Aveiro, Departamento de Matemática, P-3810 Aveiro, Portugal This paper was done while the author was a recipient of a PRAXIS XXI-scholarship of the Fundação para a Ciência e a Tecnologia visiting the Universidade de Aveiro in Portugal.

ISSN 0232-2064 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Berlin

2. Preliminaries

Let e_{1}, …, e_{n} be an orthonormal basis of ℝ^{n}. The Clifford algebra Cℓ_{0,n} is the free algebra over ℝ^{n} generated modulo the relation

x^{2} = −|x|^{2}e_{0},

where e_{0} is the identity of Cℓ_{0,n}. For the algebra Cℓ_{0,n} we have the anticommutative relationship

e_{i}e_{j} + e_{j}e_{i} = −2δ_{ij}e_{0},

where δ_{ij} is the Kronecker symbol. In the following we will identify the Euclidean space ℝ^{n} with ⋀^{1} Cℓ_{0,n}, the space of all vectors of Cℓ_{0,n}. This means that each element x of ℝ^{n} may be represented by

x = ∑_{i=1}^{n} x_{i}e_{i}.

From an analysis viewpoint one extremely crucial property of the algebra Cℓ_{0,n} is that each non-zero vector x ∈ ℝ^{n} has a multiplicative inverse given by \frac{−x}{|x|^{2}} . Up to a sign this inverse corresponds to the Kelvin inverse of a vector in Euclidean space.

For all what follows let Ω ⊂ ℝ^{n} be a bounded domain with a sufficiently smooth boundary Γ = ∂Ω. Then any function f : Ω ↦ Cℓ_{0,n} has a representation f = ∑_{A} e_{A}f_{A} with ℝ-valued components f_{A}. Thus notations f ∈ C^{k}(Ω, Cℓ_{0,n}) (k ∈ N ∪ {0}) and f ∈ L_{p}(Ω, Cℓ_{0,n}) (1 ≤ p) might be understood both coordinatewisely and directly. For instance, f ∈ L_{p}(Ω, Cℓ_{0,n}) means that {f_{k}} ⊂ L_{p}(Ω, ℝ) or, equivalently, that ∫_{Ω} |f(x)|^{p}dx < ∞. All these sets are Cℓ_{0,n}-bi-modules. L_{2}(Ω, Cℓ_{0,n}) can be converted into a Hilbert Cℓ_{0,n}-module, namely, an inner product can be defined as

⟨f, g⟩ := ∫_{Ω} \overline{f(x)} · g(x) dx

and thus L_{2}(Ω, Cℓ_{0,n}) becomes a right Cℓ_{0,n}-module.

In the following we use the short notation L_{p}(Ω), C^{k}(Ω), etc., instead of L_{p}(Ω, Cℓ_{0,n}), C^{k}(Ω, Cℓ_{0,n}).

We now introduce the Dirac operator D = ∑_{i=1}^{n} e_{i} \frac{∂}{∂x_{i}}. This operator is a hypercomplex analogue to the complex Cauchy-Riemann operator. In particular, we have D^{2} = −Δ, where Δ is the Laplacian over ℝ^{n}. A function f : Ω ↦ Cℓ_{0,n} is said to be left-monogenic if it satisfies the equation (Df)(x) = 0 for each x ∈ Ω. A similar definition can be given for right-monogenic functions. Basic properties of the Dirac operator and left-monogenic functions one can find in [2, 6].

An important example of a left-monogenic function is the so-called generalized Cauchy kernel

e(x) = \frac{1}{ω} \frac{−x}{|x|^{n}} ,

where ω denotes the surface area of the unit ball in ℝ^{n}. This function is a fundamental solution of the Dirac operator which we can use to introduce the following integral operators.

On a Direct Decomposition of the Space L_{p}(Ω) 841

Definition 1. Let us denote by α(y) the outward pointing normal vector to Γ at y. For f ∈ C^{0,β}(Ω) (0 < β ≤ 1) we have

• the Teodorescu transform (or T -operator)

T f(x) = ∫_{Ω} e(x − y)f(y) dΩ_{y},

• the boundary operator F_{Γ}

F_{Γ}f(x) = ∫_{Γ} e(x − y)α(y)f(y) dΓ_{y}

with x ∉ Γ, and

• the singular integral operator S_{Γ}

Sf(x) = 2 ∫_{Γ} e(x − y)α(y)f(y) dΓ_{y}

where x ∈ Γ.

These integral operators have the following mapping properties in scales of Sobolev spaces (k ∈ N ∪ {0}, 1 < p < ∞):

• T : W_{p}^{k}(Ω) ↦ W_{p}^{k+1}(Ω)

• F_{Γ} : W_{p}^{k+1− \frac{1}{p}} (Γ) ↦ W_{p}^{k+1}(Ω)

• S_{Γ} : L_{p}(Γ) ↦ L_{p}(Γ).

For the proof we refer to [6].

In this paper we need an additional mapping property for the T -operator.

Theorem 1. The operator

T : W_{p}^{−1}(Ω) ↦ L_{p}(Ω) (1 < p < ∞)

is bounded.

Proof. Based on the fact that this operator is defined by the acting of its symbol to the Fourier transform of the function, the usual way for the proof would be the investigation of the symbol, i.e. the Fourier transform of the kernel, but here we use another way. This idea is more suitable for us due to the fact that our function spaces are modules, and it demonstrates the way one can work directly with this kind of spaces. Each element f ∈ W_{p}^{−1}(Ω) has a representation in the form

f = f_{0} + ∑_{k=1}^{n} ∂_{k}f_{k}

with f_{0}, f_{k} ∈ L_{p}(Ω). If we choose

v = v_{0} + ∑_{k=1}^{n} ∂_{k}v_{k}

with v_{0}, v_{k} ∈ C_{0}^{∞}(Ω), then we can consider

T v = ∫_{Ω} e(x − y)v(y) dΩ_{y}.

We have

\begin{aligned}T v & = ∫_{Ω} e(x − y) (v_{0}(y) + ∑_{k=1}^{n} ∂_{k}v_{k}(y)) dΩ_{y} \\ & = ∫_{Ω} e(x − y)v_{0}(y)dΩ_{y} + ∑_{k=1}^{n} ∫_{Ω} ∂_{k}e(x − y)v_{k}(y) dΩ_{y}. \end{aligned}

The integrals ∫_{Ω} ∂_{k}e(x−y)v_{k}(y)dΩ_{y} are strongly singular integrals of Calderon-Zygmund type for all k [6], i.e.

∥T v∥_{L_{p}(Ω)} ≤ ∥∫_{Ω} e(x − y)v_{0}(y) dΩ_{y}∥_{L_{p}(Ω)} + ∑_{k=1}^{n} ∥∫_{Ω} ∂_{k}e(x − y)v_{k}(y) dΩ_{y}∥_{L_{p}(Ω)}

≤ C_{0}∥v_{0}∥_{L_{p}(Ω)} + ∑_{k=1}^{n} C_{k}∥v_{k}∥_{L_{p}(Ω)},

where C_{k} (k = 0, …, n) are constant. Using the density of C_{0}^{∞}(Ω) in L_{p}(Ω) in the case of f ∈ W_{p}^{−1}(Ω) we get

∥T f∥_{L_{p}(Ω)} ≤ C (∥f_{0}∥_{L_{p}(Ω)} + ∑_{k=1}^{n} ∥f_{k}∥_{L_{p}(Ω)})

for any arbitrary representation of f. From this ∥T f∥_{L_{p}(Ω)} ≤ Ĉ∥f∥_{W_{p}^{−1}(Ω)} follows

Proposition 1. Due to the fact that the Laplace operator Δ : W_{p}^{1}(Ω) ↦ W_{p}^{−1}(Ω) is bounded we get that the operator D : L_{p}(Ω) ↦ W_{p}^{−1}(Ω) is bounded because of D = −ΔT.

Furthermore, we have the following properties (for proofs we refer again to [6]):

Theorem 2. Let f ∈ L_{p}(Ω) (1 < p < ∞). Then

DT f = f,

i.e. the T -operator is a right inverse to D. In the case of f ∈ C^{0,β}(\overline{Ω}) (0 < β ≤ 1)

DT f(x) = \{ \begin{matrix}f(x) & for x ∈ Ω \\ 0 & for x ∈ ℝ^{n} \ \overline{Ω} \end{matrix}

holds.

Theorem 3 (Borel-Pompeiu formula). For f ∈ W_{p}^{1}(Ω) (1 < p < ∞)

F_{Γ}f + T Df = f

holds.

Theorem 4 (Plemelj-Sokhotzki formulae). Let f ∈ W_{p}^{1− \frac{1}{p}} (Γ) (1 < p < ∞). Then

tr F_{Γ}f = \frac{1}{2}f + \frac{1}{2}S_{Γ}f  and  tr^{−}F_{Γ}f = −\frac{1}{2}f + \frac{1}{2}S_{Γ}f,

where tr^{−} denotes the trace operator in the exterior domain ℝ^{n} \ \overline{Ω}.

From these Plemelj-Sokhotzki formulae we can derive the projections P_{Γ} = \frac{1}{2}I + \frac{1}{2}S_{Γ} and Q_{Γ} = −\frac{1}{2}I + \frac{1}{2}S_{Γ}. Hereby, P_{Γ} is the projection onto the space of all Cℓ_{0,n}- valued functions which are left-monogenic extendable into the domain Ω and Q_{Γ} is the projection onto the space of Cℓ_{0,n}-valued functions which are left-monogenic extendable into the domain ℝ^{n} \ \overline{Ω} and vanish at infinity.

3. A direct L_{p}-decomposition

As we have mentioned earlier the following theorem is proved in [6].

Theorem 5. The Hilbert module L_{2}(Ω) allows the orthogonal decomposition

L_{2}(Ω) = ker D(Ω) ∩ L_{2}(Ω) ⊕ D(W̊_{2}^{1}(Ω)).

The proof of this theorem is based on the existence of an inner product in L_{2}(Ω) and the properties of the boundary projections P_{Γ} and Q_{Γ}. Due to the fact that there is no inner product in L_{p}(Ω) for p ≠ 2 we cannot use the same basic ideas in our attempt to extend the above theorem to the case of L_{p}(Ω). Nevertheless, by using the fact that the Dirichlet problem of the Poisson equation with homogeneous boundary data

\begin{matrix}−Δu & = f in Ω \\ u & = 0 on Γ \end{matrix} \}

for f ∈ W_{p}^{−1}(Ω) (1 < p < ∞) has a unique solution [9], whereby the solution operator will be denoted with Δ_{0}^{−1}, we can prove the following theorem.

Theorem 6. The space L_{p}(Ω) (1 < p < ∞) allows the direct decomposition

L_{p}(Ω) = ker D(Ω) ∩ L_{p}(Ω) + · D(W̊_{p}^{1}(Ω)).  (2)

Proof. As a first step let us take a look at the intersection of the two subspaces ker D(Ω) ∩ L_{p}(Ω) and D(W̊_{p}^{1}(Ω)). Suppose f ∈ ker D(Ω) ∩ L_{p}(Ω) ∩ D(W̊_{p}^{1}(Ω)). Obviously, we have Df = 0.

Moreover, because of f ∈ D(W̊_{p}^{1}(Ω)) there exists a function g ∈ W̊_{p}^{1}(Ω) with Dg = f and Δg = 0. From the uniqueness of Δ_{0}^{−1} we get g = 0 and, consequently, f = 0, i.e. the intersection of these subspaces contains only the zero function. Therefore, our sum is a direct one.

Now let f ∈ L_{p}(Ω). Then f_{2} = DΔ_{0}^{−1}Df ∈ D(W̊_{p}^{1}(Ω)). Let us now apply D to the function f_{1} = f − f_{2}. This results in

Df_{1} = Df − Df_{2} = Df − DDΔ_{0}^{−1}Df = Df + ΔΔ_{0}^{−1}Df = Df − Df = 0,

i.e. Df_{1} ∈ ker D(Ω) ∩ L_{p}(Ω). Because f ∈ L_{p}(Ω) was arbitrary chosen (2) is a decomposition of the space L_{p}(Ω)

Starting from this decomposition we get the projections

P : L_{p}(Ω) ↦ ker D ∩ L_{p}(Ω)
Q : L_{p}(Ω) ↦ D(W̊_{p}^{1}(Ω)).

For p = 2 these are orthoprojections.

Now of interest is the question about representation formulae of our projections. Directly from our proof of decomposition (2) we get

Qf = DΔ_{0}^{−1}Df.

An interesting aspect is that this representation is independent of p. Because of L_{q}(Ω) ⊂ L_{p}(Ω) for q ≥ p this means that Q as well as P maps the subset L_{q}(Ω) of L_{p}(Ω) into itself. Later on we will get other representations of P and Q, but now let us take a look at the applications of this direct decomposition.

4. Applications

As a first application we can derive a very interesting property of our T -operator.

Theorem 7. Let f ∈ L_{p}(Ω) (1 < p < ∞), then tr T f = 0 holds if and only if f belongs to the space imQ.

Proof. If f ∈ imQ, then there exists u ∈ W̊_{p}^{1}(Ω) with f = Du. From the Borel-Pompeiu formula we know u = F_{Γ}u + T Du = T Du = T f. Because of u ∈ W̊_{p}^{1}(Ω) we have tr T f = 0.

Now, let tr T f = 0. Using our projections we can decompose f = Pf + Qf. This results in tr T Pf +tr T Qf = 0. As we have previously shown, always tr T Qf = 0 holds. Therefore, tr T Pf = 0. Moreover, T Pf is harmonic and, due to the uniqueness of the Dirichlet problem of the Laplace equation, equals zero. Applying the Dirac operator D we get 0 = DT Pf = Pf, i.e. f ∈ imQ

The basic ideas of this proof one can find already in [6], were the case of p = 2 was proved.

As we mentioned before, in [6] the orthogonal decomposition (1) is used to solve boundary value problems of mathematical physics. We will illustrate the same applicability of our direct decomposition with the example of a Stokes system.

As a first step we have to take a closer look to the Dirichlet problem of the Poisson equation. From [9] we know that the problem

\begin{matrix}−Δu & = f in Ω \\ u & = 0 on Γ \end{matrix} \}

has a unique solution for f ∈ W_{q}^{−1}(Ω) (1 < q < ∞). This solution can be represented in the form u = T QT f, i.e. we get the identity Δ_{0}^{−1}f = T QT f. Therefore, we get for the general problem

\begin{matrix}−Δu & = f in Ω \\ u & = g on Γ \end{matrix} \}

with f ∈ W_{q}^{k}(Ω) and g ∈ W_{q}^{k+ \frac{3}{2}} (Γ) (k ≥ −1, 1 < q < ∞) the representation

u = F_{Γ}g + T PDh + T QT f,

where h is a W_{q}^{k+2}-extension of g into the domain Ω. The proof is the same as in [6] for the case q = 2.

With the help of this representation we can prove the following

Theorem 8. Suppose k ∈ N and 1 < q < ∞. Then

tr T F_{Γ} : W_{q}^{k− \frac{1}{q}} (Γ) ∩ imP_{Γ} ↦ W_{q}^{k+1− \frac{1}{q}} (Γ) ∩ imQ_{Γ}

is an isomorphism.

Proof. Obviously,

(tr T F_{Γ})(W_{q}^{k− \frac{1}{q}} (Γ)) ⊂ W_{q}^{k+1− \frac{1}{q}} (Γ).

Let us first show that ker tr T F_{Γ} = \{0\}. For this we suppose v ∈ W_{q}^{k− \frac{1}{q}} (Γ) ∩ imP_{Γ} with tr T F_{Γ}v = 0. Because of DDT F_{Γ}v = 0 we get from the uniqueness of the solution of the Dirichlet problem of the Laplace equation T F_{Γ}v = 0. The operator T has only a trivial kernel, i.e. F_{Γ}v = 0, and due to v ∈ imP_{Γ} we have v = 0.

Now, let w ∈ imQ_{Γ}, i.e. w = Q_{Γ}v. From the representation of the solution of the boundary value problem

\begin{matrix}DDu & = 0 in Ω \\ u & = w on Γ \end{matrix} \}

we obtain u = F_{Γ}w + T PDh, where h is a W_{q}^{k+2}-extension of w in Ω. Because of w ∈ imQ_{Γ}, it follows F_{Γ}w = 0 and T PDh = T F_{Γ}PDh. Therefore, we have w ∈ im tr T F_{Γ}.

From the proved injectivity and surjectivity of the mapping tr T F_{Γ} we get our statement

The above used idea of generalizing the corresponding proof in the case of W_{2}^{k}(Ω) to the case of W_{p}^{k}(Ω) (1 < p < ∞) was already outlined in [6].

As a proposition from this theorem we get for our projection P the representation

Pf = F_{Γ}(tr T F_{Γ})^{−1}tr T f

for f ∈ W_{q}^{k}(Ω) (k ∈ N, 1 < q < ∞). In the case of q = 2 this representation is proved in [6].

Now, let us take a look at our example. We consider the system

−Δu + \frac{1}{η} Dp = \frac{ρ}{η}f in Ω  (3)
Sc Du = 0 in Ω  (4)
u = 0 on Γ.  (5)

This system describes the stationary flow of a homogeneous viscuous incompressible fluid for small Reynold numbers. Hereby, u is the velocity, p the pressure, ρ the density, and η the viscosity of the fluid.

In the case of q = 2 this system is investigated in [6], but in these investigations orthogonality conditions are widely used, which make a direct translation to our case impossible. Nevertheless, let us start our investigations with the following

Lemma 1. Suppose f ∈ W_{q}^{−1}(Ω) and p ∈ L_{q}(Ω) (1 < q < ∞). Then for each solution of system (3) − (5) we have the representation

u = \frac{ρ}{η}T QT f − \frac{1}{η} T Qp.

Proof. Let p_{n} ∈ W_{q}^{1}(Ω) with p_{n} → p in L_{q}(Ω). Using the Borel-Pompeiu formula we get

T QT (Dp_{n}) = T Q(p_{n} − F_{Γ}p_{n}) = T Qp_{n}.

From the density of W_{q}^{1}(Ω) in L_{q}(Ω), T QT Dp = T Qp follows. Therefore, for u ∈ W̊_{q}^{1}(Ω) and p ∈ L_{q}(Ω) we have

T QT (\frac{ρ}{η}f) = T QT (DDu + \frac{1}{η} Dp) = u + \frac{1}{η} T Qp

whereas u = \frac{ρ}{η} T QT f − \frac{1}{η} T Qp follows

This result means that our system (3) - (5) is equivalent to the system

u + \frac{1}{η} T Qp =\frac{ρ}{η}T QT f  (6)
\frac{1}{η} Sc Qp =\frac{ρ}{η} Sc QT f  (7)

whereby D applied to the first equation results in

Du + \frac{1}{η} Qp = \frac{ρ}{η}QT f.

For solving our problem it is enough to consider system (6) - (7).

Suppose u ∈ W̊_{q}^{1}(Ω) ∩ ker div, f ∈ W_{q}^{−1}(Ω) and p ∈ L_{q}(Ω, ℝ) (1 < q < ∞). First, let us consider the question if the sum in the above equation represents a direct sum. Hereby, it follows from Du = Qp that Sc Qp = 0 holds, because of Qp = Dv with v ∈ W̊_{q}^{1}(Ω) and Sc Du = 0. Therefore, it means that u = 0 and p = 0, i.e. Du + Qp is a direct sum, which is a subset of imQ.

Therefore, the question is: Does a functional H ∈ (L_{q}(Ω) ∩ imQ)′ with H(Du) = 0 and H(Qp) = 0, but H(QT f) ≠ 0, exists?

Equivalently, one can ask if there exists h ∈ W_{r}^{−1}(Ω) (\frac{1}{r} + \frac{1}{q} = 1) such that

(Du, QT h)_{Sc} = 0  for all u ∈ W̊_{q}^{1}(Ω) ∩ ker div
(Qp, QT h)_{Sc} = 0  for all p ∈ L_{q}(Ω)

but (QT f, QT h)_{Sc} ≠ 0, whereby

(w, v)_{Sc} = Sc ∫_{Ω} \overline{w}vdΩ

with w ∈ L_{q}(Ω) and v ∈ L_{r}(Ω) (\frac{1}{q} + \frac{1}{r} = 1)? Hereby, we are using that (L_{q}(Ω)∩imQ)′ = L_{r}(Ω) ∩ imQ (\frac{1}{q} + \frac{1}{r} = 1) holds.

Let us now consider the system

\begin{matrix}(Du, QT h)_{Sc} & = 0  for all u ∈ W̊_{q}^{1}(Ω) ∩ ker div \\ (Qp, QT h)_{Sc} & = 0  for all p ∈ L_{q}(Ω, ℝ) \end{matrix} \}

with h ∈ W_{r}^{−1}(Ω). Here

(Du, QT h)_{Sc} = (u, DQT h)_{Sc} = (u, h)_{Sc} = 0

holds. From this h = grad g = Dg with g ∈ L_{r}(Ω, ℝ) follows (c.f. [4: Chapter III/Lemma 1.1]). Furthermore, we have

(Qp, QT h)_{Sc} = (Qp, QT Dg)_{Sc} = (Qp, Q(g − F_{Γ}g))_{Sc} = (Qp, Qg)_{Sc} = 0

for all p ∈ L_{q}(Ω, ℝ). Therefore, Qg = 0 as well as h = DQg = 0 and we get

(QT f, QT h)_{Sc} = 0 for all f ∈ W̊_{q}^{1}(Ω).

The conclusion is that we can decompose each function QT f (f ∈ W_{q}^{−1}(Ω), 1 < q < ∞) in the form

Du + \frac{1}{η} Qp = \frac{ρ}{η}QT f

with u ∈ W̊_{q}^{1}(Ω) ∩ ker div and p ∈ L_{q}(Ω, ℝ). Moreover, applying the T -operator

u + \frac{1}{η} T Qp = \frac{ρ}{η}T QT f

follows.

We are now proceed to establish the following

Theorem 9. The Stokes system (3) − (5) has a unique solution {u, p} in the form

u + \frac{1}{η} T Qp = \frac{ρ}{η}T QT f.

Hereby, the hydrostatic pressure p is unique up to a constant.

Remark 1. The uniqueness of the solution can be shown exactly in the same way as in [6].

Now, suppose f ∈ W_{q}^{k}(Ω) (k ≥ 0, 1 < q < ∞). Then the unique solution {u, p} of problem (3) - (5) has the representation

u = \frac{ρ}{η}T Vec T f − \frac{ρ}{η}T Vec F_{Γ}(tr T Vec F_{Γ})^{−1}tr T Vec T f
p = ρ Sc T f − ρ Sc F_{Γ}(tr T Vec F_{Γ})^{−1}tr T Vec T f.

Consequently, we have u ∈ W_{q}^{k+2}(Ω) ∩ W̊_{q}^{1}(Ω) and p ∈ W_{q}^{k+1}(Ω, ℝ). The proof of this representation is analogous to the proof of a similar result in [6] for the case q = 2.

Remark 2. System (3) - (5) should really be considered as an example for the general way of treating such kind of problems by the help of our direct decomposition. In the same way it is also possible to investigate all the other problems from [6 - 8], like the Navier-Stokes problem, Helmholtz equation, Lamé equations, or the biharmonic equation.

References

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[2] Brackx, F., Delanghe, R., and F. Sommen: Clifford Analysis (Research Notes in Mathematics: Vol. 76). Boston: Pitman-Longman 1982.

[3] Dubinski, J. A.: On a nonlinear analytic problem. Russian Acad. Sci. Dokl. Math. 48 (1994), 370 – 375.

[4] Galdi, G. P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. New York: Springer 1994.

[5] Gürlebeck, K., Kähler, U., Ryan, J., and W. Sprößig: Clifford analysis over unbounded domains. Adv. Appl. Math. 19 (1997), 216 – 239.

[6] Gürlebeck, K. and W. Sprößig: Quaternionic and Clifford Calculus for Engineers and Physicists. Chichester: John Wiley &. Sons 1997.

[7] Kähler, U.: Elliptic boundary value problems in bounded and unbounded domains. In: Dirac Operators in Analysis (eds.: J. Ryan and D. Struppa). Boston: Pitman-Longman (to appear).

[8] Kravchenko, V. V. and M. V. Shapiro: Integral Representations for Spatial Models of Mathematical Physics (Research Notes in Mathematics: Vol. 351). Boston: Pitman-Longman 1996.

[9] Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. Berlin: Dt. Verlag Wiss. 1978.

Received 20.11.1998; in revised form 14.05.1999