Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 18 (1999), No. 2, 193–204

Oblique Derivative Problems

for Elliptic Systems of Second Order Equations in Infinite Domains

H. Begehr and G. C. Wen

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

Abstract. There are many problems in mechanics and physics, the mathematical models of which are some boundary value problems for nonlinear elliptic systems of first and second order equations in multiply connected domains including infinity. In this paper, we discuss oblique derivative problems for systems of second order equations.

Keywords: Complex partial differential equations, oblique derivative problems

AMS subject classification: 35 J 25, 35 J 60, 35 J 65

1. Formulation of the problems

Let D be an (N + 1)-connected domain in ℂ including infinity, with boundary Γ = ∪_{j=0}^{N}Γ_{j} ∈ C_{α}^{2} (0 < α < 1). Without loss of generality we may assume that D is a circular domain in {z ∈ ℂ : |z| > 1}, whose boundary consists of N + 1 circles Γ_{0} = Γ_{N+1} = {z ∈ ℂ : |z| = 1} and Γ_{j} = {z ∈ ℂ : |z − z_{j}| = γ_{j}} (j = 1, …, N), where z_{j} ∈ ℂ are given points, 0 < γ_{j} ∈ ℝ are given constants (see, e.g., [2, 3]).

We consider the nonlinear elliptic system of second order equations in complex form

\begin{matrix}w_{zz̄} & = F (z, w, w_{z}, \overline{w}_{z}, w_{zz}, \overline{w}_{zz}) \\ F & = Q_{1}w_{zz} + Q_{2}\overline{w}_{zz} + A_{1}w_{z} + A_{2}\overline{w}_{z} + A_{3}w + A_{4} \\ Q_{j} & = Q_{j}(z, w, w_{z}, \overline{w}_{z}, w_{zz}, \overline{w}_{zz}) (j = 1, 2) \\ A_{j} & = A_{j}(z, w, w_{z}, \overline{w}_{z}) (j = 1, 2, 3, 4). \end{matrix} \}  (1.1)

Suppose that (1.1) satisfies the following conditions (C)_{1} - (C)_{3}.

(C)_{1} Q_{j}(z, w, w_{z}, \overline{w}_{z}, U, V) and A_{j}(z, w, w_{z}, \overline{w}_{z}) are continuous in w, w_{z}, \overline{w}_{z} ∈ ℂ for almost every z ∈ D and all U, V ∈ ℂ, and Q_{j} = 0 and A_{j} = 0 for z ∉ D.

H. Begehr: Freie Univ. Berlin, I. Math. Inst., Arnimallee 3, D-14195 Berlin G. C. Wen: Peking University, Dept. Math., Beijing 100871, P. R. China. This research was carried out while the author was visiting FU Berlin on the basis of a DAAD–K.C. Wong Fellowship during March to August 1997.

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

(C)_{2} Q_{j}(z, w, w_{z}, \overline{w}_{z}, U, V) and A_{j}(z, w, w_{z}, \overline{w}_{z}) are measurable in z ∈ D for all continuously differentiable functions w = w(z) on \overline{D} and all measurable functions U, V ∈ L_{p_{0},2}(\overline{D}), and satisfy

L_{p,2}[A_{j}(z, w, w_{z}, \overline{w}_{z}), \overline{D}] ≤ k_{j−1}  (j = 1, ..., 4)  (1.2)

in which p_{0} and p with 2 < p_{0} ≤ p and k_{j} (j = 0, 1, 2, 3) are non-negative constants.

(C)_{3} System (1.1) satisfies for any functions w ∈ C^{1}(\overline{D}) and constants U^{j}, V^{j} ∈ ℂ (j = 1, 2) the uniform ellipticity condition

\begin{matrix}|F (z, w, w_{z}, \overline{w}_{z}, U^{1}, V^{1}) − F (z, w, w_{z}, \overline{w}_{z}, U^{2}, V^{2})| \\ ≤ q_{1}|U^{1} − U^{2}| + q_{2}|V^{1} − V^{2}| \end{matrix}  (1.3)

for almost every point z ∈ D, where q_{1} ≥ 0 and q_{2} ≥ 0 are constants with q_{1} + q_{2} < 1.

Now we formulate the oblique derivative problem, i.e. the Poincaré boundary value problem as follows (compare [5]).

Problem (P). In the domain D, find a solution w = w(z) of system (1.1), which is continuously differentiable on \overline{D} and satisfies the boundary condition

\begin{matrix}Re [\overline{λ_{1}(z)}w_{z} + a_{11}(z)w] & = a_{12}(z) \\ Re [\overline{λ_{2}(z)}wz̄ + a_{21}(z)w] & = a_{22}(z) \end{matrix} \}  (z ∈ Γ)  (1.4)

where λ_{j} with |λ_{j}(z)| = 1 and a_{jk} (j, k = 1, 2) are known functions, which satisfy the conditions

C_{α}[λ_{j}, Γ] ≤ k_{0},  C_{α}[a_{j1}, Γ] ≤ k_{1},  C_{α}[a_{j2}, Γ] ≤ k_{4}  (1.5)

in which α with \frac{1}{2} < α < 1 and k_{0}, k_{1}, k_{4} are non-negative constants.

Denote

K_{j} = \frac{1}{2π} Δ_{Γ} arg λ_{j}(z)  (j = 1, 2).  (1.6)

[K_{1}, K_{2}] is called the index of Problem (P). When K_{1} < 0 and K_{2} < 0, then Problem (P) may not be solvable. Further, when K_{1} ≥ 0 and K_{2} ≥ 0, then the solution of Problem (P) is not necessarily unique. Hence we consider the well-posedness of Problem (P) with modified boundary conditions (see [1, 4]).

Problem (Q). Find a continuous solution [w, U, V] of the complex system

\begin{matrix}Uz̄ & = F (z, w, U, V, U_{z}, V_{z}) \\ F & = Q_{1}U_{z} + Q_{2}V_{z} + A_{1}U + A_{2}V + A_{3}w + A_{4} \\ Vz̄ & = Ū_{z} \end{matrix} \}  (1.7)

satisfying the boundary condition

Re [\overline{λ_{j}(z)}U_{j}(z) + a_{j1}(z)w(z)] = a_{j2}(z) + h_{j}(z)  (j = 1, 2; z ∈ Γ)  (1.8)

and the relation

w(z) = − ∫_{1}^{z} [\frac{U(ζ)}{ζ^{2}} dζ + \frac{\overline{V (ζ)}}{ζ̄^{2}} dζ̄ − ∑_{m=1}^{N} \frac{d_{m}z_{m}}{ζ(ζ − z_{m})} dζ] + c_{0}  (1.9)

where U_{1} = U, U_{2} = V and d_{m} are appropriate real constants such that the function determined by the integral in (1.9) is single-valued in D, and the undetermined functions h_{j} are of the form

h_{j}(z) =
0  if z ∈ Γ (K_{j} ≥ N)
h_{jk}  if z ∈ Γ_{k} ( \begin{matrix}k = 1, …, N − K_{j} \\ 0 ≤ K_{j} < N \end{matrix} )

0  if z ∈ Γ_{k} ( \begin{matrix}k = N − K_{j} + 1, …, N + 1 \\ 0 ≤ K_{j} < N \end{matrix} )

h_{jk}  if z ∈ Γ_{k} ( \begin{matrix}k = 1, …, N \\ K_{j} < 0 \end{matrix} )

h_{j0} + Re ∑_{m=1}^{−K_{j}−1}(h_{jm}^{+} + ih_{jm}^{−}) z^{m} if z ∈ Γ_{0} (K_{j} < 0),
(1.10)

Here h_{jk} and h_{jm}^{±} are unknown real constants to be determined appropriately. In addition, for K_{j} ≥ 0 (j = 1, 2) the solution w is assumed to satisfy the point conditions

\begin{matrix}Im[\overline{λ_{j}(a_{k})} U_{j}(a_{k}) + a_{j1}w(a_{k})] = b_{jk} (j = 1, 2; k ∈ J_{j}) \\ J_{j} = \{ \begin{matrix}\{1, …, 2K_{j} − N + 1\} & if K_{j} ≥ N \\ \{N − K_{j} + 1, …, N + 1\} & if 0 ≤ K_{j} < N \end{matrix} \end{matrix} \}  (1.11)

where a_{k} ∈ Γ_{k} (k = 1, …, N) and a_{k} ∈ Γ_{0} (k = N + 1, …, 2K_{j} − N + 1, with K_{j} ≥ N for j = 1, 2) are distinct points and b_{jk} are all real constants satisfying the conditions

∑_{j=1,2; k∈J_{j}} |b_{jk}| ≤ k_{5}  (1.12)

with a non-negative constant k_{5} such that |c_{0}| ≤ k_{5}.

Problem (Q)_{0}. This is a special case of Problem (Q), namely with A_{4} = 0, a_{j2} = 0, b_{jk} = 0 (j = 1, 2; k ∈ J_{j}) and c_{0} = 0.

In order to prove the uniqueness of solutions for Problem (Q), we need to add the condition that for any functions U^{j}, V^{j}, w^{j} ∈ C̃(\overline{D}) (j = 1, 2) with U_{z}^{1}, V_{z}^{1} ∈ L_{p_{0},2}(\overline{D}) the equality

\begin{matrix}F (z, w^{1}, U^{1}, V^{1}, U_{z}^{1}, V_{z}^{1}) − F (z, w^{2}, U^{2}, V^{2}, U_{z}^{1}, V_{z}^{1}) \\ = Ã_{1}(U^{1} − U^{2}) + Ã_{2}(V^{1} − V^{2}) + Ã_{3}(w^{1} − w^{2}) \end{matrix}  (1.13)

holds in almost every point z ∈ D, where L_{p_{0},2}[Ã_{j}, \overline{D}] < ∞ (j = 1, 2, 3).

2. A priori estimates for solutions of problem (Q)

In oder to prove the solvability of Problem (Q), we need to give some estimates of its solutions.

Theorem 2.1. Suppose that Conditions (C)_{1} - (C)_{3} hold and the constants q_{2} and k_{1}, k_{2} in (1.2), (1.3) and (1.5) are small enough. Then any solution [w, U, V] of Problem (Q) satisfies the estimates

\begin{matrix}L_{1} & = L(U) = C_{β}[U, \overline{D}] + L_{p_{0},2}[|Uz̄| + |U_{z}|, \overline{D}] ≤ M_{1} \\ L_{2} & = L(V) ≤ M_{1} \end{matrix} \}  (2.1)

and

S = S(w) = C_{β}^{1}[w, \overline{D}] + L_{p_{0},2}[|w_{zz̄}| + |w_{zz}| + |\overline{w}_{zz}|, \overline{D}] ≤ M_{2}  (2.2)

where β = min(α, 1 − \frac{2}{p_{0}} ), p_{0} with 2 < p_{0} ≤ p, M_{j} = M_{j}(q_{0}, p_{0}, k, α, K, D) (j = 1, 2; k = (k_{0}, …, k_{5})) are non-negative constants and K = (K_{1}, K_{2}).

Proof. Let the solution [w, U, V] of Problem (Q) be substituted into system (1.7), the boundary conditions (1.8) and (1.11), and relation (1.9). It is clear that (1.7) and (1.8) can be rewritten in the form

\begin{matrix}Uz̄ − Q_{1}U_{z} − A_{1}U = A \\ A = Q_{2}V_{z} + A_{2}V + A_{3}w + A_{4} \\ Uz̄ = \overline{V}_{z} \end{matrix} \}  in D  (2.3)

and

\begin{matrix}Re[\overline{λ_{j}(z)}U_{j}(z)] = r_{j}(z) + h_{j}(z) \\ with r_{j}(z) = a_{j2}(z) − Re[a_{j1}(z)w(z)] \\ Im[\overline{λ_{j}(a_{k})}U_{j}(a_{k})] = s_{jk} \\ with s_{jk} = b_{jk} − Im[\overline{λ_{j}(a_{k})}(a_{k})] \end{matrix} \}  (j = 1, 2; k ∈ J_{j}; z ∈ Γ) (2.4)

where A and r_{j}, s_{jk} satisfy the inequalities

\begin{aligned}L_{p_{0},2}[A, \overline{D}] & ≤ q_{2}L_{p_{0},2}[V_{z}, \overline{D}] + L_{p_{0},2}[A_{2}, \overline{D}]C[V, \overline{D}] \\ & + L_{p_{0},2}[A_{3}, \overline{D}]C[w, \overline{D}] + L_{p_{0},2}[A_{4}, \overline{D}] \\ & ≤ q_{2}L_{2} + k_{1}L_{2} + k_{2}S_{1} + k_{3} \end{aligned}  (2.5)

and

\begin{matrix}C_{α}[r_{j}, Γ] & ≤ C_{α}[a_{j1}, Γ]C[w, Γ] + C_{α}[a_{j2}, Γ] ≤ k_{1}S_{1} + k_{4} \\ |s_{jk}| & ≤ k_{1}S_{1} + k_{5} \end{matrix} \} (j = 1, 2; k ∈ J_{j}) (2.6)

Oblique Derivative Problems 197

in which S_{1} = C[w, \overline{D}]. In accordance with the estimates on Problem B for (2.3) in [4], we obtain

\begin{aligned}L_{1} & ≤ M_{3}[(q_{2} + k_{1})L_{2} + k_{2}S_{1} + k_{3} + 2k_{1}S_{1} + k_{4} + k_{5}] \\ & = M_{3}[(q_{2} + k_{1})L_{2} + (k_{2} + 2k_{1})S_{1} + k_{3} + k_{4} + k_{5}] \end{aligned}  (2.7)

where M_{3} = M_{3}(q_{0}, p_{0}, k, α, K, D). Moreover, noting that V is a solution of the modified Riemann-Hilbert problem for Uz̄ = \overline{V}_{z}, we have

L_{2} ≤ M_{3}[L_{1} + 2k_{1}S_{1} + k_{4} + k_{5}].  (2.8)

In addition, from (1.9) it can be derived that

S_{1} = C[w, \overline{D}] ≤ k_{5} + M_{4}[C(U, \overline{D}) + C(V, \overline{D})] ≤ k_{5} + M_{4}(L_{1} + L_{2})  (2.9)

where M_{4} = M_{4}(D). Combining (2.7) - (2.9), it is derived that

\begin{aligned}L_{2} & ≤ M_{3}\{M_{3}[(q_{2} + k_{1})L_{2} + (k_{2} + 2k_{1})(k_{5} + M_{4}(L_{1} + L_{2})) + k_{3} + k_{4} + k_{5}] \\ & + 2k_{1}(k_{5} + M_{4}(L_{1} + L_{2})) + k_{4} + k_{5}\} \\ & ≤ M_{3}\{(q_{2} + k_{1})M_{3}L_{2} + (k_{2} + 2k_{1})(1 + M_{3})M_{4}(L_{1} + L_{2}) \\ & + k_{5}(k_{2} + 2k_{1})(1 + M_{3}) + (k_{3} + k_{4} + k_{5})(1 + M_{3})\}. \end{aligned} (2.10)

Provided that the constants q_{2} and k_{1}, k_{2} are sufficiently small, for instance, when

M_{3}[(q_{2} + k_{1})M_{3} + (k_{2} + 2k_{1})(1 + M_{3})M_{4}] < \frac{1}{2},

we thus have

\begin{aligned}L_{2} & ≤ 2M_{3}[(k_{2} + 2k_{1})(1 + M_{3})M_{4}L_{1} \\ & + k_{5}(k_{2} + 2k_{1})(1 + M_{3}) + (k_{3} + k_{4} + k_{5})(1 + M_{3})] \\ & = M_{5}L_{1} + M_{6}. \end{aligned}  (2.11)

Substituting (2.11) and (2.9) into (2.7), it can be obtained that

\begin{aligned}L_{1} & ≤ M_{3}[(q_{2} + k_{1})(M_{5}L_{1} + M_{6}) + (k_{2} + 2k_{1})M_{4}(L_{1} + L_{2}) \\ & + k_{5}(k_{2} + 2k_{1}) + k_{3} + k_{4} + k_{5}] \\ & ≤ M_{3}\{[(q_{2} + k_{1})M_{5} + (k_{2} + 2k_{1})M_{4}(1 + M_{5})]L_{1} \\ & + (q_{2} + k_{1})M_{6} + (k_{2} + 2k_{1})M_{4}M_{6} + k_{5}(k_{2} + 2k_{1}) + k_{3} + k_{4} + k_{5}\}. \end{aligned}  (2.12)

198 H. Begehr and G. C. Wen

Moreover, choose q_{2} and k_{1}, k_{2} small enough such that

M_{3}[(q_{2} + k_{1})M_{5} + (k_{2} + 2k_{1})(1 + M_{5})M_{4}] < \frac{1}{2}.

Then it can be concluded that

\begin{aligned}L_{1} & ≤ 2M_{3}[(q_{2} + k_{1})M_{6} + (k_{2} + 2k_{1})M_{4}M_{6} + k_{5}(k_{2} + 2k_{1}) + k_{3} + k_{4} + k_{5}] \\ & = M_{7} \end{aligned} (2.13)

and

L_{2} ≤ M_{5}M_{7} + M_{6} ≤ M_{1} = max(M_{7}, M_{5}M_{7} + M_{6}).  (2.14)

Furthermore, from (1.9) it follows that (2.2) holds From Theorem 2.1 we can derive the following result. Theorem 2.2. Under the conditions of Theorem 2.1, any solution [w, U, V] of

Problem (Q) satisfies the estimates

\begin{matrix}L_{1} & = L(U) ≤ M_{8}k^{∗} \\ L_{2} & = L(V) ≤ M_{8}k^{∗} \end{matrix} \}  (2.15)

and

S = S(w) ≤ M_{9}k^{∗}  (2.16)

where k^{∗} = k_{3} + k_{4} + k_{5} and M_{j} = M_{j}(q_{0}, p_{0}, k_{0}, α, K, D) (j = 8, 9).

Proof. If k^{∗} = 0, i.e. k_{3} = k_{4} = k_{5} = 0, the estimates in (2.15) can be derived by Theorem 3.1 below. If k^{∗} > 0, it is clear that the system of functions [w^{∗}, U^{∗}, V^{∗}] = [\frac{w^{∗}}{k^{∗}} , \frac{U^{∗}}{k^{∗}} , \frac{V^{∗}}{k^{∗}}] is a solution of the boundary value problem

\begin{matrix}U ∗ ; z̄ & = Q_{1}U_{z}^{∗} + Q_{2}V_{z}^{∗} + A_{1}U^{∗} + A_{2}V^{∗} + A_{3}w^{∗} + \frac{A_{4}}{k^{∗}} \\ V ∗ ; z̄ & = \overline{U^{∗}}_{z} \end{matrix} \}  (2.17)
Re[\overline{λ_{j}(z)}U^{∗}(z) + a_{j1}(z)w^{∗}(z)] = \frac{a_{j2}(z) + h_{j}(z)}{k^{∗}}  (z ∈ Γ)  (2.18)
Im[\overline{λ_{j}(z)}U^{∗}(z) + a_{j1}(z)w^{∗}(z)]|_{z=a_{k}} = \frac{b_{jk}}{k^{∗}}  (2.19)

where j = 1, 2 and k ∈ J_{j}, and

w^{∗}(z) = − ∫_{0}^{z} [\frac{U^{∗}(ζ)}{ζ^{2}} dζ − ∑_{m=1}^{N} \frac{d_{m}z_{m}}{k^{∗}ζ(ζ − z_{m})} dζ + \frac{\overline{V^{∗}(ζ)}}{ζ̄^{2}} dζ̄] + \frac{c_{0}}{k^{∗}} .  (2.20)

From (1.2), (1.5) and (1.12) we see that

L_{p,2}[\frac{A_{4}}{k^{∗}} , \overline{D}] ≤ 1, C_{α}[\frac{a_{j2}}{k_{∗}} , Γ] ≤ 1, ∑_{j=1,2; k∈J_{j}} \frac{|b_{jk}|}{k_{∗}} ≤ 1, \frac{|c_{0}|}{k_{∗}} ≤ 1.

On the basis of the estimates in Theorem 2.1, we obtain for the solution [w^{∗}, U^{∗}, V^{∗}] of the boundary value problem (2.17) - (2.20) the estimate

L(U^{∗}) ≤ M_{8}, L(V^{∗}) ≤ M_{8}, S(w^{∗}) ≤ M_{9}. (2.21)

From the above estimates it immediately follows that estimates (2.15) and (2.16) hold

Remark. Through the mapping z = z(ζ) = \frac{1}{ζ} the complex equation (1.1) can be reduced to the form

\begin{matrix}w_{ζζ̄} & = G(z, w, w_{ζ}, \overline{w}_{ζ}, w_{ζζ}, w̄_{ζζ}) \\ G & = Q̃_{1}w_{ζζ} + Q̃_{2}\overline{w}_{ζζ} + Ã_{1}w_{ζ} + Ã_{2}\overline{w}_{ζ} + Ã_{3}w + Ã_{4} \end{matrix} \} (z ∈ D̃ = ζ(D)) (2.22)

in which

Q̃_{j} = \frac{Q_{j}ζ^{2}}{ζ̄^{2}} , Ã_{j} = −\frac{A_{j}}{ζ̄^{2}} (j = 1, 2) and Ã_{j} = \frac{A_{j}}{|ζ|^{4}} (j = 3, 4) (ζ ∈ D̃)

and ζ = ζ(z) = \frac{1}{z}. By Condition (C), the above coefficients satisfy the conditions

|Q̃_{1}| + |Q̃_{2}| ≤ q_{0} (ζ ∈ D̃) and L_{p,2}[Ã_{j}, \overline{D̃}] ≤ k_{j−1} (j = 1, 2, 3, 4).  (2.23)

If the function w is a solution of the complex equation (1.1) with Condition (C) in D, then w(z) = w[z(ζ)] = w[\frac{1}{ζ}] is a solution of the complex equation (2.22) in D̃. Noting that w_{zz̄} = |ζ|^{4}w_{ζζ̄} and w_{zz} = ζ^{4}w_{ζζ}, we see that if w(z) ∈ W_{p_{0},4}^{2}(D) (2 < p_{0} ≤ p), then w[z(ζ)] ∈ W_{p_{0}}^{2}(D̃). The inverse result is also true.

Moreover, denoting U(z) = U[z(ζ)] = U(\frac{1}{ζ}), we have Uz̄ = −ζ̄^{2}Uζ̄ and U_{z} = −ζ^{2}U_{ζ}, and we see that if U(z) ∈ W_{p_{0},2}^{1}(D) (2 < p_{0} ≤ p), then U[z(ζ)] ∈ W_{p_{0}}^{1}(D̃). The inverse result is also true.

If f(z) ∈ L_{p_{0},2}(\overline{D}), then

\begin{matrix}T f & = −\frac{1}{π} ∫∫_{D} \frac{f(ζ)}{ζ − z} dσ_{ζ} = −\frac{1}{π} ∫∫D̃ \frac{f(\frac{1}{ζ})}{ζ̄^{2}ζ(1 − ζz)} dσ_{ζ} = S(0) − S(\frac{1}{z}) \\ S(z) & = −\frac{1}{π} ∫∫_{D} \frac{f̃(ζ)}{ζ − z} dσ_{ζ},  f̃(ζ) = \frac{f(\frac{1}{ζ})}{ζ̄^{2}}. \end{matrix} \} (2.24)

This shows that f̃(z) ∈ L_{p_{0}}(\overline{D̃}). Hence

\begin{aligned}C̃_{α}[S(z), D̃] & ≤ ML_{p_{0}}[f̃(z), \overline{D̃}] \\ C̃_{α}[S(0) − S(\frac{1}{z}), D̃] & ≤ ML_{p_{0}}[f̃(z), \overline{D̃}] \end{aligned}  (2.25)

in which α = 1 − \frac{2}{p_{0}} and M = M(p_{0}). Thus by using the method of continuity and the contracting mapping principle, we can prove that there exist the solutions ψ = T f and ϕ = T g ∈ W_{p_{0},2}^{1}(\overline{D}) of

ψz̄ = Q_{1}ψ_{z} + A_{1}ψ + A, A = Q_{2}V_{z} + A_{2}V + A_{3}w + A_{4}  (2.26)
ϕz̄ = Q_{1}ϕ_{z} + A_{1}  (2.27)

in D. Moreover, we can also find the solution χ(z) = \frac{1}{z} + T h of the equation

Wz̄ = QW_{z} or h(z) = Q(z)[ − \frac{1}{z^{2}} + Πh].  (2.28)

It is clear that −\frac{Q(z)}{z^{2}} ∈ L_{p,2}(\overline{D}), and then h(z) ∈ L_{p_{0},2}(\overline{D̃}). Due to the fact that the function χ(\frac{1}{z}) = z + S(0) − S(z) = z + S(0) − T [h̃] is a solution of

h̃(z) = \frac{Q̃(z)z^{2}}{z̄^{2}[1 + Πh̃]} in D̃

where h̃(ζ) = \frac{h( \frac{1}{ζ})}{ζ̄^{2}}, the above function χ(\frac{1}{z}) is a homeomorphism in D̃. Obviously, χ(z) is also a homeomorphism in D.

From Theorem 2.1 we see that the solution w = w(z) satisfies the estimate

U(z), V (z) = O(|z|^{\frac{2}{p_{0}} −1}) as z → ∞  and  ∫_{Γ}[U(z) dz + V dz̄] = 0

where Γ̃ = \{z ∈ ℂ : |z| = R\}. Herein R is a sufficiently large number. Hence w is in \overline{D} continuously differentiable.

3. Solvability of boundary value problems

On the basis of proper a priori estimates nonlinear problems are often solved by the Leray-Schauder technique. This method is extensively used in [2, 3] for different problems. In this way here the solvability of problems (P) and (Q) are discussed.

Theorem 3.1. If Conditions (C)_{1} - (C)_{3} and (1.13) hold, and the constants q_{2} and k_{1}, k_{2} in (1.2), (1.3) and (1.5) are small enough, then the solution [w, U, V] of Problem (Q) is unique.

Proof. Denote by [w^{j}, U^{j}, V^{j}] (j = 1, 2) two solutions of Problem (Q) and substitute them into (1.7) - (1,9) and (1.11). Then [w, U, V] = [w^{1} − w^{2}, U^{1} − U^{2}, V^{1} − V^{2}] is a solution of the homogeneous boundary value problem

\begin{matrix}Uz̄ & = Q̃_{1}U_{z} + Q̃_{2}V_{z} + Ã_{1}U + Ã_{2}V + Ãw \\ Vz̄ & = Ū_{z} \end{matrix} \}  (3.1)

Re[\overline{λ_{j}}(z)U_{j}(z) + a_{j1}(z)w(z)] = h_{j}(z)  (z ∈ Γ)  (3.2)
Im[\overline{λ_{j}(z)}U(z) + a_{j1}(z)w(z)]|_{z=a_{k}} = 0  (j = 1, 2; k ∈ J_{j})  (3.3)

w(z) = − ∫_{1}^{z} [\frac{U(ζ)}{ζ^{2}} dζ − ∑_{m=1}^{N} \frac{d_{m}z_{m}}{ζ(ζ − z_{m})} dζ + \frac{\overline{V (ζ)}}{ζ̄^{2}} dζ̄]  (3.4)

the coefficients of which satisfy conditions (1.7) - (1.9) and (1.11), but k_{3} = k_{4} = k_{5} = 0. On the basis of Theorem 2.2, provided q_{2} and k_{1}, k_{2} are sufficiently small, we can derive that U = V = w = 0 on D̄, i.e. w^{1} = w^{2}, U^{1} = U^{2} and V^{1} = V^{2} on D̄

In the following, we use the foregoing estimates of solutions and the Leray-Schauder theorem to prove the solvability of Problem (Q) for the nonlinear elliptic system.

Theorem 3.2. Suppose that the conditions of Theorem 2.1 are satisfyed. Then Problem (Q) is solvable.

Proof. First of all, we assume that F (z, w, U, V, U_{z}, V_{z}) = 0 from (1.7) in the neighbourhood D^{∗} of the boundary Γ, namely

\begin{matrix}U ∗ ; z̄ & = t F^{∗}(z, w, U, V, U_{z}^{∗}, V_{z}^{∗}) \\ V ∗ ; z̄ & = t \overline{U^{∗}}_{z} \end{matrix} \}  (0 ≤ t ≤ 1).  (3.5)

We introduce the Banach space

B = W_{p_{0},2}^{1}(D) × W_{p_{0},2}^{1}(D) × C^{1}(\overline{D}) (2 < p_{0} ≤ p).

Denote by B_{M} the set of triples of continuous functions ω = [w, U, V] satisfying the inequalities

\begin{matrix}L(U) = C_{β}[U, \overline{D}] + L_{p_{0},2}[|Uz̄| + |U_{z}|, \overline{D}] & < M_{10} \\ L(V) & < M_{10} \\ C^{1}[w(z), \overline{D}] & < M_{11} \end{matrix} \}  (3.6)

where M_{10} = M_{1} + 1 and M_{11} = M_{2} + 1, with β and M_{1}, M_{2} being non-negative constants as stated in (2.1) and (2.2). It is evident that B_{M} is a bounded open set in B.

Next, we arbitrarily select a system of functions ω = [w, U, V] ∈ B_{M} and substitute it into the appropriate positions of (1.7) - (1.9) and (1.11), and then consider the boundary value problem (Q)′ with parameter t ∈ [0, 1]

\begin{matrix}U ∗ ; z̄ & = t F^{∗}(z, w, U, V, U_{z}^{∗}, V_{z}^{∗}) \\ V ∗ ; z̄ & = t \overline{U^{∗}}_{z} \end{matrix} \}  in D  (3.7)

Re [\overline{λ_{j}(z)}U^{∗}(z) + ta_{j1}(z)w(z)] = a_{j2}(z) + h_{j}(z)  (z ∈ Γ)  (3.8)
Im [\overline{λ_{j}(z)}U^{∗}(z) + ta_{j1}(z)w(z)]|_{z=a_{k}} = b_{jk}  (j = 1, 2; k ∈ J_{j})  (3.9)

w^{∗}(z) = − ∫_{0}^{z} [\frac{U^{∗}(ζ)}{ζ^{2}} − ∑_{m=1}^{N} \frac{d_{m}z_{m}}{ζ(ζ − z_{m})}]dζ + \frac{\overline{V^{∗}(ζ)}}{ζ̄^{2}} dζ̄  (3.10)

where w, U, V are known functions as stated before. Noting that Problem (Q)′ consists of two modified Riemann-Hilbert boundary value problems for elliptic complex equations of first order and applying [4: Theorem 3.2], there exist the solution U^{∗}, V^{∗} ∈ W_{p_{0},2}^{1}(D) (2 < p_{0} ≤ p). From (3.10), the single-valued function w^{∗} on \overline{D} is determined. Denote by ω^{∗} = [w^{∗}, U^{∗}, V^{∗}] = T (ω, t) (0 ≤ t ≤ 1) this mapping from ω onto ω^{∗}. According to Theorem 2.1, if ω = [w, U, V] = T (ω, t), then ω = [w, U, V] satisfies estimates (2.1) and (2.2), consequently ω ∈ B_{M}. Setting B_{0} = B_{M} × [0, 1], we shall verify that the mapping ω^{∗} = T (ω, t) satisfies the three conditions of the Leray-Schauder theorem:

(1) When t = 0, by Theorem 2.1, it is evident that ω^{∗} = T (ω, t) ∈ B_{M}.

(2) As stated before, the solution ω = [w, U, V] of the functional equation ω = T (ω, t) satisfies estimates (2.1) and (2.2) which shows that ω = T (ω, t) does not have a solution ω = [w, U, V] on the boundary ∂B_{M} = \overline{B_{M}} \ B_{M}.

(3) ω^{∗} = T (ω, t) continuously maps the Banach space B into itself, and is completely continuous on B_{M}. Besides, for ω ∈ \overline{B_{M}}, T (ω, t) is uniformly continuous with respect to t.

In fact, let us choose any sequence \{ω_{n}\}_{n∈ℕ} = \{[w_{n}, U_{n}, V_{n}]\}_{n∈ℕ} ⊂ \overline{B_{M}}. By Theorem 2.1, it is not difficult to see that ω_{n}^{∗} = [w_{n}^{∗}, U_{n}^{∗}, V_{n}^{∗}] = T (ω_{n}, t) (0 ≤ t ≤ 1) satisfies the estimates

L(U_{n}^{∗}) ≤ M_{12},  L(V_{n}^{∗}) ≤ M_{12},  S(w_{n}^{∗}) ≤ M_{13}  (3.11)

where M_{j} = M_{j}(q_{0}, p_{0}, k, α, K, D, M) (j = 12, 13). Hence there can be selected subsequences of \{w_{n}^{∗}\}, \{U_{n}^{∗}\} and \{V_{n}^{∗}\}, which uniformly converge to w_{0}^{∗}, U_{0}^{∗} and V_{0}^{∗} on \overline{D}, and {U_{nz}^{∗}}, {U_{nz̄}^{∗}} and {V_{nz}^{∗}}, {V_{nz̄}^{∗}} in D weakly converge to U_{0z}^{∗}, U_{0z̄}^{∗} and V_{0z}^{∗}, V_{0z̄}^{∗}, respectively. For convenience, denote by the same symbols as before these subsequences. From ω_{n}^{∗} = T (ω_{n}, t) and ω_{0}^{∗} = T (ω_{0}, t) (0 ≤ t ≤ 1) we obtain

\begin{matrix}U_{nz̄}^{∗} − U_{0z̄}^{∗} & = t[F (z, w_{n}, U_{n}, V_{n}, U_{nz}^{∗}, V_{nz}^{∗}) \\ & − F (z, w_{n}, U_{n}, V_{n}, U_{0z}^{∗}, V_{0z}^{∗}) + c_{n}] \\ c_{n} & = F (z, w_{n}, U_{n}, V_{n}, U_{0z}^{∗}, V_{0z}^{∗}) − F (z, w_{0}, U_{0}, V_{0}, U_{0z}^{∗}, V_{0z}^{∗}) \\ V_{nz̄}^{∗} − V_{0z̄}^{∗} & = t [\overline{U}_{nz}^{∗} − \overline{U}_{0z}^{∗}] \end{matrix} \} (z ∈ D)

(3.12)
Re [\overline{λ_{j}(z)}(U_{n}^{∗} − U_{0}^{∗}) + ta_{j1}(z)(w_{n} − w_{0})] = h_{j}(z)  (z ∈ Γ)  (3.13)
Im [\overline{λ_{j}}(U_{n}^{∗} − U_{0}^{∗}) + ta_{j1}(w_{n} − w_{0})]|_{z=a_{k}} = b_{jk}  (j = 1, 2; k ∈ J_{j}) (3.14)

w_{n}^{∗}(z) − w_{0}^{∗}(z) = − ∫_{1}^{z} [\frac{U_{n}^{∗}(ζ) − U_{0}^{∗}(ζ)}{ζ^{2}} − ∑_{m=1}^{N} \frac{d_{m}z_{m}}{ζ(ζ − z_{m})}]dζ

+ [\frac{\overline{V_{n}^{∗}(ζ)} − \overline{V_{0}^{∗}(ζ)}}{ζ̄^{2}}]dζ̄.  (3.15)

It is not difficult to see that c_{n} → 0 for almost every point z ∈ D as n → ∞. Hence we can prove that L_{p_{0}}[c_{n}, \overline{D}] → 0 for n → ∞ as follows: Choosing two arbitrary sufficiently small positive constants ε_{1} and ε_{2}, there exist a subset D_{∗} ⊂ D and a sufficiently large positive integer N such that meas D_{∗} < ε_{1} and |c_{n}| < ε_{2} on \overline{D} \ D_{∗} for n > N. By the Hölder and Minkowski inequalities we have

\begin{matrix}L_{p_{0},2}[c_{n}, \overline{D}] & ≤ L_{p_{0},2}[c_{n}, D_{∗}] + L_{p_{0},2}[c_{n}, \overline{D} \ D_{∗}] \\ & ≤ L_{p_{1},2}[c_{n}, D_{∗}]L_{p_{2},2}[1, D_{∗}] + ε_{2}L_{p_{0},2}[1, \overline{D} \ D_{∗}] \\ & ≤ M_{14}ε_{1}^{1/p_{2}} + ε_{2}π^{1/p_{0}} \\ & = ε \end{matrix} (n > N)

where p_{2} = \frac{p_{0}p_{1}}{p_{1}−p_{0}} , 2 < p_{0} < p_{1} < p_{2} < ∞ and M_{14} is a non-negative constant. On the basis of Theorem 2.2, it can be derived that

\begin{matrix}L(U_{n} − U_{0}) \\ L(V_{n} − V_{0}) \\ S(w_{n} − w_{0}) \end{matrix} \} → 0 (n → ∞).

Because of the completeness of the Banach space B, there exists a system of functions ω_{0} = [w_{0}, U_{0}, V_{0}] ∈ B such that

\begin{matrix}L(U_{n} − U_{0}) \\ L(V_{n} − V_{0}) \\ S(w_{n} − w_{0}) \end{matrix} \} → 0 (n → ∞).

This shows the complete continuity of ω^{∗} = T (ω, t) (0 ≤ t ≤ 1) on \overline{B_{M}}. By a similar method we can also prove that ω^{∗} = T (ω, t) continuously maps \overline{B_{M}} into B and T (ω, t) is uniformly continuous with respect to t for ω ∈ \overline{B_{M}}.

Hence by the Leray-Schauder theorem, we see that the functional equation ω = T (ω, t) (0 ≤ t ≤ 1) with t = 1, i.e. Problem (Q) has a solution.

Finally, we can eliminate the assumption of F (z, w, U, V, U_{z}, V_{z}) = 0 in D^{∗} and prove the solvability of Problem (Q) for the general nonlinear elliptic system (1.7) in D. This completes the proof

Theorem 3.3. Under the same conditions as in Theorem 3.2, the result of solvability of Problem (P) for the complex equation is as follows:

(1) If K_{n} = \frac{1}{2π} Δ_{Γ}arg λ_{n}(z) ≥ N (n ∈ ℕ), then Problem (P) has 2N solvability conditions, and the general solution depends on 2(K_{1} + K_{2} − 2N + 2) arbitrary real constants.

(2) If 0 ≤ K_{j} < N (j = 1, 2), the total number of solvability conditions of Problem (P) is not greater than 4N − K_{1} − K_{2}, and the general solution depends on K_{1} + K_{2} + 4 arbitrary real constants.

(3) If K_{j} < 0 (j = 1, 2), then Problem (P) has 4N − 2K_{1} − 2K_{2} − 2 solvability conditions, and the general solution depends on two real constants.

We can also write solvability conditions of Problem (P) in other cases.

Proof. We only discuss the case 0 ≤ K_{j} < N (j = 1, 2). Let the solution [w, U, V] of Problem (Q) be substituted into (1.7) - (1.9) and (1.11). The functions h_{j} (j = 1, 2) and the complex constants d_{m} (m = 1, …, N) are then determined. If the functions and the constants are equal to zero, namely

h_{j}(z) = h_{jk} (j = 1, …, N − K_{j})  when 0 ≤ K_{j} < N (j = 1, 2)  (3.16)

and

d_{m} = 0  (m = 1, …, N),  (3.17)

then w_{z} = U(z) and \overline{w}_{z} = V (z), and w is a solution of Problem (P). Hence when 0 ≤ K_{j} < N (j = 1, 2), Problem (P) has 4N − K_{1} − K_{2} solvability conditions. In addition, the real constants b_{jk} (k = N − K_{j} + 1, …, N + 1; j = 1, 2) in (1.11) and the complex constant c_{0} in (1.9) may be arbitrary. This shows that the general solution of Problem (P) (0 ≤ K_{j} < N; j = 1, 2) depends on K_{1} +K_{2} +4 arbitrary real constants

References

[1] Vekua, I. N.: Generalized Analytic Functions. Oxford: Pergamon Press 1962.

[2] Wen, G. C. and H. Begehr: Boundary Value Problems for Elliptic Equations and Systems. Harlow: Longman Sci. and Techn. 1990.

[3] Begehr, H. and G. C. Wen: Nonlinear Elliptic Boundary Value Problems and Their Applications. Harlow: Longman Sci. and Techn. 1996. 1996.

[4] Wen, G. C.: The Riemann-Hilbert boundary value problem for nonlinear elliptic complex equations of first order in multiply connected infinite domains. In: Partial Differential and Integral Equations (eds.: H. Begehr, R. P. Gilbert and G.-C. Wen). Dordrecht: Kluwer 1999, pp. 85 – 95.

[5] Wolfersdorf, L. von: On the theory of nonlinear generalized Poincare problems for harmonic functions. In: Continuum Mechanics and Related Problems of Analysis (eds.: M. Balavadze et al.). Tbilisi: Metsniereba 1993, pp. 330 – 337.

Received 09.07.1998; in revised form 16.12.1998