On the Riemann-Hilbert problem in multiply connected domains

1 Introduction

This note is a continuation of the paper [1] where the Riemann-Hilbert problem was solved in these general settings for simply connected domains. In the present paper, on the basis of [1] and a theorem due to Poincare, see e.g. Section VI.1 in [2], it is given a solution of the problem for finitely connected domains.

Recall that boundary value problems for analytic functions are due to the Riemann dissertation (1851), also to works of Hilbert (1904, 1912, 1924) and Poincaré (1910). The Riemann dissertation contained a general setting of a problem on finding analytic functions with a connection between their real and imaginary parts on the boundary.

The first concrete problem of such a type has been proposed by Hilbert (1904) and called by the Hilbert problem or the Riemann-Hilbert problem. That consists in finding an analytic function f inside of a domain bounded by a rectifiable Jordan curve C with the boundary condition

where it was assumed by him that the functions λ and φ are continuously differentiable with respect to the natural parameter s on C and, moreover, |λ| ≠ 0 everywhere on C. Hence without loss of generality one can assume that |λ| ≡ 1 on C.

The first way for solving this problem based on the theory of singular integral equations was proposed by Hilbert (1904), see [3]. This attempt was not quite successful because the theory of singular integral equations had not been developed enough at that time yet. However, just that way became the main approach in this research direction, see e.g. the monographs [4–6]. In particular, the existence of solutions to this problem was in that way proved for Hölder continuous λ and φ, see e.g. [4]. But subsequent weakening conditions on λ and φ in this way led to strengthening conditions on the contour C, say to the Lyapunov curves or to the Radon condition of bounded rotation or even to smooth curves.

However, Hilbert (1905) has solved his problem with the above settings to (1) in the second way based on the reduction it to solving the corresponding two Dirichlet problems, see e.g. [7]. It has been recently shown in [1] that the latter approach makes possible to obtain perfectly general results in the problem for the arbitrary Jordan domains with coefficients λ and boundary data φ that are only measurable with respect to the harmonic measure.

The key was the following Gehring result on the Dirichlet problem for harmonic functions: if φ: ℝ → ℝ is 2π-periodic, measurable and finite a.e. with respect to the Lebesgue measure, then there is a harmonic function in the unit disk 𝔻 = {z ∈ ℂ : |z| < 1} such that u(z) → φ(ϑ) for a.e. ϑ as z → e^{i ϑ} along any nontangential path, see [8], see also [9]. But the way of the reduction of the Riemann-Hilbert problem to the corresponding 2 Dirichlet problems was original in [1].

2 The case of circular domains

Let us start from the simplest kind of multiply connected domains. Recall that a domain D in ℂ¯=ℂ∪{∞} is called circular if its boundary consists of finite number of mutually disjoint circles and points. We call such a domain nondegenerate if its boundary consists only of circles.

Proof. Indeed, by the Poincare theorem, see e.g. Theorem VI.1 in [2], there is a locally conformal mapping g of the unit disc 𝔻 = {z ∈ ℂ : |z| < l} onto 𝔻_*. Let h : 𝔻_* → 𝔻 be the corresponding multivalent analytic function that is inverse to g. 𝔻_* without a finite number of cuts is simply connected and hence h has there only single-valued branches that are extended to the boundary by the Caratheodory theorem.

By Section VI.2 in [2], ∂𝔻 without a countable set of its points consists of a countable collection of arcs every of which is a one-to-one image of a circle in ∂𝔻_* without its one point under every extended branch of h. Note that by the reflection principle g is conformally extended into a neighborhood of every such arc and, thus, nontangential paths to its points go into nontangential paths to the corresponding points of circles in ∂𝔻_* and inversely.

Setting Λ = λ ∘ g and Φ = φ ∘ g with the extended g on the given arcs of ∂𝔻 we obtain measurable functions on ∂𝔻. Thus, by Theorem 2.1 in [1] there exist analytic functions F : 𝔻 → ℂ such that

along any nontangential path to a.e. η ∈ ∂𝔻. By the above arguments, we see that f = F ∘ h are desired multivalent analytic solutions of (2). □

In particular, choosing λ ≡ 1 in (2), we obtain the following statement.

3 The case of rectifiable Jordan curves

To resolve the Riemann-Hilbert problem in the case of domains bounded by a finite number of rectifiable Jordan curves we should extend to this case the known results of Caratheodory (1912), Lindelöf (1917), F. and M. Riesz (1916) and Lavrentiev (1936) for Jordan’s domains.

Proof. (i) Indeed, we are able to transform 𝔻_* into a simply connected domain 𝔻_* through a finite sequence of cuts. Thus, we come to the desired conclusion applying the Caratheodory theorems to simply connected domains 𝔻_* and D^* ≔ ω⁻¹ (𝔻^*), see e.g. Theorem 9.4 in [10] and Theorem II.C.1 in [11].

(ii) In the construction from the previous item, we may assume that the point ζ is not the end of the cuts in D generated by the cuts in 𝔻_* under the extended mapping ω⁻¹. Thus, we come to the desired conclusion twice applying the Caratheodory theorems, the reflection principle for conformal mappings and the Lindelöf theorem for the Jordan domains, see e.g. Theorem II.C.2 in [11].

Points (iii) and (iv) are proved similarly to the last item on the basis of the corresponding results of F. and M. Riesz and Lavrentiev for Jordan domains with rectifiable boundaries, see e.g. Theorem II.D.2 in [11], and [12], see also the point III.1.5 in [13]. □

Proof. This case is reduced to the case of a bounded nondegenerate circular domain 𝔻_* in the following way. First, there is a conformal mapping ω of D onto a circular domain 𝔻_*, see e.g. Theorem V.6.2 in [2]. Note that 𝔻_* is not degenerate because isolated singularities of conformal mappings are removable due to the well-known Weierstrass theorem, see e.g. Theorem 1.2 in [10]. Without loss of generality, we may assume that 𝔻_* is bounded.

By point (i) in Lemma 3.1ω can be extended to a homeomorphisms of D¯ onto 𝔻*¯. If ∂D is rectifiable, then by point (iii) in Lemma 3.1 length ω⁻¹ (E) = 0 whenever E ⊂ ∂𝔻_* with |E| = 0, and by (iv) in Lemma 3.1, conversely, |ω(ℰ)| = 0 whenever ℰ ⊂ ∂𝔻 with length ℰ = 0.

In the last case ω and ω⁻¹ transform measurable sets into measurable sets. Indeed, every measurable set is the union of a sigma-compact set and a set of measure zero, see e.g. Theorem III(6.6) in [14], and continuous mappings transform compact sets into compact sets. Thus, a function φ : ∂D → ℝ is measurable with respect to the natural parameter on ∂D if and only if the function Φ = φ ∘ ω⁻¹ : ∂𝔻_* → ℝ is measurable with respect to the natural parameter on ∂𝔻_*.

By point (ii) in Lemma 3.1, if ∂D has a tangent at a point ζ ∈ ∂D, then arg [ω(ζ) – ω(z)] – arg [ζ – z] → const as z → ζ. In other words, the conformal images of sectors in D with a vertex at ζ is asymptotically the same as sectors in 𝔻_* with a vertex at w = ω(ξ). Thus, nontangential paths in D are transformed under ω into nontangential paths in 𝔻_* and inversely. Finally, a rectifiable Jordan curve has a tangent a.e. with respect to the natural parameter and, thus, Theorem 3.2 follows from Theorem 2.1. □

In particular, choosing λ ≡ 1 in (5), we obtain the following statement.

4 The case of arbitrary Jordan curves

The conceptions of a harmonic measure introduced by R. Nevanlinna in [15] and a principal asymptotic value based on one nice result of F. Bagemihl [16] make possible with a great simplicity and generality to formulate the existence theorems for the Dirichlet and Riemann-Hilbert problems.

First of all, given a measurable set E ⊆ ∂𝔻 and a point z ∈ 𝔻, a harmonic measure of E at z relative to 𝔻 is the value at z of the bounded harmonic function u in 𝔻 with the boundary values 1 a.e. on E and a.e on ∂𝔻 \ E. In particular, by the mean value theorem for harmonic functions, the harmonic measure of E at 0 relative to 𝔻 is equal to |E|2π. In general, the geometric sense of the harmonic measure of E at z₀ relative to 𝔻 is the angular measure of view of E from the point z₀ in radians divided by 2π. Hence the harmonic measure on ∂𝔻 has also the corresponding probabilistic interpretation. The harmonic measure in domains D bounded by finite collections of Jordan curves is defined in a similar way.

Next, a Jordan curve generally speaking has no tangents. Hence we need a replacement for the notion of a nontangential limit. In this connection, recall Theorem 2 in [16], see also Theorem III.1.8 in [17], stating that, for any function Ω:𝔻→ℂ¯, for all pairs of arcs γ₁ and γ₂ in 𝔻 terminating at ζ ∈ ∂𝔻, except a countable set of ζ ∈ ∂𝔻,

where C{Ω, γ) denotes the cluster set of Ω at ζ along γ, i.e.,

Applying the Poincare mapping, branches of its inverse mapping and their boundary behavior, see e.g. Theorem VI.1 and Section VI.2 in [2], we extend this result to arbitrary domains D bounded by a finite number of Jordan curves, cf. the proof of Theorem 2.1.

Now, given a function Ω:D→ℂ¯ and ζ ∈ ∂𝔻, denote by P(Ω, ζ) the intersection of all cluster sets C(Ω, γ) for arcs γ in D terminating at ζ. Later on, we call the points of the set P(Ω, ζ) principal asymptotic values of Ω at ζ. Note that, if Ω has a limit along at least one arc in D terminating at a point ζ ∈ ∂𝔻 with the property (7), then the principal asymptotic value is unique.

Proof. By the reasons of the first item in the proof of Theorem 3.2, there is a conformal mapping ω of D onto a bounded nondegenerate circular domain 𝔻_* in ℂ. Set Λ = λ ∘ Ω and Φ = φ ∘ Ω where Ω ≔ ω⁻¹ extended to ∂𝔻_* by point (i) in Lemma 3.1.

Note that harmonic measure zero is invariant under conformal mappings. Thus, arguing as in the third item of the proof to Theorem 3.2, we conclude that the functions Λ and Φ are measurable with respect to harmonic measures in 𝔻_*.

By Theorem 2.1 there exist multivalent analytic functions F : 𝔻_* → ℂ such that

along any nontangential path to a.e. η ∈ ∂𝔻_*.

By the construction the functions f ≔ F ∘ ω are desired multivalent analytic solutions of (8) in view of the Bagemihl result. □

In particular, choosing λ ≡ 1 in (8), we obtain the following consequence.

5 On dimension of spaces of solutions

Recall that by the Lindelöf maximum principle, see e.g. Lemma 1.1 in [18], it follows the uniqueness theorem for the Dirichlet problem in the class of bounded harmonic functions on the unit disk. Our multivalent analytic solutions are generally speaking not bounded and the latter explains the following fact.

Proof. By Theorem 5.1 in [1] the space of solutions of the problem (3) has the infinite dimension. Thus, the conclusion follows by the construction of these solutions in the given theorems through the reduction to (3). □