Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 18 (1999), No. 4, 827–837

Spatial Riemann Problem for Analytic Functions of Two Complex Variables

H. Begehr and D. Q. Dai

Abstract. The Riemann boundary value problem is discussed for analytic functions in polydiscs. Necessary and sufficient conditions for the existence of a finite number of solutions and a finite number of solvability conditions are derived.

Keywords: Several complex variables, analytic functions, Riemann problem

AMS subject classification: 30 E 25, 32 A 07

1. Introduction

Let

D^{++} = {(z, w) ∈ ℂ^{2} : |z| < 1 and |w| < 1}
D^{−+} = {(z, w) ∈ ℂ^{2} : |z| > 1 and |w| < 1}
D^{+−} = {(z, w) ∈ ℂ^{2} : |z| < 1 and |w| > 1}
D^{−−} = {(z, w) ∈ ℂ^{2} : |z| > 1 and |w| > 1}

and let

T^{2} = {(t, ω) ∈ ℂ^{2} : |t| = |ω| = 1}

be the characteristic boundary. In this paper we shall be concerned with the spatial Riemann problem

Problem (R_{2}). Determine four analytic functions φ^{±±} on D^{±±} such that

\begin{aligned}R_{2}φ : & = A(t, ω)φ^{++}(t, ω) + B(t, ω)φ^{−+}(t, ω) \\ & + C(t, ω)φ^{+−}(t, ω) + D(t, ω)φ^{−−}(t, ω) \\ & = F (t, ω) \end{aligned}  (1)

on T^{2}, where A, B, C, D ∈ W and F ∈ W, W being the Wiener algebra of functions f defined on T^{2} such that

f(t, ω) = ∑_{(p,q)∈ℤ^{2}} f_{n,m}t^{p}ω^{q}

H. Begehr: Freie Univ., I. Math. Inst., Arnimallee 3, D-14195 Berlin Dao-Qing Dai: Zhongshan Univ., Dept. Math., Guangzhou, 510275 China ISSN 0232-2064 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Berlin

is continuous on T^{2}, with norm

∥f∥_{T^{2}} = ∑_{(p,q)∈ℤ^{2}} |f_{p,q}| < +∞,

where ℤ^{2} = {(p, q) : p and q are integers}.

For the Riemann problem for analytic functions of one complex variable, due to a technique of canonical decomposition, complete solutions have been obtained (see, e.g., [9, 13]). This problem is also solved for generalized analytic functions, as well as linear and nonlinear elliptic and other systems of equations in the plane [1, 3, 4, 6, 7, 15, 16]. Spatial Riemann problems were investigated, e.g., in [8, 10 - 12, 14] (for more references one is referred to [2]).

In this paper, we require the unknown functions to satisfy the Cauchy conditions

\begin{matrix}φ^{±−}(z, ∞) & = 0  for all |z| < > 1 \\ φ^{−±}(∞, w) & = 0  for all |w| < > 1 \end{matrix} \}  (2)

so that the homogeneous problem (1) with F = 0 and A = B = C = D = 1 has only the trivial solution. We restrict ourself to the case that

\begin{matrix}A & = 1 \\ B & = t^{l}ω^{λ} \\ C & = t^{m}ω^{μ} \\ D & = t^{n}ω^{ν} \end{matrix} \}  (3)

where (l, λ), (m, μ), (n, ν) ∈ ℤ^{2}. In the general case, that is without condition (3), let us first review the main results concerning the one-dimensional Riemann problem

R_{1}Φ := Φ^{+} + GΦ^{−} = g  on Γ,  (4)

where Γ is a closed smooth curve in ℂ, G ≠ 0 on Γ and Hölder continuous. The function G has a canonical decomposition (see [9, 13])

G(t) = t^{κ} \frac{X^{+}(t)}{X^{−}(t)},  (5)

where X^{+} and X^{−} are respectively boundary values of functions holomorphic in the interior and exterior domains determined by Γ. Inserting (5) into (4) and setting

Φ0^{{+ \atop −}} (z) = \frac{φ^{{+ \atop −}} (z)}{X^{{+ \atop −}} (z)}

problem (4) is reduced to

Φ_{0}^{+}(t) + t^{κ}Φ_{0}^{−}(t) = \frac{g(t)}{X^{+}(t)}  on Γ.  (6)

Two consequences of (6) are now at hand (see [9, 13]):

(i) Problem (R_{1}, g) is solvable in the sense of Noether.

(ii) The number of solutions and of solvability conditions is determined by the index κ, not on other properties of the function G.

Result (i) is, in general, not true for problem (1) - (2). For Result (ii), it was observed in [10] that the solvability of problem (1), i.e. R_{2}φ = F, depends on the coefficients of the operator R_{2}, which means dependence not only on the partial indices of the coefficients A, B, C, D, but also on other analytical properties, e.g. on being the boundary values of analytic functions. Some necessary and sufficient conditions for the solvability of problem (1) with special coefficients, for example when B = C = 0, were obtained in [10: Theorems 3 - 7].

We remark that, for any positive integers a_{i}, b_{i}, c_{i}, d_{i} (i = 1, 2) and any continuous functions a, b, c, d on T^{2}, the problem

a(t, ω)f^{++} + b(t, ω)f^{−+} + c(t, ω)f^{+−} + d(t, ω)f^{−−} = F_{0}(t, ω) on Γ,

where

F_{0}(t, ω) = a(t, ω)t^{a_{1}}ω^{a_{2}} + b(t, ω)t^{−b_{1}}ω^{b_{2}} + c(t, ω)t^{c_{1}}ω^{−c_{2}} + d(t, ω)t^{−d_{1}}ω^{−d_{2}},

has the solution

f^{++}(z, w) = z^{a_{1}}w^{a_{2}}_{f^{−+}(z, w) = z^{−b_{1}}w^{b_{2}}}_{f^{+−}(z, w) = z^{c_{1}}w^{−c_{2}}}_{f^{−−}(z, w) = z^{−d_{1}}w^{−d_{2}}} for (z, w) ∈ D^{++}

for (z, w) ∈ D^{−+}
for (z, w) ∈ D^{+−}
for (z, w) ∈ D^{−−}.

This example shows that it might be necessary to distinguish statements for the following two subjects:

(a) The solvability (in some sense) of the problem R_{2}φ = F for a given function F (b) The solvability (in some sense) of the problem R_{2}φ = F for any function F

which are equivalent for the Noether solvability of R_{1}Φ = g and were ignored in [10]. We do not develop any further along this direction in this paper and confine ourselves to case (3).

For problem (1) - (3), in [10: Theorem 2, p. 223] it was shown that its solvability depends on no more than a countable number of necessary and sufficient conditions imposed on the free term. In [11: Theorem 2, p. 36] necessary conditions were found for the homogeneous problem to have a finite set of linearly independent solutions, and for the inhomogeneous problem to have a finite set of solvability conditions. Readers should be aware that the conditions stated in [11: p. 35] in Theorem 2 there are not correct, which might be due to misprints.

We shall find necessary and sufficient conditions for the solvability of problem (1) - (3). It should be pointed out that the methods used in this paper could be generalized to higher dimensions. This will be carried out in a forthcoming publication.

830 H. Begehr and D. Q. Dai

2. The homogeneous problem

In this section, we consider the homogeneous problem (R_{2}), that is, F = 0 in (1). We introduce four subsets of ℤ^{2} by

d_{A} = {(p, q) ∈ ℤ^{2} : p ≥ 0 and q ≥ 0}
d_{B} = {(p, q) ∈ ℤ^{2} : p < l and q ≥ λ}
d_{C} = {(p, q) ∈ ℤ^{2} : p ≥ m and q < μ}
d_{D} = {(p, q) ∈ ℤ^{2} : p < n and q < ν}.

Let f ∈ W with

f(t, ω) = ∑_{(p,q)∈ℤ^{2}} f_{p,q}t^{p}ω^{q},

and let X be a subset of ℤ^{2}. We define an operator P_{X} : W → W by

(P_{X}f)(t, ω) = ∑_{(p,q)∈X} f_{p,q}t^{p}ω^{q}.  (7)

Then we have the following consequences.

Lemma 1. The operator P_{X} defined by (7) satisfies:

(i) P_{X}^{2} = P_{X}.

(ii) P_{X}P_{Y} = P_{X∩Y} for all X, Y ∈ ℤ^{2}.

Proof. For any f ∈ W with f(t, ω) = ∑_{(p,q)∈ℤ^{2}} f_{p,q}t^{p}ω^{q}, we have

P_{X}^{2}f = P_{X}(P_{X}f) = P_{X}( ∑_{(p,q)∈X} f_{(p,q)}t^{p}ω^{q}) = ∑_{(p,q)∈X} f_{p,q}t^{p}ω^{q} = P_{X}f,

hence P_{X}^{2} = P_{X}, which proves (i). Moreover, we have

P_{X}P_{Y} f = P_{X}(P_{Y} f) = P_{X}( ∑_{(p,q)∈Y} f_{p,q}t^{p}ω^{q}) = ∑_{(p,q)∈X∩Y} f_{p,q}t^{p}ω^{q} = P_{X∪Y} f.

Hence P_{X}P_{X} = P_{X∩Y}, which proves (ii)

For the homogeneous problem (1) - (3), we are interested in finding non-trivial solutions. We have

Theorem 1. The homogeneous problem (1) − (3) has non-trivial solutions if and only if there exist α, β ∈ {A, B, C, D}, α ≠ β, such that

d_{α} ∩ d_{β} ≠ ∅.  (8)

Proof. Assume that (8) holds and let X = d_{α} ∩ d_{β}. Without loss of generality, we assume that α = A and β = B. Then the functions

\begin{aligned}φ_{p,q}^{++}(z, w) & = z^{p}w^{q} \\ φ_{p,q}^{−+}(z, w) & = −z^{p−l}w^{q−λ} \\ φ_{p,q}^{+−}(z, w) & = 0 \\ φ_{p,q}^{−−}(z, w) & = 0 \end{aligned}  \begin{matrix}for (z, w) & ∈ D^{++} \\ for (z, w) & ∈ D^{−+} \\ for (z, w) & ∈ D^{+−} \\ for (z, w) & ∈ D^{−−} \end{matrix} \}

for (p, q) ∈ X are solutions of the homogeneous problem (1) - (3).

Conversely, if (8) does not hold, then d_{α} ∩ d_{β} = ∅ for all α, β ∈ {A, B, C, D}, α ≠ β, and, by virtue of Lemma 1, P_{d_{α}}P_{d_{β}} = 0. By applying P_{d_{D}} to (1) we get t^{n}ω^{ν}φ^{−−} = 0. It then follows that φ^{−−} = 0. Similarly, we can show that φ^{++} = 0, φ^{−+} = 0 and φ^{+−} = 0. That is the homogeneous problem has only the trivial solution

Let

Σ = (d_{A} ∩ d_{B}) ∪ (d_{A} ∩ d_{C}) ∪ (d_{D} ∩ d_{B}) ∪ (d_{D} ∩ d_{C}).  (9)

From Theorem 1, if, for example, d_{A} ∩d_{B} ≠ ∅, since its cardinal number will be infinite, the homogeneous problem (1) - (3) will have an infinite number of linearly independent solutions. We now derive conditions for finiteness of the number of solutions.

Lemma 2. The number of solutions of the homogeneous problem (1) − (3) is finite if and only if Σ = ∅, where Σ is defined by (9).

Proof. Let Σ be not empty. Then there exists (p_{0}, q_{0}) ∈ Σ. Without loss of generality, let (p_{0}, q_{0}) ∈ d_{A} ∩ d_{B}. Then {(p, q) ∈ ℤ^{2} : p = p_{0} and q ≥ q_{0}} ⊂ d_{A} ∩ d_{B}. From the proof of Theorem 1, problem (1) - (3) has an infinite number of linearly independent solutions. Hence Σ = ∅.

Conversely, let Σ = ∅. Since d_{A} ∩ d_{B} = ∅ and d_{A} ∩ d_{C} = ∅, by applying P_{d_{A}} to (1) we get φ^{++}(t, ω) + P_{d_{A}}(t^{n}ω^{ν}φ^{−−}) = 0 on T^{2}. From P_{d_{A}\d_{D}}(t^{n}ω^{ν}φ^{−−}) = 0, we get P_{d_{A}\d_{D}}φ^{++} = 0. Noticing that d_{A} ∩ d_{D} is at most a finite subset of ℤ^{2}, the Fourier coefficients of φ^{++} contains only at most finitely many terms. One can argue similarly for φ^{−+}, φ^{+−} and φ^{−−}. Hence problem (1) - (3) has at most finitely many solutions

Lemma 3. Let Σ be defined by (9). Then Σ is empty if and only if

\begin{matrix}l & ≤ 0 \\ μ & ≤ 0 \\ λ & ≥ ν \\ m & ≥ n. \end{matrix} \}  (10)

Proof. From

d_{A} ∩ d_{B} = {(p, q) ∈ ℤ^{2} : 0 ≤ p < l and q ≥ max(λ, 0)}
d_{A} ∩ d_{C} = {(p, q) ∈ ℤ^{2} : p ≥ max(m, 0) and 0 ≤ q < μ}
d_{D} ∩ d_{B} = {(p, q) ∈ ℤ^{2} : p < min(n, l) and λ ≤ q < ν}
d_{D} ∩ d_{C} = {(p, q) ∈ ℤ^{2} : m ≤ p < n and q < min(μ, ν)}

832 H. Begehr and D. Q. Dai

condition (10) follows.

Lemma 4. Let Σ be defined by (9) and Σ = ∅. Then if the homogeneous problem (1) − (3) has a non-trivial solution, we have either

d_{A} ∩ d_{D} ≠ ∅  (11)

or

d_{B} ∩ d_{C} ≠ ∅.  (12)

Proof. If d_{A} ∩ d_{D} = ∅ and d_{B} ∩ d_{C} = ∅, by virtue of Theorem 1, the homogeneous problem (1) - (3) would have only the trivial solution

Lemma 5. Conditions (11) and (12) are equivalent to

\begin{matrix}n & > 0 \\ ν & > 0 \end{matrix} \}  (13)

and

\begin{matrix}l & > m \\ μ & > λ \end{matrix} \} ,  (14)

respectively.

Proof. From

\begin{aligned}d_{A} ∩ d_{D} & = \{(p, q) ∈ ℤ^{2} : 0 ≤ p < n and 0 ≤ q < ν\} \\ d_{B} ∩ d_{C} & = \{(p, q) ∈ ℤ^{2} : m ≤ p < l and λ ≤ q < μ\} \end{aligned}

conditions (13) and (14) follow

Lemma 6. Let Σ be defined by (9) and Σ = ∅. Then if (13) or (14) hold, the homogeneous problem (1)−(3) has respectively nν and (l−m)(μ−λ) linearly independent solutions, which are given by

\begin{aligned}φ_{p,q}^{++}(z, w) & = z^{p}w^{q} \\ φ_{p,q}^{−+}(z, w) & = 0 \\ φ_{p,q}^{+−}(z, w) & = 0 \\ φ_{p,q}^{−−}(z, w) & = −z^{p−n}w^{q−ν} \end{aligned}  \begin{matrix}for (z, w) & ∈ D^{++} \\ for (z, w) & ∈ D^{−+} \\ for (z, w) & ∈ D^{+−} \\ for (z, w) & ∈ D^{−−} \end{matrix} \}  (15)

for all (p, q) ∈ d_{A} ∩ d_{D}, or by

\begin{aligned}φ_{p,q}^{++}(z, w) & = 0 \\ φ_{p,q}^{−+}(z, w) & = z^{p−l}w^{q−λ} \\ φ_{p,q}^{+−}(z, w) & = −z^{p−m}w^{q−μ} \\ φ_{p,q}^{−−}(z, w) & = 0 \end{aligned}  \begin{matrix}for (z, w) & ∈ D^{++} \\ for (z, w) & ∈ D^{−+} \\ for (z, w) & ∈ D^{+−} \\ for (z, w) & ∈ D^{−−} \end{matrix} \}  (16)

for all (p, q) ∈ d_{B} ∩ d_{C}.

Proof. We prove only (15). When (13) holds, we have for (p, q) ∈ d_{A} ∩ d_{D}

p ≥ 0, q ≥ 0  and  p − n < 0, q − ν < 0.

Hence the functions z^{p}w^{q} and z^{p−n}w^{q−ν} are holomorphic in D^{++} and D^{−−}, respectively. Moreover, we have for (t, ω) ∈ T^{2}

1 · φ_{p,q}^{++}(t, ω) + t^{l}ω^{λ}φ_{p,q}^{−+}(t, ω) + t^{m}ω^{μ}φ_{p,q}^{+−}(t, ω) + t^{n}ω^{ν}φ_{p,q}^{−−}(t, ω)
= 1 · t^{p}ω^{q} + t^{l}ω^{λ} · 0 + t^{m}ω^{μ} · 0 + t^{n}ω^{ν}(−t^{p−n}ω^{q−ν})
= 0.

Hence (15) is a solution of the homogeneous problem (1) - (3).

For different (p_{1}, q_{1}), (p_{2}, q_{2}) ∈ d_{A} ∩ d_{D} it is clear that φ_{p_{1},q_{1}}, φ_{p_{2},q_{2}} are linearly independent. By calculating the cardinality of d_{A} ∩ d_{D}, the homogeneous problem (1) − (3) has therefore nν linearly independent solutions

Theorem 2. The homogeneous problem (1) − (3) has a finite number of non-trivial solutions if and only if

\begin{matrix}l & ≤ 0 \\ μ & ≤ 0 \\ λ ≥ ν & > 0 \\ m ≥ n & > 0 \end{matrix} \}  (17)

or

\begin{matrix}0 & ≥ l > m ≥ n \\ 0 & ≥ μ > λ ≥ ν. \end{matrix} \}  (18)

Moreover, if (17) − (18) are satisfied, the linearly independent solutions are given by (15) and (16), respectively.

Proof. Combining Lemmas 2 - 5, we get (17) and (18). The rest follows from Lemma 6

Remarks. It seems worth of mentioning the symmetry of the roles of t and ω in problem (1) - (3). We denote the indicies in (3) by (0, 0), (l, λ), (m, μ) and (n, ν). After the transform t → ω, ω → t, the indicies become correspondingly (0, 0), (μ, m), (λ, l) and (ν, n) since B and C are interchanged with one another. In conditions (17) and (18) this symmetry is well-preserved. This remark applies also to (26) and (27) below.

3. The inhomogeneous problem

For the solvability of problem (R_{2}), we have the following necessary condition.

Lemma 7. If problem (R_{2}) is solvable for any F ∈ W, then

d_{A} ∪ d_{B} ∪ d_{C} ∪ d_{D} = ℤ^{2}.  (19)

Proof. Let

X = ℤ^{2}\(d_{A} ∪ d_{B} ∪ d_{C} ∪ d_{D}),  (20)

suppose that X ≠ ∅ and consider the operator P_{X}. From

X = (ℤ^{2}\d_{A}) ∩ (ℤ^{2}\d_{B}) ∩ (ℤ^{2}\d_{C}) ∩ (ℤ^{2}\d_{D}),

by virtue of Lemma 1, we have

P_{X} = P_{ℤ^{2}\d_{A}}P_{ℤ^{2}\d_{B}}P_{ℤ^{2}\d_{C}} P_{ℤ^{2}\d_{D}}

and

P_{ℤ^{2}\d_{A}}φ^{++} = P_{ℤ^{2}\d_{B}}(t^{l}ω^{λ}φ^{−+}) = P_{ℤ^{2}\d_{C}} (t^{m}ω^{μ}φ^{+−}) = P_{ℤ^{2}\d_{D}}(t^{n}ω^{ν}φ^{−−}) = 0.

Applying P_{X} to both sides of (1), we get 0 = P_{X}F. Thus problem (R_{2}) is not solvable for F satisfying P_{X}F ≠ 0.

Supposing now that (19) holds we seek a particular solution to problem (R_{2}). We shrink the sets d_{A}, d_{B}, d_{C}, d_{D} to d′_{A}, d′_{B}, d′_{C}, d′_{D} so that d′_{A} ∪ d′_{B} ∪ d′_{C} ∪ d′_{D} = ℤ^{2} and d′_{α} ∩ d′_{β} = ∅ for all α, β ∈ {A, B, C, D}, α ≠ β. Let the free term F have the Fourier series representation

F (t, ω) = ∑_{(p,q)∈ℤ^{2}} F_{p,q}t^{p}ω^{q}.

We define

\begin{aligned}φ_{0}^{++}(z, w) & = ∑_{(p,q)∈d′_{A}}F_{p,q}z^{p}w^{q} \\ φ_{0}^{−+}(z, w) & = ∑_{(p,q)∈d′_{B}}F_{p,q}z^{p−l}w^{q−λ} \\ φ_{0}^{+−}(z, w) & = ∑_{(p,q)∈d′_{C}} F_{p,q}z^{p−m}w^{q−μ} \\ φ_{0}^{−−}(z, w) & = ∑_{(p,q)∈d′_{D}}F_{p,q}z^{p−n}w^{q−ν} \end{aligned}  \begin{matrix}for (z, w) ∈ D^{++} \\ for (z, w) ∈ D^{−+} \\ for (z, w) ∈ D^{+−} \\ for (z, w) ∈ D^{−−}. \end{matrix} \}

From d′_{B} ⊂ d_{B}, it follows that, for all (p, q) ∈ d′_{B}, we have p < l and q ≥ λ. Hence φ_{0}^{−+} satisfies (2). Similarly, we can show that the rest meet with (2), too. Thus {φ_{0}^{±±}} is a solution of problem (R_{2})

From Lemma 7, we therefore have

Theorem 3. Problem (R_{2}) is solvable for any F ∈ W if and only if (19) holds.

If the set X defined by (20) is not empty, then by virtue of Lemma 7 problem (R_{2}) is not solvable. Moreover, when X is an infinite set, the free term F has to satisfy an infinite number of solvability conditions. We now seek conditions so that only a finite number of solvability conditions is needed.

Lemma 8. If problem (R_{2}) has only a finite number of solvability conditions, then

\begin{matrix}l & ≥ 0 \\ ν & ≥ λ \\ n & ≥ m \\ μ & ≥ 0. \end{matrix} \}  (21)

Proof. From the proof of Lemma 7, the set

X = ℤ^{2}\(d_{A} ∪ d_{B} ∪ d_{C} ∪ d_{D})

must be finite. If l < 0, there would be an infinite gap

{(p, q) ∈ ℤ^{2} : l ≤ p < 0 and q > max(λ, 0)}

between d_{A} and d_{B} for whatever λ is. This gap can not be filled up by d_{C} and d_{D} since for (p, q) ∈ d_{C} or (p, q) ∈ d_{D}, we have q < μ or q < ν. Hence we must have l ≥ 0. By considering d_{B} ∩ d_{D}, d_{C} ∩ d_{D} and d_{C} ∩ d_{A}, respectively, we get the rest of (21)

Lemma 9. If problem (R_{2}) has a non-empty finite set of solvability conditions, then

\begin{matrix}λ & ≠ 0 \\ m & ≠ 0 \\ n & ≠ 0. \end{matrix} \}  (22)

Proof. If λ = 0, by virtue of Lemma 8, we have

d_{A} ∪ d_{B} = {(p, q) ∈ ℤ^{2} : −∞ < p < +∞ and q ≥ 0} = ℤ_{+}^{2}.  (23)

Let ℤ_{−}^{2} = ℤ\ℤ_{+}^{2}. Then from

d′′_{C} = d_{C} ∩ ℤ_{−}^{2} = {(p, q) ∈ ℤ^{2} : p ≥ m and q < min(μ, 0)}
= {(p, q) ∈ ℤ^{2} : p ≥ m and q < 0}

and

d′′_{D} = d_{D} ∩ ℤ_{−}^{2} = {(p, q) ∈ ℤ^{2} : p < n and q < min(ν, 0)}
= {(p, q) ∈ ℤ^{2} : p < n and q < 0}

we have, in view of (21),

d′′_{C} ∪ d′′_{D} = ℤ_{−}^{2}.  (24)

From (23) - (24) we get

d_{A} ∪ d_{B} ∪ d_{C} ∪ d_{D} = ℤ^{2}.  (25)

By Lemma 7, (25) implies that the set of solvability conditions is empty. Hence λ ≠ 0. Similarly, we have m ≠ 0 and n ≠ 0

836 H. Begehr and D. Q. Dai

Lemma 10. Under the assumption of Lemma 9, we have:

(i) If λ > 0, then n < 0 and λ > μ (and m ≤ n < 0). (ii) If λ < 0 then m > l(≥ 0) and ν < 0 (and n ≥ m > 0). Proof. We prove only the first statement. Suppose that n ≥ 0. From ν ≥ λ, we

have

d_{B} ∪ d_{D} ⊃ {(p, q) ∈ ℤ^{2} : p < min(l, n) and q ≥ λ}
∪ {(p, q) ∈ ℤ^{2} : p < min(l, n) and q < ν}
= {(p, q) ∈ ℤ^{2} : p < min(l, n) and − ∞ < q < +∞}
⊃ {(p, q) ∈ ℤ^{2} : p < 0 and − ∞ < q < +∞}.

Since μ ≥ 0 and ν ≥ λ > 0, we have

d_{D} ∪ d_{C} ⊃ {(p, q) ∈ ℤ^{2} : p < n and q < 0} ∪ {(p, q) ∈ ℤ^{2} : p ≥ m and q < 0}
⊃ {(p, q) ∈ ℤ^{2} : −∞ < p < +∞ and q < 0}.

Hence d_{A} ∪ d_{B} ∪ d_{C} ∪ d_{D} = ℤ^{2}, which shows that n < 0. Similarly, we can show that λ > μ

From Lemmas 8 - 11, we get Lemma 11. Problem (R_{2}) has a finite number of solvablity conditions only if

\begin{matrix}l & ≥ 0 \\ ν ≥ λ > μ & ≥ 0 > n ≥ m \end{matrix} \}  (26)

or

\begin{matrix}n ≥ m > l & ≥ 0 > ν ≥ λ \\ μ & ≥ 0. \end{matrix} \}  (27)

Let X_{1}, X_{2} ⊂ ℤ^{2} be defined by

X_{1} = {(p, q) ∈ ℤ^{2} : n ≤ p < 0 and μ ≤ q < λ}  (28)
X_{2} = {(p, q) ∈ ℤ^{2} : l ≤ p < m and ν ≤ q < 0}.  (29)

The cardinalities of these sets are respectively (μ − λ)n and (l − m)ν. Moreover, we have

Lemma 12. Let conditions (26) or (27) be satisfied. Then

ℤ^{2}\(d_{A} ∪ d_{B} ∪ d_{C} ∪ d_{D}) = \{ \begin{matrix}X_{1} & if (26) is satisfied \\ X_{2} & if (27) is satisfied \end{matrix}

where X_{1} and X_{2} are defined by (28) and (29).

From Lemmas 11 and 12, we get Theorem 4. Problem (R_{2}) has a finite number of solvability conditions if and only

if (26) or (27) hold, i.e. P_{X_{1}}F = 0 or P_{X_{2}}F = 0.

Acknowledgments. The second-named author is partly supported by the Alexander von Humboldt Stiftung and the National Science Foundation of China.

Spatial Riemann Problem 837

References

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Received 04.02.1999; in revised form 24.06.1999