Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 18 (1999), No. 3, 611–624

The Generalized Riemann-Hilbert Boundary Value Problem for Non-Homogeneous Polyanalytic Differential Equation of Order n in the Sobolev Space W_{n,p}(D)

Ali Seif Mshimba

Abstract. Given is a nonlinear non-homogeneous polyanalytic differential equation of order n in a simply-connected domain D in the complex plane. Initially we prove (under certain conditions) the existence of its general solution in W_{n,p}(D) by first transforming it into a system of integro-differential equations. Next we prove the solvability of a generalized Riemann-Hilbert problem for the differential equation. This is effected by first reducing the boundary value problem posed to a corresponding one for a polyanalytic function. The latter is then transformed into n classical Riemann-Hilbert problems for holomorphic functions, whose solutions are known in the literature.

Keywords: Polyanalytic functions, generalized Cauchy-Pompeiu integral operators of higher order, Riemann-Hilbert problem

AMS subject classification: 30 G 30, 35 J 40, 47 G 10

1. Introduction

We consider the following non-homogeneous polyanalytic differential equation of order n in a given simply-connected bounded domain D in the complex plane ℂ:

\begin{aligned}\frac{∂^{n}w}{∂\overline{z}^{n}} & = F (z, w, \{\frac{∂^{m+k}w}{∂z^{m}∂\overline{z}^{k}} \}) \\ n ≥ m, k ∈ ℕ_{0}, m + k & ≤ n, (0, 0) ≠ (m, k) ≠ (0, n), n ∈ ℕ. \end{aligned}  (1)

The right-hand side is a continuous function of its variables z ∈ D, w and the partial derivatives of w of order not exceeding n and excluding \frac{∂^{n}w}{∂\overline{z}^{n}}, which are denoted here by \{\frac{∂^{m+k}w}{∂z^{m}∂\overline{z}^{k}} \}. Following [5, 6] the general solution of equation (1) may be expressed in

Ali Seif Mshimba: University of Dar es Salaam, Department of Mathematics, P.O. Box 35062, Dar es Salaam, Tanzania; e-mail: amshimba@cs.udsm.ac.tz

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

the form

\begin{aligned}w(z) & = Φ(z) + T_{0,n,D}F (ζ, w(ζ), \{\frac{∂^{m+k}w}{∂ζ^{m}∂\overline{ζ}^{k}} \})(z) \\ & = Φ(z) + ∫∫_{D} K_{0,n}(z − ζ)F (ζ, w(ζ), \{\frac{∂^{m+k}w}{∂ζ^{m}∂\overline{ζ}^{k}} \})dξdη \end{aligned}  (2)

where Φ is a polyanalytic function of order n in D, and T_{0,n,D} is a generalized Cauchy-Pompeiu type singular integral operator:

K_{m,n}(z) = \{ \begin{matrix}\frac{(−m)!(−1)^{m}}{(n − 1)!π} z^{m−1}\overline{z}^{n−1} & if m ≤ 0 \\ \frac{(−n)!(−1)^{n}}{(m − 1)!π} z^{m−1}\overline{z}^{n−1} & if n ≤ 0 \\ \frac{z^{m−1}\overline{z}^{n−1}}{(m − 1)!(n − 1)!π} ( log |z|^{2} − ∑_{r=1}^{m−1} \frac{1}{r} − ∑_{s=1}^{n−1} \frac{1}{s}) & if m, n ∈ ℕ. \end{matrix} (3)

When m = 1 or n = 1, the corresponding summation in the formula is dropped. The kernel K_{m,n} of the integral operator T_{m,n,D} has no singularity on D, except possibly at the origin. Moreover, it follows from the properties of the operators T_{m,k,D} (m+k ≤ n) that T_{0,n,D}f ∈ W_{n,p}(D), if f ∈ L_{p}(D) (1 < p < ∞) (cf. [2, 5, 6]).

Suppose w ∈ W_{n,p}(D) is a solution of equation (1). Thus w may be expressed in the form (2), and hence we obtain

\begin{aligned}\frac{∂w}{∂z} & = Φ_{z} + T_{−1,n,D}F , \frac{∂w}{∂\overline{z}} = Φ\overline{z} + T_{0,n−1,D}F \\ \frac{∂^{k}w}{∂z^{k}} & = \frac{∂^{k}Φ}{∂z^{k}} + T_{−k,n,D}F , \frac{∂^{k}w}{∂\overline{z}^{k}} = \frac{∂^{k}Φ}{∂\overline{z}^{k}} + T_{0,n−k,D}F (0 ≤ k ≤ n) \end{aligned}

and, in general,

\frac{∂^{m+k}w}{∂z^{m}∂\overline{z}^{k}} = \frac{∂^{m+k}Φ}{∂z^{m}∂\overline{z}^{k}} + T_{−m,n−k,D}F (n ≥ m, k; m + k ≤ n).

Consequently, we arrive at the following result (cf. [12, 15, 21])).

Theorem 1. The function w ∈ W_{n,p}(D) (2 < p < ∞) defined by equation (2) is a general solution of the non-homogeneous polyanalytic equation (1) if and only if for a given in the domain D polyanalytic function Φ ∈ W_{n,p}(D) of order n, (w, {h_{m,k}}) is a solution of the system

\begin{matrix}w(z) = Φ(z) + T_{0,n,D}F (ζ, w(ζ), \{h_{m,k}(ζ)\})(z) \\ h_{m,k}(z) = \frac{∂^{m+k}Φ}{∂z^{m}∂\overline{z}^{k}} + T_{−m,n−k,D}F (ζ, w(ζ), \{h_{m,k}(ζ)\})(z) \\ n ≥ m, k ∈ ℕ_{0}, m + k ≤ n, (0, 0) ≠ (m, k) ≠ (0, n), n ∈ ℕ. \end{matrix} \}  (4)

We note in passing that the integral operators T_{m,k,D} (m + k = 0 < m^{2} + k^{2}) are of singular Calderon-Zygmund type, and may be viewed as analogues of Vekua-type integral operators Π_{D} and \overline{Π}_{D} defined by

\begin{aligned}Π_{D}f(z) & = −\frac{1}{π} ∫∫_{D} \frac{f(ζ)}{(ζ − z)^{2}} dξdη \\ \overline{Π}_{D}f(z) & = −\frac{1}{π} ∫∫_{D} \frac{f(ζ)}{(\overline{ζ} − \overline{z})^{2}} dξdη \end{aligned}

(cf. [8, 11, 17, 18, 22]). They are singular and must be understood in the sense of Cauchy’s principal value. Moreover, they satisfy the Calderon-Zygmund inequality (cf. [5, 6, 8, 17, 18, 22])

∥T_{m,n,D}f∥_{p,D} ≤ A_{p}∥f∥_{p,D}  (5)

where

A_{p} = ∥T_{m,n,D}∥_{p}, A_{p} ≥ A_{2} = 1 (1 < p < ∞).

On the other hand, if m + k > 0, then T_{m,k,D} are regular or weakly singular integral operators, and they may be viewed as generalizations of the Cauchy-Pompeiu integral operators T_{D}, \overline{T}_{D}, T_{D}^{∗} and the potential operator P_{D} given by

\begin{aligned}T_{D}f(z) & = −\frac{1}{π} ∫∫_{D} \frac{f(ζ)}{ζ − z} dξdη,  \overline{T}_{D}f(z) = −\frac{1}{π} ∫∫_{D} \frac{f(ζ)}{\overline{ζ} − \overline{z}} dξdη \\ T_{D}^{∗}f(z) & = −\frac{1}{π} ∫∫_{D} \frac{f(ζ)}{|ζ − z|} dξdη,  P_{D}f(z) = \frac{2}{π} ∫∫_{D} f(ζ) log |ζ − z| dξdη. \end{aligned}

Moreover, since ∥K_{m,n}∥_{1,D} ≤ C(m, n, D) =const, it follows from the convolution theorem of W. H. Young (see [17], for instance) that T_{m,k,D} maps the Banach space L_{p}(D) (1 ≤ p ≤ ∞) into itself, and the estimate

∥T_{m,k,D}f∥_{p,D} ≤ C(m, k, D) ∥f∥_{p,D}  (1 ≤ p ≤ ∞, m + k > 0)  (6)

holds.

2. Existence of the general solution

We make the following assumptions on the right-hand side of equation (1):

(A1) F (z, w, {h_{m,k}}) is a continuous function of its variables z ∈ D, w and the partial derivatives of w of order not exceeding n and excluding \frac{∂^{n}w}{∂\overline{z}^{n}}, which are denoted here by {h_{m,k}}.

(A2) There exists a tupel (w^{∗}, {h_{m,k}^{∗}}) (w^{∗}, h_{m,k}^{∗} ∈ L_{p}(D), 2 < p < ∞) such that

F (z, w^{∗}, {h_{m,k}^{∗}}) ∈ L_{p}(D). (A3) F (z, w, {h_{m,k}}) satisfies a Lipschitz condition of the form

|F (z, w(z), {h_{m,k}(z)}) − F (z, w̃(z), {h̃_{m,k}(z)}|

≤ L_{1} max { max_{m+k<n} |h_{m,k}(z) − h̃_{m,k}(z)|, |w(z) − w̃(z)|}

+ L_{2} max_{m+k=n} |h_{m,k}(z) − h̃_{m,k}(z)|

almost everywhere on D. While 0 < L_{2} < 1, L_{1} may take any positive value.

Note. It follows from assumptions (A2) and (A3) that F (z, w, {h_{m,k}}) ∈ L_{p}(D) (2 < p < ∞) if w and all the elements of {h_{m,k}} belong to L_{p}(D).

We introduce the following Banach space L_{p}(D) (2 < p < ∞):

L_{p}(D) = {(w, {h_{m,k}})| w, h_{m,k} ∈ L_{p}(D)}
n ≥ m, k ∈ ℕ_{0}, m + k ≤ n, (0, 0) ≠ (m, k) ≠ (0, n), n ∈ ℕ.

∥(w, {h_{m,k}})∥ = max {γ∥w∥_{p,D}, γ max_{m+k<n} ∥h_{m,k}∥_{p,D}, max_{m+k=n} ∥h_{m,k}∥_{p,D}} (γ > 0).

Next we define a mapping ℙ in L_{p}(D) through the right-hand side of (4). For any tuple (w, {h_{m,k}}) ∈ L_{p}(D) we set

(W, {H_{m,k}}) = ℙ(w, {h_{m,k}})
W (z) = Φ(z) + T_{0,n,D}F (ζ, w(ζ), {h_{m,k}(ζ)})(z)
H_{m,k}(z) = \frac{∂^{m+k}}{∂z^{m}∂\overline{z}^{k}} Φ + T_{−m,n−k,D}F (ζ, w(ζ), \{h_{m,k}(ζ)\})(z)  (7)
n ≥ m, k ∈ ℕ_{0}, m + k ≤ n, (0, 0) ≠ (m, k) ≠ (0, n), n ∈ ℕ
Φ ∈ W_{n,p}(D) (2 < p < ∞) a polyanalytic function in D.

It follows immediately from the preceding discussion that ℙ maps L_{p}(D) (2 < p < ∞) into itself.

We next show that, under certain conditions, ℙ is contractive in L_{p}(D), so that we can apply the Banach fixed point theorem. To this end we consider the images (W, {H_{m,k}}), (W̃, {H̃_{m,k}}) of (w, {h_{m,k}}), (w̃, {h̃_{m,k}}) ∈ L_{p}(D), respectively, under the mapping ℙ. We then have

γ∥W − W̃∥_{p} ≤ γ∥T_{0,n,D}∥_{p}∥F (z, w, {h_{m,k}}) − F (z, w̃, {h̃_{m,k}})∥_{p,D}

≤ γ∥T_{0,n,D}∥_{p}(L_{1} max { max_{0<m+k<n} ∥h_{m,k} − h̃_{m,k}∥_{p,D}, ∥w − w̃∥_{p,D}}

+ L_{2} max_{m+k=n} ∥h_{m,k} − h̃_{m,k}∥_{p,D})

≤ ∥T_{0,n,D}∥_{p}(L_{1} + γL_{2})∥(w, {h_{m,k}}) − (w̃, {h̃_{m,k}})∥.

Similarly we obtain

\begin{aligned}γ∥H_{m,k} − H̃_{m,k}∥_{p,D} & ≤ ∥T_{−m,n−k,D}∥_{p}(L_{1} + γL_{2})∥(w, \{h_{m,k}\}) − (w̃, \{h̃_{m,k}\})∥ \\ ∥H_{α,β} − H̃_{α,β}∥_{p} & ≤ ∥T_{−α,n−β,D}∥_{p}(\frac{1}{γ} L_{1} + L_{2})∥(w, \{h_{m,k}\}) − (w̃, \{h̃_{m,k}\})∥ \end{aligned}

for 0 < m + k < n and α + β = n with (α, β) ≠ (0, n). On account of the relations

∥T_{−m,n−k,D}∥ = \{ \begin{matrix}C(m, k, D) & for 0 < m + k < n \\ ∥Π_{D}∥_{p} & for m + k = n \end{matrix}  (1 < p < ∞),  (8)

where Π_{D} is the strongly singular Vekua-type integral operator, we arrive at the estimate

∥(W, {H_{m,k}}) − (W̃, {H̃_{m,k}})∥

≤ (\frac{1}{γ} L_{1} + L_{2}) max \{γ∥T_{0,n,D}∥_{p}, γ max_{m+k<n} ∥T_{−m,n−k,D}∥_{p}, ∥Π_{D}∥_{p}\}

× ∥(w, {h_{m,k}}) − (w̃, {h̃_{m,k}})∥

and ℙ is contractive in L_{P} (D) (2 < p < ∞) if

(\frac{1}{γ} L_{1} + L_{2}) max \{γ∥T_{0,n,D}∥_{p}, γ max_{m+k<n} ∥T_{−m,n−k,D}∥_{p}, ∥Π_{D}∥_{p}\} < 1.  (9)

This condition may be satisfied if the constants L_{1}, L_{2} and γ can be chosen properly and the domain D made sufficiently small. It is known (cf. [5, 6, 8, 12, 17, 18], for instance) that

∥Π_{D}∥_{p} ≥ 1 (1 < p < ∞)  and  ∥Π_{D}∥_{2} = 1.

Thus for a chosen L_{p}(D) (2 < p < ∞) we need L_{2}, 0 < L_{2} < 1, such that L_{2}∥Π_{D}∥_{p} < 1. Next we choose the constant γ > 0 large enough so that, for the given L_{1} > 0, (\frac{1}{γ} L_{1} + L_{2})∥Π_{D}∥_{p} < 1 also holds. Finally, since ∥T_{0,n,D}∥_{p} and ∥T_{−m,n−k,D}∥_{p}, 0 < m + k < n, vary directly with the area of the domain D, we may satisfy estimate (9) eventually be reducing the size of D.

If estimate (9) is realized, then ℙ has a unique fixed element (ẇ, {ḣ_{m,k}}) ∈ L_{p}(D) (2 < p < ∞) and ẇ is the general solution of equation (1) corresponding to the given in D polyanalytic function Φ ∈ W_{n,p}(D) of order n. Moveover, ẇ ∈ W_{n,p}(D) (2 < p < ∞):

ẇ(z) = Φ(z) + T_{0,n,D}F (ζ, ẇ(ζ), {ḣ_{m,k}(ζ)})(z)

ḣ_{m,k}(z) = \frac{∂^{m+k}Φ}{∂z^{m}∂\overline{z}^{k}} + T_{−m,n−k,D}F (ζ, ẇ(ζ), \{ḣ_{m,k}(ζ)\})(z)

n ≥ m, k ∈ ℕ_{0}, m + k ≤ n, (0, 0) ≠ (m, k) ≠ (0, n), n ∈ ℕ.

Theorem 2. Under assumptions (A1) - (A3) and (9) the non-homogeneous polyanalytic differential equation (1) admits a uniquely defined solution w ∈ W_{n,p}(D) (2 < p < ∞) given by equation (2) for every prescribed in the domain D polyanalytic function Φ ∈ W_{n,p}(D). This solution defines a mapping from Φ ⟶ w = R(Φ).

3. The generalized Riemann-Hilbert problem for polyanalytic functions

We consider the following boundary value problem for a polyanalytic function Φ of order n:

\frac{∂^{n}Φ}{∂\overline{z}^{n}} = 0  on D = \{z : |z| < 1\}
Re [(a_{k} + ib_{k})\frac{∂^{n−1}Φ}{∂x^{n−k}∂y^{k−1}} ](t) = c_{k}(t)  on ∂D (k = 1, …, n)

(10)

where a_{k}, b_{k}, c_{k} ∈ W_{1− \frac{1}{p} ,p}(∂D) (2 < p < ∞) are prescribed real-valued functions on ∂D. Moreover, (a_{k} + ib_{k})(t) ≠ 0 for all t ∈ ∂D.

A polyanalytic function Φ of order n may be expressed as

Φ = Φ(z, \overline{z}) = ∑_{ρ=0}^{n−1} \overline{z}^{ρ}φ_{ρ}(z) (φ_{ρ} holomorphic).

Thus

\frac{∂^{n−1}Φ}{∂x^{n−k}∂y^{k−1}} = i^{k−1}(\frac{∂}{∂z} + \frac{∂}{∂\overline{z}} )^{n−k}(\frac{∂}{∂z} − \frac{∂}{∂\overline{z}} )^{k−1} ∑_{ρ=0}^{n−1} \overline{z}^{ρ}φ_{ρ}(z)

= i^{k−1} ∑_{α=0}^{n−k} ∑_{β=0}^{k−1}(−1)^{β}({n − k \atop α} )({k − 1 \atop β})

× ∑_{ρ=α+β}^{n−1} \frac{ρ!}{(ρ − α − β)!} \overline{z}^{ρ−α−β} \frac{d^{n−α−β−1}}{dz^{n−α−β−1}} φ_{ρ}(z).

Since on ∂D \overline{t} = \frac{1}{t} we shall replace \overline{z} by \frac{1}{z} in the expression above and then reduce the given boundary conditions for the polyanalytic function Φ to n eqivalent Riemann-Hilbert boundary value problems for some holomorphic functions G_{k} (k = 1, …, n) which are defined in terms of the holomorphic functions φ_{ρ} (ρ = 0, …, n − 1). Thus

\begin{aligned}\frac{∂^{n−1}Φ}{∂x^{n−k}∂y^{k−1}} & = i^{k−1} ∑_{α=0}^{n−k} ∑_{β=0}^{k−1}(−1)^{β}({n − k \atop α} )({k − 1 \atop β}) \\ & × ∑_{ρ=α+β}^{n−1} \frac{ρ!}{(ρ − α − β)!}z^{α+β−ρ} \frac{d^{n−α−β−1}}{dz^{n−α−β−1}} φ_{ρ}(z). \end{aligned}

Hence

Re [(a_{k} + ib_{k})\frac{∂^{n−1}Φ}{∂x^{n−k}∂y^{k−1}} ](t)

= Re[(a_{k} + b_{k})(t) t^{1−n}i^{k−1} ∑_{α=0}^{n−k} ∑_{β=0}^{k−1}(−1)^{β}({n − k \atop α} )({k − 1 \atop β})

× ∑_{ρ=α+β}^{n−1} \frac{ρ!}{(ρ − α − β)!} t^{n+α+β−ρ−1} \frac{d^{n−α−β−1}}{dt^{n−α−β−1}} φ_{ρ}(t)],

i.e.

Re [(a_{k} + ib_{k})(t) i^{k−1}t^{1−n}G_{k}(t)] = c_{k}(t)  (k = 1, …, n)  (11)

where

\begin{aligned}G_{k}(z) & = ∑_{α=0}^{n−k} ∑_{β=0}^{k−1}(−1)^{β}({n − k \atop α} )({k − 1 \atop β}) \\ & × ∑_{ρ=α+β}^{n−1} \frac{ρ!}{(ρ − α − β)!} z^{n+α+β−ρ−1} \frac{d^{n−α−β−1}}{dz^{n−α−β−1}} φ_{ρ}(z). \end{aligned}  (12)

The solution of the Riemann-Hilbert problem (11) is known (see [9, 11, 14, 16]). If κ_{k} := index [(a_{k} − ib_{k}), ∂D] ≥ 0, then the general solution G_{k} is given with the aid of the Schwarz integral as

z^{1−n}G_{k}(z) = \frac{X_{k}(z)}{2πi}[ ∫_{∂D} \frac{i^{1−k}c_{k}(t)}{(a_{k} + ib_{k})(t) X_{k}^{+}(t)} \frac{t + z}{(t − z)t} dt + P_{κ_{k}}(z)]  (13)

where P_{κ_{k}} is a polynomial of degree not exceeding κ_{k} and X_{κ} is the canonical solution of the corresponding homogeneous problem

X_{k}(z) = z^{κ_{k}} exp Γ_{k}(z)
Γ_{k}(z) = \frac{1}{4πi} ∫_{∂D} log ((−1)^{k−2}t^{−2κ_{k}} \frac{(a_{k} − ib_{k})(t)}{(a_{k} + ib_{k})(t)})\frac{t + z}{(t − z)t} dt .

If any of the κ_{k} is negative, then the corresponding Riemann-Hilbert problem has a unique solution, bounded at infinity for instance, if and only if the conditions

∫_{∂D} \frac{c_{k}(t)}{(a_{k} + ib_{k})(t) X_{k}^{+}(t)} t^{j}dt = 0  (j = 0, …, −2κ_{k} − 2)  (14)

are fulfilled, and in that case the solution is given by (13) as well, with the obvious modification that we set P_{κ_{k}}(z) ≡ 0 (cf. [9, 11, 16]).

We investigate the possibility for the satisfaction of the solvability conditions (14). For this purpose we consider the modified Riemann-Hilbert problem (cf. [4, 25])

Re [i^{k−1}t^{1−n}(a_{k} + ib_{k})(t) G_{k}(t)] = c_{k}(t) − ∑_{s=κ_{k}+1}^{−κ_{k}−1} λ_{s}t^{s}  on ∂D  (11)′

where λ_{−s} = \overline{λ_{s}} are constants yet to be determined appropriately. The modified problem is uniquely solvable for κ_{k} < 0, and the solution G_{k} to the original Riemann-Hilbert problem (11) has the representation

z^{1−n}G_{k}(z) = \frac{X_{k}(z)}{2πi} ∫_{∂D} \frac{i^{1−k}}{(a_{k} + ib_{k})(t)X_{k}^{+}(t)}(c_{k}(t − ∑_{s=κ_{k}+1}^{−κ_{k}−1} λ_{s}t^{s})\frac{t + z}{(t − z)t} dt (13)′

(cf. [4, 25]).

In order that a Riemann-Hilbert problem with non-negative index to be uniquely solvable 2κ_{k} + 1 point conditions need to be imposed on the solution G_{k}. These conditions can be expressed in terms of the solution Φ of the given polyanalytic equation (10). Suppose r among the n Riemann-Hilbert problems (11) have non-negative indices, whose sum is N. Then, we demand that

\begin{matrix}Im [i^{k−1}(a_{k} + ib_{k})(τ_{j}) τ_{j}^{1−n}G_{k}(τ_{j})] & = dj (j = 1, 2, …, N + r) \\ τ_{j} ∈ ∂D, τ_{m} ≠ τ_{n} for m & ≠ n, d_{j} ∈ ℝ. \end{matrix}  (15)

It can be shown that z^{1−n}G_{k}(z) ∈ W_{1,p}(0) (2 < p < ∞, k = 1, …, n) and estimates of the form

\begin{matrix}∥z^{1−n}G_{k}∥_{p,D} & ≤ C_{k}(a_{k}, b_{k}, p, D) ∥c_{k}∥_{p,∂D} \\ ∥z^{1−n}G_{k}∥_{1,p,D} & ≤ K_{k}(a_{k}, b_{k}, p, D) ∥c_{k}∥_{1− \frac{1}{p} ,p,∂D} \end{matrix} (k = 1, …, n; 2 < p < ∞) (16)

hold (cf. [12, 14, 15]).

Suppose we have determined all n holomorphic functions G_{k} (k = 1, …, n) uniquely. We proceed to compute the required polyanalytic function Φ by expressing the holomorphic functions φ_{p} (p = 0, …, n − 1) in terms of G_{k}. We shall make use of the following three facts:

1. Derivatives of Φ with respect to x, y can easily be expressed by the holomorphic functions G_{k}. It follows from (12) that

\frac{∂^{n−j}Φ}{∂x^{n−q−ν−j}∂y^{q+ν}} = i^{q+ν}(\frac{∂}{∂z} + \frac{∂}{∂\overline{z}} )^{n−q−ν−j}(\frac{∂}{∂z} − \frac{∂}{∂\overline{z}} )^{q+ν} ∑_{ρ=0}^{n−1} \overline{z}^{ρ}φ_{ρ}(z)

= i^{q+ν} ∑_{α=0}^{n−q−ν−j} ∑_{β=0}^{q+ν}(−1)^{β}({n − q − ν − j \atop α} )({q + ν \atop β} )

× ∑_{ρ=α+β}^{n−1} \frac{ρ!}{(ρ − α − β)!} \overline{z}^{ρ−α−β} \frac{d^{n−α−β−1}}{dz^{n−α−β−1}} φ_{ρ}(z)

= i^{q+ν}z^{1−n}G_{q+ν+j}(z)

on ∂D (i.e. \overline{z} = \frac{1}{z}).

2. Derivatives of Φ with respect to z, \overline{z} can be expressed by the derivatives with respect to x, y, and hence in terms of G_{k}. Indeed, on ∂D we have

\begin{aligned}\frac{∂^{n−j}Φ}{∂z^{n−k}∂\overline{z}^{k−j}} & = 2^{j−n}(\frac{∂}{∂x} − i\frac{∂}{∂y} )^{n−k}(\frac{∂}{∂x} + i\frac{∂}{∂y} )^{k−j}Φ \\ & = 2^{j−n} ∑_{q=0}^{n−k} ∑_{ν=0}^{k−j}(−1)^{q}i^{q+ν}({n − k \atop q} )({k − j \atop ν} )\frac{∂^{n−j}Φ}{∂x^{n−q−ν−j}∂y^{q+ν}} \\ & = 2^{j−n} ∑_{q=0}^{n−k} ∑_{ν=0}^{k−j}(−1)^{q}i^{q+ν}({n − k \atop q} )({k − j \atop ν} )i^{q+ν}z^{1−n}G_{q+ν+j}(z) \\ & = 2^{j−n}z^{1−n} ∑_{q=0}^{n−k} ({n − k \atop q} ) ∑_{ν=0}^{k−j}(−1)^{ν}({k − j \atop ν} )G_{q+ν+j}(z) \end{aligned} (17)

for n ≥ k ≥ j.

3. The holomorphic functions φ_{ρ} can be expressed by the derivatives of the polyanalytic function Φ with respect to z and \overline{z}.

Thus

\frac{∂^{n−1}Φ}{∂\overline{z}^{n−1}} = \frac{∂^{n−1}}{∂\overline{z}^{n−1}} ∑_{ρ=0}^{n−1} \overline{z}^{ρ}φ_{ρ}(z) = (n − 1)!φ_{n−1}(z).

On the other hand, it follows from (17), with k = n and j = 1, that

\frac{∂^{n−1}Φ}{∂\overline{z}^{n−1}} = (2z)^{1−n} ∑_{ν=0}^{n−1}(−1)^{ν}({n − 1 \atop ν})G_{ν+1}(z) on ∂D.

Hence we may conclude that

φ_{n−1}(z) = \frac{1}{(n − 1)!}(2z)^{1−n} ∑_{ν=0}^{n−1}(−1)^{ν}({n − 1 \atop ν})G_{ν+1}(z) on ∂D.

Next we have, on the one hand,

\begin{aligned}\frac{∂^{n−2}Φ}{∂\overline{z}^{n−2}} & = \frac{∂^{n−2}}{∂\overline{z}^{n−2}} (\overline{z}^{n−2}φ_{n−2}(z) + \overline{z}^{n−1}φ_{n−1}(z)) \\ & = (n − 2)! φ_{n−2}(z) + (n − 1)!\overline{z} φ_{n−1}(z). \end{aligned}

On the other hand, we deduce from (17), with k = n and j = 2, that

\frac{∂^{n−2}Φ}{∂\overline{z}^{n−2}} = 2^{2−n}z^{1−n} ∑_{ν=0}^{n−2}(−1)^{ν}({n − 2 \atop ν})G_{ν+2}(z) on ∂D.

So we can obtain for φ_{n−2} the representation

φ_{n−2}(z) = \frac{1}{(n − 2)!}[2^{2−n}z^{1−n} ∑_{ν=0}^{n−2}(−1)^{ν}({n − 2 \atop ν})G_{ν+2}(z) − (n − 1)! \overline{z} φ_{n−1}(z)]

on ∂D.

Similarly we compute φ_{n−3}, …, φ_{1}, φ_{0}. Suppose we have computed φ_{n−1}, φ_{n−2}, …, φ_{n−j+1}. Then we compute φ_{n−j} as

\frac{∂^{n−j}Φ}{∂\overline{z}^{n−j}} = \frac{∂^{n−j}}{∂\overline{z}^{n−j}} ∑_{ρ=n−j}^{n−1} \overline{z}^{ρ}φ_{ρ}(z) = (n − j)!φ_{n−j}(z) + ∑_{ρ=n−j+1}^{n−1} φ_{ρ}(z)\frac{∂^{n−j}\overline{z}^{ρ}}{∂\overline{z}^{n−j}} .

On the other hand, for k = n formula (17) yields

\frac{∂^{n−j}Φ}{∂\overline{z}^{n−j}} = 2^{j−n}z^{1−n} ∑_{ν=0}^{n−j}(−1)^{ν}({n − j \atop ν} )G_{ν+j}(z).

We thus arrive at the general representation

φ_{n−j}(z) = \frac{1}{(n − j)!}[2^{j−n}z^{1−n} ∑_{ν=0}^{n−j}(−1)^{ν}({n − j \atop ν} )G_{ν+j}(z) − ∑_{ρ=n−j+1}^{n−1} φ_{ρ}(z) \frac{∂^{n−j}\overline{z}^{ρ}}{∂\overline{z}^{n−j}} ]

for φ_{n−j} (j = 1, …, n). Hence all n holomorphic functions φ_{j} (j = 0, …, n − 1) are uniquely determinable, and with them the polyanalytic function Φ as well. Furthermore, since a_{k}, b_{k}, c_{k} ∈ W_{1− \frac{1}{p} ,p}(∂D) (2 < p < ∞), we conclude that z^{1−n}G_{k}(z) ∈ W_{1,p}(D) (cf. [1, 10, 12, 14, 15, 19, 20]). It thus follows from (12) that

t^{1−n}G_{1}(t) = ∑_{α=0}^{n−1} ({n − 1 \atop α}) ∑_{j=α}^{n−1} \frac{j!}{(j − α)!} t^{α−j} \frac{d^{n−α−1}}{dt^{n−α−1}} φ_{j}(t) ∈ W_{1− \frac{1}{p} ,p}(∂D)

and, in particular,

\frac{d^{n−α−1}}{dt^{n−α−1}} φ_{j}(t) ∈ W_{1− \frac{1}{p} ,p}(∂D) (j, α = 0, …, n − 1).

Hence

\frac{d^{n−1}}{dt^{n−1}} φ_{j} ∈ W_{1− \frac{1}{p} ,p}(∂D) (j = 0, 1, …, n − 1).

It now follows from the properties of traces of functions that

φ_{j} ∈ W_{n− \frac{1}{p} ,p}(∂D) and φ_{j}, Φ ∈ W_{n,p}(D) (j = 1, …, n)

and the estimates

\begin{matrix}∥Φ∥_{p,D} & ≤ C_{1}(p, D) max_{k} ∥c_{k}∥_{p,∂D} \\ ∥Φ∥_{j,p,D} & ≤ C_{2}(p, D) max_{k} ∥c_{k}∥_{1− \frac{1}{p} ,p,∂D} \end{matrix} \}  (j, k = 1, …, n; 2 < p < ∞)  (18)

hold (cf. [1, 10, 12, 14, 15, 19, 20]).

4. The generalized Riemann-Hilbert problem for equation (1)

We now take up the following boundary value problem for the function w:

\frac{∂^{n}w}{∂\overline{z}^{n}} = F (z, w, \{\frac{∂^{m+k}w}{∂z^{m}∂\overline{z}^{k}} \}) on D

Re [(a_{k} + ib_{k}) \frac{∂^{n−1}w}{∂x^{n−k}∂y^{k−1}} ](t) = c_{k}(t) on ∂D  (k = 1, …, n)  (19)

n ≥ m, k ∈ ℕ_{0}, m + k ≤ n, (0, 0) ≠ (m, k) ≠ (0, n), n ∈ ℕ where a_{k}, b_{k}, c_{k} ∈ W_{1− \frac{1}{p} ,p}(∂D) (2 < p < ∞) are prescribed real-valued functions on ∂D with (a_{k} + ib_{k})(t) ≠ 0 for all t ∈ ∂D.

It was shown earlier that for every polyanalytic function Φ ∈ W_{n,p}(D) (2 < p < ∞) there exists a unique solution w ∈ W_{n,p}(D) to the partial differential equation (1). This solution is represented by (2). We shall now exploit the arbitrariness of the polyanalytic function Φ to construct the solution of the boundary value problem (1), (19). For this purpose we shall write Φ as

Φ = Φ_{c} + Φ_{(w,h)}

where Φ_{c}, Φ_{(w,h)} are solutions of the boundary value problems

Re [(a_{j} + ib_{j}) \frac{∂^{n−1}Φ_{c}}{∂x^{n−j}∂y^{j−1}} ](t) = c_{j}(t) on ∂D

Re [(a_{j} + ib_{j}) \frac{∂^{n−1}}{∂x^{n−j}∂y^{j−1}} Φ_{(w,h)}](t)  (20)

= −Re [(a_{j} + ib_{j}) \frac{∂^{n−1}}{∂x^{n−j}∂y^{j−1}} T_{0,n,D}F (·, w, \{h_{m,k}\})](t)

:= g_{w,h),j}(t) on ∂D

for j = 1, …, n. Since F (z, w, {h_{m,k}}) ∈ L_{p}(D) (2 < p < ∞), then T_{0,n,D}F ∈ W_{n,p}(D) (cf. [5, 6]). Moreover,

\frac{∂^{n−1}}{∂x^{n−k}∂y^{k−1}} T_{0,n,D}F (z)

= i^{k−1} ∑_{α=0}^{n−k} ({n − k \atop α} ) ∑_{β=0}^{k−1}(−1)^{β}({k − 1 \atop β})\frac{∂^{n−1}}{∂z^{n−α−β−1}∂\overline{z}^{α+β}} T_{0,n,D}F (z)

= i^{k−1} ∑_{α=0}^{n−k} ({n − k \atop α} ) ∑_{β=0}^{k−1}(−1)^{β}({k − 1 \atop β})T_{α+β+1−n,n−α−β,D}F (z)

∈ W_{1,p}(D),

i.e. g_{(w,h),j} ∈ W_{1− \frac{1}{p} ,p}(∂D) (2 < p < ∞; j = 1, …, n) (cf. [1, 10, 11, 12, 19, 20]).

Polyanalytic functions which satisfy boundary conditions of the form (20) have been constructed earlier, and we deduce from there that Φ_{c}, Φ_{(w,h)} ∈ W_{n,p}(D) (2 < p < ∞) and, in particular, the estimates

\begin{aligned}∥Φ_{(w,h)}∥_{p,D} & ≤ C(p, D) ∥g_{(w,h)j}∥_{p,∂D} ≤ C_{1}(p, D) ∥T_{0,n,D}F ∥_{1,p,D} \\ ∥Φ_{(w,h)}∥_{k,p,D} & ≤ C_{2}(p, D) ∥T_{0,n,D}F ∥_{n+1−k,p,D} \end{aligned}  (21)

hold for k, j = 1, …, n and 2 < p < ∞.

We now define a mapping ℚ in the Banach space L_{p}(D) (2 < p < ∞). For any tuple (w, {h_{m,k}}) ∈ L_{p}(D) we set

(W, {H_{m,k}}) = ℚ(w, {h_{m,k}})

where

W (z) = Φ_{c}(z) + Φ_{(w,h)}(z) + T_{0,n,D}F (ζ, w(ζ), {h_{m,k}(ζ)})(z)

H_{m,k}(z) = \frac{∂^{m+k}}{∂z^{m}∂\overline{z}^{k}} (Φ_{c}(z) + Φ_{(w,h)}(z)) + T_{−m,n−k,D}F (ζ, w(ζ), \{h_{m,k}(ζ)\})(z)

n ≥ m, k ∈ ℕ_{0}, m + k ≤ n, (0, 0) ≠ (m, k) ≠ (0, n), n ∈ ℕ.

The operator ℚ is uniquely defined, and it maps the Banach space L_{p}(D) (2 < p < ∞) into itself. Moreover, the following result holds.

Theorem 3. If (w, {h_{m,k}}) is a fixed point of the operator ℚ, then w is the solution of the given differential equation (1) which also satisfies the boundary conditions (19).

We next derive the conditions to be imposed in order that ℚ has a fixed point. Suppose (W, {H_{m,k}}), (W̃, {H̃_{m,k}}) are the respective images of (w, {h_{m,k}}), (w̃, {h̃_{m,k}}) ∈ L_{p}(D) (2 < p < ∞). If we set

φ = Φ_{(w,h)} − Φ_{(_{w},_{h})}  and  f = F (z, w, {h_{m,k}}) − F (z, w̃, {h̃_{m,k}}),

then

W − W̃ = φ + T_{0,n,D}f, H_{m,k} − H̃_{m,k} = \frac{∂^{m+k}φ}{∂z^{m}∂\overline{z}^{k}} + T_{−m,n−k,D}F

and

γ ∥W − W̃∥_{p,D}

≤ γ(C_{1}(p, D) ∥T_{0,n,D}∥_{1,p} + ∥T_{0,n,D}∥_{p})∥f∥_{p,D}

≤ (L_{1} max { max_{m+k<n} ∥h_{m,k} − h̃_{m,k}∥_{p,D}, ∥w − w̃∥_{p,D}}

+ L_{2} max_{m+k=n} ∥h_{m,k} − h̃_{m,k}∥_{p,D})γ(C_{1}(p, D) ∥T_{0,n,D}∥_{1,p} + ∥T_{0,n,D}∥_{p})

≤ (C_{1}(p, D) ∥T_{0,n,D}∥_{1,p} + ∥T_{0,n,D}∥_{p})(L_{1} + γL_{2})∥(w, {h_{m,k}}) − (w̃, {h̃_{m,k}})∥.

Similary we arrive at

\begin{aligned}γ ∥H_{m,k} − H̃_{m,k}∥_{p,D} & ≤ (C_{2}(p, D) ∥T_{0,n,D}∥_{n−m−k,p} + ∥T_{−m,n−k,D}∥_{p}) \\ & × (L_{1} + γL_{2}) ∥(w, \{h_{m,k}\}) − (w̃, \{h̃_{m,k}\})∥ \end{aligned}

and

\begin{aligned}∥H_{α,β} − H̃_{α,β}∥_{p,D} & ≤ (C_{3}(p, D) ∥T_{0,n,D}∥_{1,p} + ∥T_{−α,n−β,D}∥_{p}) \\ & × (\frac{1}{γ} L_{1} + L_{2})∥(w, \{h_{m,k}\}) − (w̃, \{h̃_{m,k}\})∥ \end{aligned}

for 0 < m + k < n, α + β = n and (α, β) ≠ (0, n). Consequently, on account of relations (8), we arrive at the estimate

∥(W, {H_{m,k}}) − (W̃, {H̃_{m,k}})∥_{p} ≤ κ ∥(w, {h_{m,k}}) − (w̃, {h̃_{m,k}})∥  (22)

The Generalized Riemann-Hilbert Boundary Value Problem 623

where (\frac{1}{γ} L_{1} + L_{2})^{−1}κ is the maximum of the three quantities

γ (C_{1}(p, D) ∥T_{0,n,D}∥_{1,p} + ∥T_{0,n,D}∥_{p})

γ max_{m+k<n} {C_{2}(p, D) ∥T_{0,n,D}∥_{n−m−k+1,p} + ∥T_{−m,n−k,D}∥_{p}}

C_{3}(p, D) ∥T_{0,n,D}∥_{1,p} + ∥Π_{D}∥_{p}

If κ < 1, then the mapping ℚ is contractive in L_{p}(D) (2 < p < ∞) and it has therefore exactly one fixed element (w, {h_{m,k}}) ∈ L_{p}(D), by the Banach fixed point theorem.

The contractiveness of ℚ imposes certain restrictions on the constants L_{1}, L_{2}, γ and the size of the domain D. Going through an argument similar to the one presented earlier for the case of the existence of a general solution, we can secure the contractiveness of ℚ, and hence the existence of a solution w ∈ W_{n,p}(D) (2 < p < ∞) of the boundary value problem posed. It is easy to establish that the solution is unique.

Theorem 4. Under the assumptions (A1) - (A3), (15) and κ < 1 the generalized Riemann-Hilbert boundary value problem (1), (19) admits a unique solution w ∈ W_{n,p}(D) (2 < p < ∞).

Acknowledgement. This work was carried out during the author’s 1998 visit to the Free University of Berlin in the Federal Republic of Germany under the DAAD sponsorship. The author wishes to thank DAAD and his host, Prof. Dr. Heinrich Begehr of the I. Mathematics Institute for the support and encouragement.

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Received 17.07.1998; in revised form 16.02.1999