Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 18 (1999), No. 2, 379–391

Nonlinear Singular Integral Equations Involving the Hilbert Transform in Clifford Analysis

S. Bernstein

Dedicated to Prof. L. von Wolfersdorf on the occasion of his 65th birthday

Abstract. We apply operator-theoretical methods for monotone and maximal monotone operators to prove the existence of solutions for nonlinear singular integral and integro-differential equations involving the Hilbert transform in the Clifford algebra Cℓ_{n,0}. Properties of the Hilbert transform are proved using Clifford analysis. We generalize well-known results concerning the complex Hilbert transform and the singular Cauchy integral operator to higher dimensions.

Keywords: Clifford analysis, Cauchy-type integrals, nonlinear integro-differential equations AMS subject classification: Primary 30 G 35, secondary 45 E 05, 45 E 10, 47 H 30

1. Introduction

Clifford analysis is a generalization of complex function theory to higher dimensions. Therefore it is natural to look for similarities. Methods of monotone operator theory as given in [6, 7] have been applied to nonlinear singular integral equations on the unit circle and on the real line. The basic property used here is the monotonicity, i.e. positivity of the Hilbert transform. To get an overview of these results we recommend the papers [17, 18].

A generalization from the real line to the complex plane is done in [1]. Singular integral operators, especially the Cauchy transform, play an important role in quaternionic and Clifford analysis. We want to give some outline about Clifford analysis and singular integral operators.

A foundation of Clifford analysis was done in [5], quaternionic analysis is treated extensively in [9] and more recently in [10, 11]. These last books explain also some relations to physical problems. Connections between harmonic and monogenic functions are discussed in [8]. The Cauchy transform and some classes of singular integral operators and associated equations in a quaternionic context were investigated in [15] concerning Fredholm property. Cauchy transform and convolution singular integral operators on Lipschitz surfaces using Clifford analytical techniques were treated in [12, 13]. A good overview of monotonicity principles and their application to operator equations is given in [19]. Special nonlinear singular integral equations were considered in [3, 4].

S. Bernstein: University of Mining and Technology, Inst. Appl. Math., D-09596 Freiberg e-mail: bernstein@math.tu-freiberg.de This paper was written when the author was visiting the University of Arkansas at Fayetteville, supported by a Feodor-Lynen fellowship of the Alexander von Humboldt foundation, Bonn, Germany ISSN 0232-2064 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Berlin

Our paper is organized as follows. First, in Section 2, we review some basic properties of Clifford algebras, Clifford function theory and related function spaces. Then, in Section 3, we prove mapping and monotonicity properties of the Hilbert transform and related operators, especially the Nemyckii operator. In the final Section 4 we apply monotone operator theory to several kind of nonlinear singular integral equations involving the Hilbert transform.

2. Preliminaries

We shall briefly review some basic definitions and properties of the function theory corresponding to the Clifford algebra. For a more detailed investigation of this matters, we refer to [5, 8 - 11].

Let {e_{1}, e_{2}, …, e_{m}} be an orthonormal basis in ℝ^{m}. Consider the 2^{m}-dimensional real Clifford algebra Cℓ_{m,0} generated by e_{1}, …, e_{m} according to the multiplication rules

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

where e_{0} is the identity of Cℓ_{m,0}. The elements e_{J} (J = {h_{1}, …, h_{k}} ⊆ {1, …, m}) define a basis of Cℓ_{m,0} where e_{J} = e_{h_{1}} ⋯ e_{h_{k}} (1 ≤ h_{1} < … h_{k} ≤ m) and e_{∅} = e_{0}. Thus, an arbitrary element a ∈ Cℓ_{m,0} can be represented as

a = ∑_{J} a_{J}e_{J} (a_{J} ∈ ℝ).

Especially, the elements x⃗ ∈ ℝ^{m} will be identified with ∑_{j=1}^{m} x_{j}e_{j} ∈ Cℓ_{m,0}. We want to denote by Sc a = a_{0}e_{0} = a_{0} the scalar part of a and by Vec a = a − Sc a the (multi−) vectorpart.

We introduce an automorphism called reversion. The reversed element â of a is given by â = ∑_{J} a_{J} ê_{J} where ê_{0} = e_{0}, ê_{j} = e_{j} and ê_{J} = ê_{h_{k}} ⋯ ê_{h_{2}} · ê_{h_{1}}. Then by

[a, b] = Sc âb = Sc ab̂ = ∑_{J} a_{J}b_{J}  and  |a|^{2} = Sc âa = Sc aâ

the scalar product and norm are given in Cℓ_{m,0} and we have especially xx̂ = ∑_{j=1}^{m} x_{j}^{2} = |x|^{2}.

We suppose G ⊂ ℝ^{m} to be a domain with a smooth boundary Γ. We consider functions f defined on G with values in Cℓ_{m,0}. These functions can be written as

f(x) = ∑_{J} f_{J}(x)e_{J} (x ∈ G).

Properties such as continuity, differentiability, integrability, and so on, which are ascribed to f have to be possessed by all components f_{J}(x). In this way the usual Banach spaces of these functions are denoted by C^{α}, L^{2}, H^{1} and H_{0}^{1}. Further,

H^{1}(G) = \{u ∈ L^{2}(G) : \frac{∂u}{∂x_{k}} ∈ L^{2}(G)\}

and H_{0}^{1}(G) is the closure of C_{0}^{∞}(G) in H^{1}. We now define the Dirac operator D by

D = ∑_{k=1}^{m} e_{k} \frac{∂}{∂x_{k}} .

We consider in G the equation

(Du)(x) = 0

and look for its solutions which are called left-monogenic functions in G.

Now we define the Cauchy kernel in ℝ^{m} by

e(x) = \frac{1}{σ_{m}} \frac{x}{|x|^{m}} (x ≠ 0) with σ_{m} = \frac{2π^{\frac{m}{2}}}{Γ (\frac{m}{2})}.

It is well known that e (a fundamental solution of D) is monogenic in ℝ^{m} \ {0}. Using the function e we introduce the integral operators

(T_{G}u)(x) = ∫_{G} e(x − y)u(y) dy  (x ∈ ℝ^{m}) (Teodorescu transform)
(F_{Γ}u)(x) = − ∫_{Γ} e(x − y)n(y)u(y) dΓ_{y} (x ∉ Γ) (Cauchy type operator)
(S_{Γ}u)(x) = − ∫_{Γ} 2e(x − y)n(y)u(y) dΓ_{y} (x ∈ Γ) (singular integral operator)

where n(y) = ∑_{i=1}^{m} e_{i}n_{i}(y) is the exterior unit vector to Γ at the point y. The integral which defines the operator S_{Γ} has to be taken in the sense of the Cauchy principal value. We remark that the operators F_{Γ}, S_{Γ}, P_{Γ} and Q_{Γ} are defined in spaces of Hölder continuous functions. It is possible to extend these operators to Sobolev spaces in the classical way by approximation (with Hölder continuous functions). We omit the detailed discussion here.

We introduce weighted L^{2}-spaces. Let G be a bounded or unbounded smooth domain in ℝ^{m} and

L^{2,α}(G, Cℓ_{m,0}) := \{u : (1 + |x|^{2})^{\frac{α}{2}} u ∈ L^{2}(G, Cℓ_{m,0})\}.

These spaces are (real) Hilbert spaces with the scalar product

(u, v)_{α} = Sc ∫_{G}(1 + |x|^{2})^{α}û(x)v(x) dx = ∫_{G}(1 + |x|^{2})^{α}[u(x), v(x)] dx

and the norm is ||u||_{α} = \sqrt{(u, u)_{α}}. We set (·, ·) = (·, ·)_{0} and || · || = || · ||_{0}. Further, we will use the weighted Sobolev spaces

\begin{matrix}H^{1,α}(G, Cℓ_{m,0}) = \\ \{u : (1 + |x|^{2})^{\frac{α}{2}} u ∈ L^{2}(G, Cℓ_{m,0}) and (1 + |x|^{2})^{\frac{α+1}{2}} Du ∈ L^{2}(G, Cℓ_{m,0})\}. \end{matrix}

It is easy to see that if G is a bounded domain, these weighted spaces coincide with L^{2}(G, Cℓ_{m,0}) and H^{1}(G, Cℓ_{m,0}), respectively.

From [2] we immediately get the following statements.

Lemma 1. Let u ∈ H^{1,α−1}(G, Cℓ_{m,0}) (−\frac{m}{2} + 1 < α < \frac{m}{2}). Then we have:

(i) F_{Γ}u + T_{G}Du = \{ \begin{matrix}u & for x ∈ G \\ 0 & for x ∈ ℝ^{m} \ \overline{G} \end{matrix} (Borel-Pompeiu formula).

(ii) DT_{G}u = \{ \begin{matrix}u & in G \\ 0 & in ℝ^{m} \ \overline{G} \end{matrix} .

(iii) DF_{Γ}u = 0 in G ∪ (ℝ^{m} \ \overline{G}).

Lemma 2 (Plemelj-Sokhotzkij formulas). Let u ∈ C^{0,α}(G, Cℓ_{m,0}) (0 < α < 1). Then we have

(i) lim_{G∋x→ξ∈Γ}(F_{Γ}u)(x) = P_{Γ}u(ξ)

(ii) lim_{ℝ^{m}\\overline{G}∋x→ξ∈Γ}(F_{Γ}u)(x) = −Q_{Γ}u(ξ)

for any ξ ∈ Γ.

The operator P_{Γ} := \frac{1}{2}(I + S_{Γ}) denotes the projection onto the space of all Cℓ_{m,0}- valued functions which have a left monogenic extension into the domain G. Further, Q_{Γ} := \frac{1}{2}(I − S_{Γ}) denotes the projection onto the space of all Cℓ_{m,0}-valued functions which have a left monogenic extension into the domain ℝ^{m} \ \overline{G} and vanish at infinity.

Corollary 1. Let u ∈ L^{2}(Γ, Cℓ_{m,0}). Then the equations

(i) S_{Γ}^{2}u = u

(ii) F_{Γ}P_{Γ}u = F_{Γ}u

(iii) P_{Γ}^{2}u = P_{Γ}u

(iv) Q_{Γ}^{2}u = Q_{Γ}u

are valid on Γ.

Corollary 2. Let u ∈ H^{1,α−1}(G, Cℓ_{m,0}) (−\frac{m}{2} + 1 < α < \frac{m}{2}). Then

(i) T_{G}Du = u in G ⟺ tr u ∈ im Q_{Γ}

(ii) T Du = u in ℝ^{m}.

3. Monogenicity and Hilbert transform

The main properties of the usual one-dimensional Hilbert transform are the relation to boundary values of holomorphic functions and the connection of conjugated harmonic functions. Both properties are also fulfilled by the Hilbert transform in Clifford analysis which has been studied be many authors, e.g., in the books [8] and [11]. We want to show that the Hilbert transform used by us connects conjugated harmonic functions and we will further use this property.

Lemma 3. Let W ∈ H^{1,α}(ℝ_{+}^{n+1}, Cℓ_{n+1,0}) be a monogenic function. Then there exist U, V ∈ H^{1,α}(ℝ_{+}^{n+1}, Cℓ_{n,0}) such that W = U + e_{n+1}V and ΔU = ΔV = 0.

Proof. Because of Cℓ_{n+1,0} = Cℓ_{n,0} + e_{n+1}Cℓ_{n,0} there exist U, V ∈ Cℓ_{n,0} such that W = U + e_{n+1}V. We want to denote by D the Dirac operator in ℝ^{n}, i.e. D = ∑_{k=1}^{n} e_{k} \frac{∂}{∂x_{k}}. Then

\begin{matrix}(D + e_{n+1} \frac{∂}{∂x_{n+1}} ) W = (D + e_{n+1} \frac{∂}{∂x_{n+1}} ) (U + e_{n+1}V) = 0 \\ ⟺ DU − e_{n+1}DV + e_{n+1} \frac{∂}{∂x_{n+1}} U + \frac{∂}{∂x_{n+1}} V = 0 \\ ⟺ \{ \begin{matrix}DU + \frac{∂}{∂x_{n+1}} V = 0 \\ \frac{∂}{∂x_{n+1}} U − DV = 0. \end{matrix} \end{matrix}  (1)

Thus ΔU = ΔV = 0 in ℝ_{+}^{n+1} where Δ is the Laplacian in ℝ^{n+1}

Let G be a bounded or unbounded smooth domain in ℝ^{n}. Then we define the Hilbert transform by

(H_{G}u)(x) = ∫_{G} 2e(x − y)u(y) dy.

If G is the hole space ℝ^{n}, we denote the Hilbert transform by H. If we interpret G ⊂ ℝ^{n} as a subset of the boundary ℝ^{n} of ℝ_{+}^{n+1} with outer normal −e_{n+1}, we get H_{G}(−e_{n+1})u = S_{G}u.

3.1 Properties of H_{G}, H and HD. Here, we want to summarize properties of the Hilbert transform H as a singular integral operator and the integro-differential operator HD.

Theorem 1. We have the following mapping properties:

(i) \{ \begin{matrix}H_{G} : L^{2,α}(G, Cℓ_{n,0}) & → L^{2,α}(G, Cℓ_{n,0}) \\ H : L^{2,α}(ℝ^{n}, Cℓ_{n,0}) & → L^{2,α}(ℝ^{n}, Cℓ_{n,0}) \end{matrix} (−\frac{n}{2} < α < \frac{n}{2}).

(ii) \{ \begin{matrix}(H_{G}u, v) & = −(u, H_{G}v) \\ (Hu, v) & = −(u, Hv) \end{matrix} for all \{ \begin{matrix}u & ∈ L^{2,α}(G, Cℓ_{n,0}) \\ v & ∈ L^{2,−α}(G, Cℓ_{n,0}) \end{matrix} (−\frac{n}{2} < α < \frac{n}{2}).

(iii) (H_{G}u, u) = 0 and (Hu, u) = 0 for all u ∈ L^{2,α}(G, Cℓ_{n,0}) (0 ≤ α < \frac{n}{2}).

(iv) H^{2} = −I on L^{2,α}(G, Cℓ_{n,0}) (−\frac{n}{2} < α < \frac{n}{2}).

Proof. Property (i) follows from the fact that H is a singular integral operator, see for example [14:p. 284] or [16:p. 205]. To prove property (ii) let x ∈ G be fixed and y ∈ G variable and put for N ≥ 1

u_{N}(y) = \{ \begin{matrix}u(y) & if |y − x| ≥ \frac{1}{N} \\ 0 & if |y − x| < \frac{1}{N}. \end{matrix}

Then ||u_{N} − u|| → 0 as N → ∞ and so ||w_{N} − w|| → 0 as N → ∞ where w_{N}(x) = ∫_{G} û_{N}(y){ \atop e(x − y)} dy and w(x) = ∫_{G} û(y){ \atop e(x − y)} dy. This and Hölders inequality lead to

∫_{G} w(x)v(x) dx = lim_{N→∞} ∫_{G} w_{N}(x)v(x) dx = lim_{N→∞} ∫_{G} ∫_{G} û_{N}(y){ \atop e(x − y)} dy v(x) dx.

In the last integral the order of integration can be reversed and because of

û_{N}(y){ \atop e(x − y)}v(x) = û(y){ \atop e(x − y)}v_{N}(x)

we get

\begin{aligned}∫_{G} w(x)v(x) dx & = lim_{N→∞} ∫_{G} ∫_{G} û(y){ \atop e(x − y)} dy v_{N}(x) dx \\ & = ∫_{G} û(y) ∫_{G} −e(y − x) v(x) dx dy. \end{aligned}

Property (iii) follows from

(H_{G}u, u) = −(u, H_{G}u) = −(H_{G}u, u).

For the second relation we use the embedding

L^{2,α}(ℝ^{n}, Cℓ_{n,0}) ⊂ L^{2}(ℝ^{n}, Cℓ_{n,0}) ⊂ L^{2,−α}(ℝ^{n}, Cℓ_{n,0}).

Thus H maps L^{2,α}(ℝ^{n}, Cℓ_{n,0}) into L^{2,−α}(ℝ^{n}, Cℓ_{n,0}) (\frac{n}{2} > α ≥ 0). To prove property (iv) we remember that S^{2} = I and thus H^{2}u = −H(−e_{n+1})H(−e_{n+1})u = −S^{2}u = −u

Lemma 4. Let u ∈ L^{2,α}(ℝ^{n}, Cℓ_{n,0}) (−\frac{n}{2} < α < \frac{n}{2}). Then there exists a v ∈ L^{2,α}(ℝ^{n}, Cℓ_{n,0}) such that w = u + e_{n+1}v ∈ im P_{ℝ^{n}}.

Proof. Assume w = u + e_{n+1}v ∈ im P_{ℝ^{n}}. Then

\begin{aligned}Sw = w on ℝ^{n} & ⟺ H(−e_{n+1})w = H(−e_{n+1})(u + e_{n+1}v) = w = u + e_{n+1}v \\ & ⟺ H(−v − e_{n+1}u) = e_{n+1}Hu − Hv = u + e_{n+1}v \\ & ⟺ \{ \begin{matrix}−Hv & = u \\ Hu & = v. \end{matrix} \end{aligned} (2)

Now, set v = Hu. Then Hv = H^{2}u = −u and going backwards inside the relations given before we obtain the desired relation Theorem 2. Let u, v ∈ H^{1,− \frac{1}{2}} (ℝ^{n}, Cℓ_{n,0}). Then

(HDu, v) = (u, DHv)  and  (−HDu, u) ≥ 0.

Proof. First of all we have

(Du, v) = Sc ∑_{k=1}^{n} ∫_{ℝ^{n}} { \atop e_{k} \frac{∂}{∂x_{k}}} u(x)v(x) dx

= Sc ∑_{k=1}^{n} ∫_{ℝ^{n}} \frac{∂}{∂x_{k}} û(x)ê_{k}v(x) dx

= −Sc ∑_{k=1}^{n} ∫_{ℝ^{n}} û(x)e_{k} \frac{∂}{∂x_{k}} v(x) dx

= −(u, Dv).

Putting this together with Theorem 1/(i) we get (HDu, v) = −(Du, Hv) = (u, DHv).

Now, let be u ∈ H^{1,− \frac{1}{2}} (ℝ^{n}, Cℓ_{n,0}). Then using Lemma 3 and Lemma 4 we have with v := Hu ∈ H^{1,− \frac{1}{2}} (ℝ^{n}, Cℓ_{n,0}) that w = u + e_{n+1}v = tr W = tr (U + e_{n+1}V) where W is a monogenic function in ℝ_{+}^{n+1}. Therefore if U, V ∈ C^{2}(\overline{ℝ_{+}^{n+1}}, Cℓ_{n,0}(ℝ)), we conclude from (1) and (2)

\frac{∂}{∂x_{n+1}} U|_{x_{n+1}=0} = tr \frac{∂}{∂x_{n+1}} U = tr DV = DV |_{x_{n+1}=0} = D tr V = Dv = DHu

and thus

(−HDu, u) = (u, −DHu)

= −(U|_{x_{n+1}=0}, \frac{∂}{∂x_{n+1}} U|_{x_{n+1}=0})

= − ∑_{J} ∫_{ℝ^{n}} U_{J} |_{x_{n+1}=0} \frac{∂}{∂x_{n+1}} U_{J} |_{x_{n+1}=0}

= ∑_{J} ∫_{ℝ_{+}^{n+1}} U_{J}ΔU_{J} dx + ∑_{k=1}^{n+1} ∑_{J} ∫_{ℝ_{+}^{n+1}} (\frac{∂}{∂x_{k}} U_{J})^{2}dx

≥ 0

because the first integral is zero due to ΔU = 0. The space C^{2}(\overline{ℝ^{n}}, Cℓ_{n,0}) is dense in H^{1,− \frac{1}{2}} (ℝ^{n}, Cℓ_{n,0}) and we get the desired relation

3.2 The Nemyckii operator. We want to study two types of non-singular integral equations. First, we require the properties of the so-called Nemyckii operator F in a Clifford-analysis context. This operator is defined as

(Φu)(x) = φ(x, u_{0}(x), u_{1}(x), …, u_{N}(x)) = φ(x, u(x)) (N = 2^{n})

with u = ∑_{J} u_{J}(x)e_{J}. We make the following assumptions:

(AC) Carathéodory condition: φ : G × Cℓ_{n,0} → Cℓ_{n,0} is a given function, where G is a non-empty, measurable set in ℝ^{n} (n ≥ 1). Moreover,

x → φ(x, u) is measurable on G for all u ∈ Cℓ_{n,0}
u → φ(x, u) is continuous on Cℓ_{n,0} for almost all x ∈ G.

We call φ a Carathéodory function if φ fulfills (AC).

(Aα) Growth condition: For all (x, u) ∈ G × Cℓ_{n,0} and α ∈ ℝ

(1 + |x|^{2})^{− \frac{α}{2}} |φ(x, u)| ≤ a(x) + b|u|(1 + |x|^{2})^{\frac{α}{2}}

where b is a fixed positive number and a ∈ L^{2,α}(G) is a real-valued non-negative function.

Proposition 1. Under the two assumptions (AC) and (Aα) the following statements are valid:

(i) (Continuity and boundedness of Φ). The Nemyckii operator Φ : L^{2,−α}(G, Cℓ_{n,0}) → L^{2,α}(G, Cℓ_{n,0}) is continuous and bounded with

||Φu||_{L^{2,α}} ≤ C (||a||_{L^{2,−α}} + ||u||_{L^{2,−α}})

and

(Φu, u) = Sc ∫_{G} { \atop φ(x, u(x))}u(x) dx for all u ∈ L^{2,−α}(G, Cℓ_{n,0}).

(ii) (Monotonicity of Φ). The function φ is monotone with respect to u, i.e.

[φ(x, u) − φ(x, v), u − v] ≥ 0

for all u, v ∈ L^{2,−α}(G, Cℓ_{n,0}) implies Φ is monotone.

(iii) (Strictly monotonicity of Φ). The function φ is strictly monotone with respect to u, i.e.

[φ(x, u) − φ(x, v), u − v] > 0

for all u, v ∈ L^{2,−α}(G, Cℓ_{n,0}) implies Φ is strictly monotone.

(iv) (Coerciveness of Φ). The inequality

[φ(x, u), u] ≥ d(1 + |x|^{2})^{−α}|u|^{2} + g(x)

where g ∈ L^{1}(G) implies Φ is coercive and (Φu, u) ≥ d||u||_{−α}^{2} + ∫_{G} g(x) dx for all u ∈ L^{2,−α}(G, Cℓ_{n,0}).

4. Monotonicity principles for integral operators

Monotonicity principles went back to Brèzis, Browder and Minty. A comprehensive description of these principles is done in [19]. In our considerations we will use the following theorem on maximal monotone operators by Browder [7].

Theorem 3. Let X be a real (separable) reflexive Banach space and A = A_{1} + A_{2} where 0 ∈ Dom(A_{1}) and A_{1} : Dom(A_{1}) ⊂ X → X^{∗} is maximal monotone, and A_{2} : X → X^{∗} is bounded, monotone, coercive and (hemi-) continuous. Then A is surjective. If A is strictly monotone, then A is injective.

This theorem also holds if A = A_{2}. In our setting X = X^{∗} = L^{2}(G, Cℓ_{n,0}) or X = L^{2,− \frac{α}{2}} (G, Cℓ_{n,0}) and X^{∗} = L^{2, \frac{α}{2}} (G, Cℓ_{n,0}).

4.1 Integral equations. To apply monotonicity principles to this integral equations we only use that the Hilbert transform is a linear, bounded and positive operator, which also implies monotonicity.

Theorem 4 (Hammerstein-type equations). Let G be a bounded or unbounded smooth domain and φ be a monotone, coercive Carathéodory function on G × Cℓ_{n,0} satisfying assumption (Aα) with α = 0 and let K be a linear bounded, positive operator from L^{2}(G, Cℓ_{n,0}) → L^{2}(G, Cℓ_{n,0}). Then

u + (λH_{G} + K)Φu = f

has a solution u ∈ L^{2}(G, Cℓ_{n,0}) for any f ∈ L^{2}(G, Cℓ_{n,0}) and each λ ∈ ℝ. If Φ or K are strictly monotone, this solution is unique.

Proof. The operator λH_{G} + K is linear bounded and monotone, the Nemyckii operator Φ is monotone, bounded and coercive. Both operators map L^{2}(G, Cℓ_{n,0}) → L^{2}(G, Cℓ_{n,0}). Now, an application of [19: Theorem 32.B] gives the desired result.

Theorem 5. Let G be a bounded or unbounded smooth domain and φ be a monotone, coercive Carathéodory function on G × Cℓ_{n,0} and K a linear bounded, positive operator from L^{2,α}(G, Cℓ_{n,0}) → L^{2,−α}(G, Cℓ_{n,0}) (\frac{n}{2} > α ≥ 0). Then

Φu + λH_{G}u + Ku = g

has a solution u ∈ L^{2,α}(G, Cℓ_{n,0}) (\frac{n}{2} > α ≥ 0) for any g ∈ L^{2,−α}(G, Cℓ_{n,0}) (\frac{n}{2} > α ≥ 0) and each fixed λ ∈ ℝ^{n}. This solution is unique if Φ + λH + K is strictly monotone.

Proof. The operator λH_{G} + K : L^{2,α}(G, Cℓ_{n,0}) → L^{2,−α}(G, Cℓ_{n,0}) (\frac{n}{2} > α ≥ 0) is linear bounded and monotone, the properties of the Carathéodory function φ imply that the Nemyckii operator Φ : L^{2,α}(G, Cℓ_{n,0}) → L^{2,−α}(G, Cℓ_{n,0}) (\frac{n}{2} > α ≥ 0) is monotone, bounded, coercive and continuous. Thus Φ + λH_{G} + K fulfills the assumptions of Theorem 3.

4.2 Maximal monotone operators and integro-differential equations. For the consideration of integro-differential equations we will use the property of maximal monotonicity.

Definition 1 (see [7]). A subset M ⊂ X × X^{∗} is said to be a monotone set if for each pairs {u_{1}, w_{1}}, {u_{2}, w_{2}} ∈ M we have (w_{2} − w_{1}, u_{2} − u_{1}) ≥ 0. Such a set M is said to be maximal monotone if it is maximal among monotone sets in the sense of inclusion, and a mapping A is said to be maximal monotone if its graph is a maximal monotone set.

Theorem 6. Let G be a smooth domain in ℝ^{n} and γ a real-valued, continuous, positive function and let C, c > 0 be constants such that

c ≤ inf_{x∈G}(1 + |x|^{2})^{\frac{1}{2}} γ(x) ≤ sup_{x∈G}(1 + |x|^{2})^{\frac{1}{2}} γ(x) ≤ C.

Then the operator

A = D + γ(x)I, Dom(A) = \{u ∈ H^{1,− \frac{1}{2}} (G, Cℓ_{n,0})| tr u ∈ im Q_{Γ}\}

is a maximal monotone mapping Dom(A) → L^{2, \frac{1}{2}} (G, Cℓ_{n,0}). If G = ℝ^{n}, then Dom(A)= H^{1,− \frac{1}{2}} (ℝ^{n}, Cℓ_{n,0}).

Proof. The operator γ(x)I maps L^{2,− \frac{1}{2}} (G, Cℓ_{n,0}) uniquely onto L^{2, \frac{1}{2}} (G, Cℓ_{n,0}) because of

||γ(x)u||2_{\frac{1}{2}} = ∫_{G}(1 + |x|^{2})^{\frac{1}{2}} |γ(x)|^{2}|u(x)|^{2}dx

≤ sup_{x∈G} \{(1 + |x|^{2})|γ(x)|^{2}\} ∫_{G}(1 + |x|^{2})^{− \frac{1}{2}} |u(x)|^{2}dx

≤ C ||u||_{− \frac{1}{2}}^{2} .

From Theorem 2 we get that (Du, u) = 0. Hence

(Du + γ(x)u, u) = (Du, u) + (γ(x)u, u) = (γ(x)u, u) = ∫_{G} γ(x)|u(x)|^{2}dx ≥ 0.

To prove maximal monotonicity we show the existence of a uniquely determined inverse operator with domain L^{2,− \frac{1}{2}} (G, Cℓ_{n,0}). From [2:p.74] we know that T_{G} maps L^{2, \frac{1}{2}} (G, Cℓ_{n,0}) into L^{2,− \frac{1}{2}} (G, Cℓ_{n,0}). It is easily seen that (T_{G}u, u) = 0. Therefore, the operator

γ^{−1}(x)I + T_{G} : L^{2, \frac{1}{2}} (G, Cℓ_{n,0}) → L^{2,− \frac{1}{2}} (G, Cℓ_{n,0})

is linear, bounded, strictly monotone and coercive due to

(γ^{−1}(x)v + T_{G}v, v) = (γ^{−1}(x)v, v)

= ∫_{G} γ^{−1}(x)(1 + |x|^{2})^{− \frac{1}{2}} (1 + |x|^{2})^{\frac{1}{2}} |v(x)|^{2}dx

≥ k ||v||2_{\frac{1}{2}} .

Now, because of tr u ∈ im Q_{Γ} we have

Du + γ(x)u = f ⟺ u + T_{G}γ(x)u = T_{G}f ⟺ (γ^{−1}(x)I + T )γ(x)u = T_{G}f.

There exists a unique v ∈ L^{2, \frac{1}{2}} (G, Cℓ_{n,0}) such that (γ^{−1}(x)I + T_{G})v = T_{G}f, and there exists a unique u ∈ L^{2,− \frac{1}{2}} (G, Cℓ_{n,0}) with u = γ^{−1}(x)v. Thus

u + T_{G}γ(x)u = T_{G}f

has a unique solution u. Moreover, u = −T_{G}(γ(x)u − f) and hence tr u ∈ imQ_{Γ} and

Du = −γ(x)u + f ∈ L^{2, \frac{1}{2}} (G, Cℓ_{n,0})

is well-defined

Theorem 7. Let G be a bounded domain and φ be a monotone, coercive Carathéodory function on G × Cℓ_{n,0} satisfying assumption (Aα) with α = 0 and K be a linear bounded, positive operator from L^{2}(G, Cℓ_{n,0}) → L^{2}(G, Cℓ_{n,0}). Then

Φu + (λH_{G} + K)u + Du + γ(x)u = g

has a unique solution u ∈ L^{2}(G, Cℓ_{n,0}) for any g ∈ L^{2}(G, Cℓ_{n,0}) and each λ ∈ ℝ^{n}.

Proof. Because G is a bounded domain we have L^{2,−α}(G, Cℓ_{n,0}) ≡ L^{2,α}(G, Cℓ_{n,0}) ≡ L^{2}(G, Cℓ_{n,0}). The operator D + γ(x)I is maximal monotone Dom(D + γ(x)I) ⊂ L^{2}(G, Cℓ_{n,0}) → L^{2}(G, Cℓ_{n,0}), the operator λH_{G} +K is a linear bounded, positive operator and the Nemyckii operator Φ is monotone, bounded, coercive and continuous. Now, apply Theorem 3.

Theorem 8. Let G be a bounded or unbounded smooth domain and φ be a monotone, coercive Carathéodory function on G × Cℓ_{n,0} satisfying assumption (Aα) with α = −\frac{1}{2} and K a linear bounded, positive operator from L^{2,− \frac{1}{2}} (G, Cℓ_{n,0}) → L^{2, \frac{1}{2}} (G, Cℓ_{n,0}). Then

Φu + Ku + Du + γ(x)u = f

has a unique solution u∈\{u ∈ H^{1,− \frac{1}{2}} (G, Cℓ_{n,0})| tr u∈im Q_{Γ}\} for any f ∈L^{2, \frac{1}{2}} (G, Cℓ_{n,0}).

Proof. We recall that the operator D + γ(x)I is maximal monotone, the operator K is supposed to be linear bounded and positive, the Nemyckii operator is, due to the properties of the Carathéodory function φ, bounded, continuous and coercive. Using Theorem 3 gives the desired result.

Theorem 9. The operator

−HD : H^{1,− \frac{1}{2}} (ℝ^{n}, Cℓ_{n,0}) ⊂ L^{2,− \frac{1}{2}} (ℝ^{n}, Cℓ_{n,0}) → L^{2, \frac{1}{2}} (ℝ^{n}, Cℓ_{n,0})

is a maximal monotone mapping.

Proof. From Theorem 2 we get the monotonicity of −HD and from Theorem 1/(iv) we know that −H is invertible and its inverse is given by H. Using Corollary 2 we get

−HDu = f ⟺ Du = Hf ⟺ u = T Du = T Hf.

Thus for arbitrary f ∈ L^{2, \frac{1}{2}} (ℝ^{n}, Cℓ_{n,0}) there exists a unique u ∈ H^{1,− \frac{1}{2}} (ℝ^{n}, Cℓ_{n,0}) such that −HDu = f

Theorem 10. Let Φ be a monotone, coercive Carathéodory function on ℝ^{n} × Cℓ_{n,0} satisfying assumption (Aα) with α = −\frac{1}{2} and K a linear bounded, positive operator from L^{2,− \frac{1}{2}} (ℝ^{n}, Cℓ_{n,0}) → L^{2, \frac{1}{2}} (ℝ^{n}, Cℓ_{n,0}). Then

Φu + Ku − HDu = f

has a solution u ∈ H^{1,− \frac{1}{2}} (ℝ^{n}, Cℓ_{n,0}) for any f ∈ L^{2, \frac{1}{2}} (ℝ^{n}, Cℓ_{n,0}). This solution is unique if Φ or K are strictly monotone.

Proof. The operator −HD : Dom(−HD) ⊂ L^{2,− \frac{1}{2}} (ℝ^{n}, Cℓ_{n,0}) → L^{2, \frac{1}{2}} (ℝ^{n}, Cℓ_{n,0}) is maximal monotone. The operator Φ + K : L^{2,− \frac{1}{2}} (ℝ^{n}, Cℓ_{n,0}) → L^{2, \frac{1}{2}} (ℝ^{n}, Cℓ_{n,0}) is bounded, monotone, coercive and continuous. Now apply Theorem 3.

Theorem 11. For μ ∈ ℝ the operator

μD − HD : H^{1,− \frac{1}{2}} (ℝ^{n}, Cℓ_{n,0}) ⊂ L^{2,− \frac{1}{2}} (ℝ^{n}, Cℓ_{n,0}) → L^{2, \frac{1}{2}} (ℝ^{n}, Cℓ_{n,0})

is a maximal monotone mapping.

Proof. We have already seen that μD − HD is monotone. Now we want to show that there exist an inverse operator. We have

μDu − HDu = (μI − H)Du = f ⟺ (μI − H)v = f in L^{2,− \frac{1}{2}} (G, Cℓ_{n,0}).

The operator μI − H is invertible and its inverse is given by \frac{1}{1+μ^{2}} (μI + H). Thus

Du = v = \frac{1}{1 + μ^{2}} (μI +H)f ⟺ u = T Du = \frac{1}{1 + μ^{2}} (μT +T H)f ∈ H^{1,− \frac{1}{2}} (ℝ^{n}, Cℓ_{n,0})

for any f ∈ L^{2,− \frac{1}{2}} (ℝ^{n}, Cℓ_{n,0})

Theorem 12. Let Φ be a monotone, coercive Carathéodory function on ℝ^{n} × Cℓ_{n,0} satisfying assumption (Aα) with α = −\frac{1}{2} and K a linear bounded, positive operator from L^{2,− \frac{1}{2}} (ℝ^{n}, Cℓ_{n,0}) → L^{2, \frac{1}{2}} (ℝ^{n}, Cℓ_{n,0}). Then

Φu + Ku + μDu − HDu = f

has a solution u ∈ H^{1,− \frac{1}{2}} (ℝ^{n}, Cℓ_{n,0}) for any f ∈ L^{2, \frac{1}{2}} (ℝ^{n}, Cℓ_{n,0}) and each μ ∈ ℝ. This solution is unique if Φ or K are strictly monotone.

Proof. The operator μD−HD is a maximal monotone operator H^{1,− \frac{1}{2}} (ℝ^{n}, Cℓ_{n,0})⊂ L^{2,− \frac{1}{2}} (ℝ^{n}, Cℓ_{n,0}) → L^{2, \frac{1}{2}} (ℝ^{n}, Cℓ_{n,0}). The remaining operator Φ + K is bounded, monotone, coercive and continuous. Thus an application of Theorem 3 completes the proof.

Remark. We dealt with the Clifford algebra Cℓ_{n,0}, i.e. e_{j}^{2} = +1. This seems to be unusual. But the operators H_{G} and H are not monotone if we use Cℓ_{0,n}. Nevertheless the operators iH_{G} and iH are monotone in spaces over the complexified Clifford algebras Cℓ_{0,n}(ℂ).

Nonlinear Singular Integral Equations 391

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392 S. Bernstein Received 09.07.1998, in revised form 29.03.1999