Boundary Value Problems for Holomorphic

Functions

Elias WegertTU Bergakademie Freiberg

Contents

1 Introduction 2

2 Function theory in the disk 22.1 Holomorphic functions and their boundary values . . . . . . . . . . . . 32.2 Hardy spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Integral representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Linear Riemann-Hilbert Problems 93.1 Problems with index zero . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Problems with positive index . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Problems with negative index . . . . . . . . . . . . . . . . . . . . . . . 153.4 Problems with piecewise continuous coefficients . . . . . . . . . . . . . 16

4 Nonlinear Riemann-Hilbert problems 184.1 Explicit Riemann-Hilbert problems . . . . . . . . . . . . . . . . . . . . 184.2 Problems with compact target manifold . . . . . . . . . . . . . . . . . . 234.3 Problems with noncompact target manifold . . . . . . . . . . . . . . . . 27

5 Applications 295.1 Boundary value problems for harmonic functions . . . . . . . . . . . . . 295.2 A problem in hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . 295.3 Design of dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . 33

6 Historical Remarks 36

1

1 Introduction

This script is devoted to boundary value problems for holomorphic functions; a livingsubject with a fascinating history and interesting applications.The main theme is a problem which is nearly as old as function theory itself and canbe traced back to Bernhard Riemann’s famous thesis “Grundlagen fur eine allgemeineTheorie der Functionen einer veranderlichen complexen Grosse”. Nowadays is usuallycalled the ‘Riemann-Hilbert problem’ and belongs to the basic problems in complexanalysis. Riemann-Hilbert problems have a rich and nice theory, interrelations withseveral other questions in operator theory, geometric function theory, and approxima-tion theory, and a number of applications in mathematical physics.These lectures are intended as introduction to linear and nonlinear scalar Riemann-Hilbert problems. Not all details and proofs are included; the interested reader isreferred to the literature quoted in the text.The plan of the paper is as follows. We start with summarizing relevant results fromcomplex analysis with emphasis on the boundary behavior of analytic functions andcontinue with a detailed study of linear Riemann-Hilbert problems with continuousand piecewise continuous coefficients. Then we consider several classes of nonlinearRiemann-Hilbert problems. Using fixed points methods, we investigate the existenceof solutions to problems with boundary conditions given in explicit form. Finallywe present results from the geometric theory of general nonlinear Riemann-Hilbertproblems.Among the various applications we sketch two: a problem in hydrodynamics and anoptimal design problem for dynamical systems.All boundary value problems are only considered for holomorphic functions in the unitdisc, but we point out that everything can be transplanted to arbitrary (smoothlybounded) Jordan domains by conformal mapping.The necessary prerequisites are kept at a minimum, however, it is supposed that thereader is familiar with basic results of functional analysis and classical function theory.

2 Function theory in the disk

In this chapter we recall some properties of holomorphic functions in the complex unitdisk D := {z ∈ C : |z| < 1} which are of particular importance for our purposes. Themajor topics are the existence of boundary values on the complex unit circle T := ∂Dand several integral representations of holomorphic functions.Since the material of this section has only auxiliary character, we do not prove theresults. Standard references are Duren [4], Garnett [6], Hoffman [12], Koosis [13],Rosenblum and Rovnyak [17], and (for the German reader) Fischer and Lieb [14].

2

2.1 Holomorphic functions and their boundary values

We denote by O(D) the linear space of holomorphic functions (≡ analytic functions)in the unit disk D. There are several concepts for defining boundary limits of functionsin O(D) at a point t on the unit circle T:The unrestricted limit, which does not take advantage of the special behavior of holo-morphic functions, is

a := limD3z→t

f(z).

For our purposes this definition is usually too strong. A concept which fits well is thenontangential limit. For t ∈ T and 0 < α < π/2 the set

S(t, α) := {z ∈ D : |arg (1− tz)| < α}

is called a Stolzian angle at t. The function f is said to have the nontangential orangular limit a at t ∈ T, if

a = limS(t,α)3z→t

f(z)

for every Stolzian angle at t. We also refer to the angular limit as the boundary valueof f at t and define f(t) := a. If the boundary values of f exist almost everywhere onT (with respect to the Lebesgue measure) the function f ∗ : T → C, t 7→ f(t) is calledthe boundary function of f . Except in this introduction the boundary function is againdenoted by f .The first fundamental result shows that a holomorphic function is already determinedby its angular limits on a set of positive measure.

Theorem 1. (Luzin-Privalov). If f ∈ O(D) has vanishing nontangential limits on asubset T ⊂ T of positive (Lebesgue-) measure then f ≡ 0.

In contrast to this situation there are nontrivial holomorphic function with vanishingradial limit

limr→1−0

f(rt) = 0

almost everywhere on T, which shows that this concept is, in a sense, too weak.

2.2 Hardy spaces

Since we shall apply functional analytic methods, appropriate spaces of holomorphicfunctions are required.Let 1 ≤ p < ∞. We say that a holomorphic function f ∈ O(D) belongs to the Hardyspace Hp ≡ Hp(D) if

‖f‖p := sup0<r<1

{ 1

2π

∫ 2π

0

|f(reiτ )|p dτ}1/p

<∞.

3

If f ∈ O(D) is bounded, we write f ∈ H∞(D) and define

‖f‖∞ := supz∈D

|f(z)|.

Equipped with the norm ‖.‖p the Hardy spaces Hp(D) are Banach spaces for all p with1 ≤ p ≤ ∞]. If 1 < p < q <∞ the embeddings H∞ ⊂ Hp ⊂ Hq ⊂ H1 are continuous.

A famous, and somewhat surprising, result is the existence of a boundary function forall f ∈ Hp(D). Let Lp(T) denote the Lebesgue spaces on the circle T with respect tothe normalized Lebesgue measure.

Theorem 2. (Fatou). Let 1 ≤ p ≤ ∞. Any function f ∈ Hp(D) has a boundaryfunction f ∗ ∈ Lp(T). The norms of f in Hp(D) and of f ∗ in Lp(T) coincide.

By Fatou’s theorem, the embedding Hp(D) → Lp(T), f 7→ f ∗ is isometric, which showsthat Hp(D) can be identified with a closed subspace Hp(T) of functions in Lp(T). Inparticular, the Hardy space H2(T), and hence H2(D), is a Hilbert space.In what follows we shall frequently not distinguish between the Hardy spaces Hp(D)and Hp(T) and denote either of the spaces by Hp. According to this convention wemay also speak of holomorphic functions on the unit circle.In this connection two natural questions arise which will be answered in the followingsubsections:

1. Which functions g ∈ Lp(T) are boundary functions of f ∈ Hp(D)?

2. How functions f in Hp(D) can be recovered from f ∗ in Hp(T)?

In order to answer the first question we associate with any function g ∈ L1(T) thesequence of its Fourier coefficients

gk :=1

2π

∫ 2π

0

g(eiτ ) e−ikτ dτ, k ∈ Z.

If f is holomorphic in D, its Taylor series

f(z) =∞∑

k=0

fk zk,

converges uniformly on compact subsets of D. The coefficients are given by Cauchy’sintegral formula

fk =1

k!f (k)(0) =

1

2πi

∫rT

f(t)

tk+1dt, 0 < r < 1, k = 0, 1, 2, . . . .

If this formula is evaluated for k < 0, then, by Cauchy’s integral theorem,

fk =1

2πi

∫rTf(t) t−k−1 dt = 0, k = −1,−2, . . . .

4

As r → 1 − 0, at least formally, the formulas for the fk turn into the formula for theFourier coefficients of the boundary function f ∗

fk =1

2πi

∫Tf ∗(t) t−(k+1) dt =

1

2π

∫ 2π

0

f ∗(eiτ ) e−ikτ dτ, k ∈ Z.

A rigorous proof of this fact requires a nontrivial result.

Theorem 3. (F. and M. Riesz) If 1 ≤ p <∞ and f ∈ Hp, the functions fr : T → Rdefined by fr(t) := f(rt) converge in Lp(T) to the boundary function f ∗.

Taking this for granted with p = 1, it shows that the Fourier coefficients fk of theboundary function f ∗ coincide with the Taylor coefficients of f if k ≥ 0 and thatfk = 0 for k < 0. In fact this condition is not only necessary, but also sufficient:

Theorem 4. Let 1 ≤ p ≤ ∞. Then g ∈ Lp(T) is the boundary function of an f inHp(D) if and only if the Fourier coefficients gk vanish for all negative k. In this casethe gk are the Taylor coefficients of f at z = 0.

Theorem 4 characterizes the space Hp(T) of holomorphic functions as the (closed)subspace of functions g ∈ Lp(T) which have vanishing Fourier coefficients gk for k <0. Accordingly we call those functions g ∈ Lp(T) for which gk = 0 for all k ≥ 0antiholomorphic and denote the corresponding subspace of Lp(T) by Hp

−(T).In the Hilbert space L2(T) the functions tk form an orthonormal basis and thus H2(T)and H2

−(T) are orthogonal. The orthogonal projection P of L2(T) onto H2(T) is thenalso defined on Lp(T) if p ≥ 2 and we may ask if it maps (continuously) into Hp(T).For 1 ≤ p < 2 the question is whether or not P extends continuously to a boundedoperator P : Lp(T) → Hp(T). With the exception of p = 1 and p = ∞ both questionsare answered in the affirmative. The resulting operator P is called the Riesz projection(or Szego projection). The complementary projection I − P is denoted by Q.

Theorem 5. Ff 1 < p <∞ the Riesz projection P : Lp(T) → Hp(T) is bounded.

2.3 Integral representations

The final topic of this section concerns the reconstruction of holomorphic functionsat inner points of D from their boundary function on T. One of several possibilitiesconsists in extending Cauchy’s integral formula to the boundary of the domain.

Theorem 6. (Cauchy’s integral formula) If f ∈ H1(D), then for all z ∈ D

f(z) =1

2πi

∫T

f ∗(t)

t− zdt.

The Cauchy integral makes sense for all density functions f ∈ Lp(T) and z ∈ C \ T,

Cau f(z) :=1

2πi

∫T

f(t)

t− zdt, z ∈ C \ T.

5

The function Cau f is holomorphic in D as well as in C \ D. The boundary behaviorof Cau f with an arbitrary density function f for angular limits z → t ∈ T from insideand outside T is described in the next theorem.

Theorem 7. (Plemel’-Sokhotski) If f ∈ Lp(T) for some p > 1, then, in the sense ofangular limits,

f+(t) := limD3z→t

Cau f(z) = Pf(t), f−(t) := limC\D3z→t

Cau f(z) = −Qf(t).

A second possibility for reconstructing holomorphic functions from their boundaryfunctions makes use of the fact that holomorphic functions are also harmonic, and canthus be determined by means of Poisson’s integral formula.

Theorem 8. (Poisson’s integral formula). If f ∈ H1(D), then for all z ∈ D

f(z) =1

2π

∫T

Ret+ z

t− zf ∗(t) |dt| ≡ 1

2πi

∫T

Ret+ z

t− zf ∗(t)

dt

t.

The kernel function in the Poisson integral is said to be the Poisson kernel (of the unitdisk) and has the alternative representations

K(t, z) := Ret+ z

t− z=

1− |z|2

|t− z|2=

1− r2

1− 2r cos(ϕ− τ) + r2,

with t = eiτ and z = reiϕ.

In what follows we will frequently use the notation f(0) to denote the mean value ofa function f on T. This notation is motivated by Poisson’s integral formula, since themean value of f coincides with the value f(0) of the harmonic extension.

If the density function is the boundary function of a holomorphic function, the Cauchyand the Poisson formula yield the same result. In order to explain their differences forgeneral densities we study the mapping properties on the monomials tk. The result is

Cauchy : tk 7→{zk if k ≥ 00 if k < 0

for the Cauchy integral formula, while for Poisson’s formula

Poisson: tk 7→{zk if k ≥ 0z−k if k < 0.

This result reflects that Poisson’s integral formula yields a function which is harmonicin D (but not necessarily holomorphic) and coincides with the density on the boundary,while Cauchy’s integral formula produces a holomorphic function with boundary valuesthat do in general not coincide with the given density function f ∗.

6

A third formula for reconstructing a holomorphic function from its boundary values, isbased on the fact that a holomorphic function w is essentially determined by the realpart of its boundary function: If w ∈ H2 then

w(t) =∞∑

k=0

cktk,

where the series converges in L2(T), and hence

Rew(eiτ ) = Re c0 +∞∑

k=1

Re ck · cos kτ −∞∑

k=1

Im ck · sin kτ.

Consequently, if Rew(t) = 0 almost everywhere on T, then ck = 0 for k ≥ 1 andRe c0 = 0. Conversely, if a real-valued function u ∈ L2(T) is given by its Fourier series

u(eiτ ) = a0 +∞∑

k=1

(ak cos kτ + bk sin kτ

)then there exists w ∈ H2 with Rew = u on T, namely

w(z) = a0 +∞∑

k=1

(ak − ibk) zk.

The function w is unique up to an imaginary constant. A more compact expression forthe Schwarz’ operator u 7→ w is given in the following theorem.

Theorem 9. (Schwarz’ integral formula) Let 1 < p <∞.

(i) If w ∈ Hp(D), then for all z ∈ D

w(z) =1

2π

∫T

t+ z

t− zRew(t) |dt|+ i Imw(0).

(ii) If u ∈ Lp(T) is real–valued, and w is defined in D by

w(z) =1

2π

∫T

t+ z

t− zu(t) |dt|,

then w ∈ Hp(D), Rew = u almost everywhere on T, and Imw(0) = 0.

Of course, the Schwarz operator also acts on complex valued functions u. In particularwe have

Schwarz : tk 7→

2 zk if k > 01 if k = 10 if k < 0,

7

which reveals some similarity with the Cauchy integral.The real part of the Schwarz kernel (t+z)/(t−z) is the Poisson kernel, already knownfrom Poisson’s integral formula, its imaginary part is said to be the conjugate Poissonkernel. With z = reiϕ and t = eiτ we have

Ret+ z

t− z=

1− r2

1− 2r cos(τ − ϕ) + r2, Im

t+ z

t− z=

2r sin(ϕ− τ)

1− 2r cos(ϕ− τ) + r2.

If z approaches a point eiσ on the boundary, we obtain the limit

sin(σ − τ)

1− cos(σ − τ)= cot

σ − τ

2,

which is said to be the Hilbert kernel. Using this kernel we define the Hilbert operator

Hf(eiτ ) :=1

2π

∫ 2π

0

f(eiσ) cotσ − τ

2dσ,

where the integral is to be understood in the Cauchy principal value sense. A straight-forward computation (with t = eiτ ) shows that

H : sin kτ 7→ cos kτ (k = 1, 2, 3, . . .), H : cos kτ 7→ − sin kτ (k = 0, 1, 2, 3, . . .),

or, in complex notation,

Hilbert : tk 7→

−itk if k < 0

0 if k = 0itk if k > 0.

Note that Hf is defined on the circle T. From the first set of formulas we also see thatH commutes with differentiation ∂τ with respect to the polar angle τ

∂τHu = H∂τu.

Theorem 10. If 1 < p <∞ the Hilbert operator H : Lp(T) → Lp(T) is bounded.

Remark. The Hilbert operator is not continuous in L1 and L∞. Also there are contin-uous functions u for which Hu is unbounded.

The most important property of the Hilbert operator is that it connects real andimaginary part of holomorphic functions on T.

Theorem 11. If w = u + iv ∈ Lp(T) with 1 < p <∞, then w ∈ Hp(T) if and only ifthe boundary functions of u and v are connected by the relations

v = −Hu+ v(0), u = Hv + u(0).

8

Since the real part of a holomorphic function is a harmonic conjugate of its imaginarypart, H is also called the conjugation operator.

For convenient reference the mapping properties of all operators in this chapter aresummarized in the table below. Here t = eit stands for a variable on T, while z ∈ D.

function Cauchy Poisson Schwarz Riesz Hilbert

tk, k > 0 zk zk 2zk tk itk

1 1 1 1 1 0tk, k < 0 0 z−k 0 0 −itk

3 Linear Riemann-Hilbert Problems

As we just saw in the preceding chapter, holomorphic functions in Hp are uniquelydetermined by their boundary values. However, the boundary function cannot beprescribed arbitrarily, since it has to have vanishing Fourier coefficients with negativeindices. In other words, the Dirichlet problem for holomorphic functions is not wellposed.This observation leads to the question of correct (‘natural’) boundary value problemsfor holomorphic functions; a problem that was already studied by Bernhard Riemannin 1851.

When looking for well-posed boundary value problems for holomorphic functions, itis helpful to keep in mind that the Laplace operator factors as a product of the twoCauchy-Riemann-type operators ∂/∂z and ∂/∂z. Since the Dirichlet problem for theLaplace equation is well posed, this leads to the idea of considering boundary valueproblems for holomorphic functions, where ‘half of the boundary function’ is given.The simplest problem of this kind was already studied in the preceding section: Finda holomorphic function w with prescribed real part on the boundary

Rew = u on T.

In fact, the solution of this problem is essentially unique (Schwarz’ integral formula).In this section we investigate more general linear boundary value problems of this type,so called Riemann-Hilbert problems.

Throughout this chapter we assume that p is a fixed number with 1 < p < ∞. Weformulate the results for arbitrary p, the proofs are usually given for p = 2 only.The notation w = u+ iv is exclusively used for a holomorphic function with real partu and imaginary part v.

The linear Riemann-Hilbert problem consists in finding all holomorphic functions w =u+iv ∈ Hp, such that the real and the imaginary part of the boundary function satisfya (real) linear relation

a(t)u(t) + b(t)v(t) = c(t) (1)

9

almost everywhere on T. Here a, b, and c are given real–valued functions on T.

Another frequently used form of writing the boundary condition is

Im (f(t)w(t)) = c(t), (2)

with a given complex-valued function f = b − ia, called the symbol of the problem(recall that f(t) denotes the complex conjugate of f(t)). Geometrically the boundarycondition requires that every boundary value w(t) must lie on a straight straight linewhich is parallel to f(t), considered as a vector in R2.We study problem (2) for continuous symbols f which have no zeros on T. Later on theresults are extended to piecewise continuous symbols. The right-hand side c is assumedto be in the Lebesgue space Lp

R(T) of real–valued functions, the solution is sought inthe Hardy space Hp.It turns out that the solvability of (2) depends on a geometric characteristic of itssymbol f . If f is continuous and nonvanishing on T we can choose a branch of theargument ϕ := arg f such that the mapping

Φ: [0, 2π] → R, τ 7→ arg f(eiτ )

is continuous. The winding number of f (about the origin) is then defined as

wind f :=1

2π

(Φ(2π)− Φ(0)

).

Note that wind f is an integer and does not depend on the special choice of the branchof the argument. Geometrically, the winding number of f indicates the number oforiented turns of the graph of f about the origin. The number wind f is also referredto as the (geometrical) index of the Riemann-Hilbert problem (1).

3.1 Problems with index zero

After dividing the boundary condition Re (fw) = c by |f |, we may assume that |f | ≡ 1on T. If the function f has winding number zero it can be written as f = exp(iϕ) with acontinuous function ϕ. We start with investigating the homogeneous Riemann-Hilbertproblem.

Theorem 12. Let f = exp(iϕ), ϕ ∈ C(T), and 1 < p <∞. Then the general solutionw ∈ Hp of the homogeneous Riemann-Hilbert problem

Im (f w) = 0 (3)

is given by w = d · w0, where d is an arbitrary real number and

w0 := exp(Hϕ+ iϕ).

10

Proof. Before we give a rigorous proof of the result, we derive a formal solution.1. The function w0 := exp (Hϕ+ iϕ) is holomorphic and satisfies

argw0 = ϕ = arg f.

Since arg (fw0) = −arg f + argw0 = 0, the function w0 is a (special) solution of thehomogeneous Riemann-Hilbert problem (3).

2. Conversely, if w is any solution of the homogeneous problem, we consider thefunction w/w0. Since Im (w/w0) = 0 on T we have w/w0 = C with a real constant C.Thus the general solution is w = C w0.

3. In order to justify these formal manipulations we need Zygmund’s lemma. Roughlyspeaking this estimate shows that Hϕ has only logarithmic singularities (of a definitestrength) if ϕ is bounded.

Lemma 1. (Zygmund) For all ϕ ∈ L∞(T) with ‖ϕ‖∞ ≤ γ < π/2 we have

‖ exp(Hϕ)‖1 ≤1

cos γ.

If ϕ ∈ C(T), then exp(Hϕ) ∈ Lp(T) for all p with 1 ≤ p <∞.

Proof. We define the holomorphic function f as the Schwarz’ integral of ϕ,

f(z) :=1

2π

∫T

t+ z

t− zϕ(t) |dt|.

Since f ∈ Hp for all p < ∞, it has angular limits almost everywhere on T and theboundary function of f equals ϕ − iHϕ. The function g := exp(if) is holomorphicin D and has angular limits almost everywhere on T. Its boundary function satisfiesg∗ = exp(if ∗) = exp(Hϕ + iϕ), i.e. |g| = exp(Hϕ) almost everywhere on T. SinceIm f(0) = 0,

Re g(0) ≤ |g(0)| = | exp(if(0))| = 1.

Moreover,|arg g(z)| = |Re f(z)| ≤ ‖ϕ‖∞ ≤ γ < π/2.

So, for each r < 1,

1 ≥ Re g(0) =1

2π

∫T

Re g(rt) |dt| = 1

2π

∫T|g(rt)| cos arg g(rt) |dt|

≥ cos γ

2π

∫T|g(rt)| |dt|.

Consequently g ∈ H1 and ‖g‖1 ≤ 1/ cos γ. Since exp(Hϕ) = |g| on T, assertion (i)follows.In order to prove the second assertion we write ϕ as a sum of a trigonometric polynomialand a function with a sufficiently small L∞–norm.

11

We use Zygmund’s lemma to finish the proof of Theorem 12. First of all we see thatthe (boundary) function

w0 = exp(Hϕ+ iϕ) = exp(Hϕ) · exp(iϕ)

belongs to Lp for all p ∈ [1,∞). The function Hϕ + iϕ extends holomorphically intoD and belongs Hp. Consequently w0, the exponential of that function, is holomorphicin D. Its nontangential limits exist almost everywhere and the boundary function is inLp, consequently we have w0 ∈ Hp. It was already shown that it is a (special) solutionof the homogeneous Riemann-Hilbert problem.By the same reasoning, 1/w0 belongs to Hq for each q ∈ [1,∞). If w ∈ Hp isany solution, then (by Holder’s inequality) w/w0 ∈ H1+ε for some positive ε. SinceIm (w/w0) = 0 on T, the function w/w0 must be a real constant.

In the next step we consider inhomogeneous problems with index zero.

Theorem 13. Let f = exp(iϕ), ϕ ∈ C(T), and 1 < p <∞. Then, for each real valuedfunction c ∈ Lp(T), the general solution of Im (f w) = c in Hp is given by

w = wc + d · w0,

where d is an arbitrary real number and

w0 := exp(Hϕ+ iϕ),

wc := w0 ·(H (c/|w0|) + i (c/|w0|)

).

The operator Lp → Hp, c 7→ wc is bounded.

Proof. (For p = 2) We start with a formal construction of the solution. What followsresembles the method of variation of constants from ordinary differential equations.

1. We seek a particular solution of the inhomogeneous problem (2) in the form

w(t) = C(t)w0(t)

with a holomorphic function C. Inserting this ansatz into the boundary relation gives

ImC(t) =c(t)

|w0(t)|

and, reconstructing the holomorphic function C from its imaginary part, we get thedesired formula for the solution.

2. It remains to justify this formal approach. We have to show that the special solutionwc of the inhomogeneous problem belongs to H2 and that ‖wc‖2 ≤ C ‖c‖2 for someC > 0. Since

|wc| =∣∣i c+ |w0|H(c/|w0|)

∣∣ ≤ |c|+ |w0|∣∣H(c/|w0|)

∣∣,12

and |w0| = exp(Hϕ), we see that it suffices to show that∥∥ exp(Hϕ) ·H(c exp(−Hϕ)

)‖2 ≤ C ‖c‖2.

This estimate follows from a result of Helson and Szego about the mapping propertiesof the Hilbert operator in weighted Lebesgue spaces

L2%(T) := {f : ‖f‖2,% := ‖%f‖2 <∞}

on T.

Theorem 14. (Helson and Szego) Let % be an almost everywhere positive weight func-tion on T. Then the Hilbert operator H is bounded in L2

%(T) if and only if % admits arepresentation

% = exp (ψ +H ϕ),

where ψ, ϕ ∈ L∞(T) and ‖ϕ‖∞ ≤ γ < π/4. If ψ = 0, then

‖%Hv‖2 ≤ C(γ) ‖% v‖2

with C(γ) := 1/(√

2 cos γ − 1).

Proof. We only prove the ‘if’ part since we do not need the converse. The functionexpψ is bounded from above and below (away from zero), and thus we can restrictattention to weight functions % := exp (H ϕ) with ‖ϕ‖∞ ≤ γ < π/4. Moreover, takinginto account that H annihilates the constants and that Hv(0) = 0, it is sufficient toshow that there exists a constant C(γ), such that for all polynomials w = u+ iv withw(0) = 0 the estimate

‖%u‖2 ≤ C(γ) ‖%v‖2

holds. The result then follows from a density argument. By Zygmund’s lemma, thefunction exp(2Hϕ) is in L1, and hence g := exp (Hϕ+ iϕ) satisfies

g2 = cos 2ϕ exp(2Hϕ) + i sin 2ϕ exp(2Hϕ) ∈ H1.

Since the function w is a polynomial, also w2g2 ∈ H1. Because w(0) = 0,

1

2π

∫Tw2(t)g2(t) |dt| = (w2g2)(0) = w2(0)g2(0) = 0.

From w2 = u2 − v2 + 2iuv we obtain that

Re (w2g2) = %2 (u2 − v2) cos 2ϕ− 2 %2 uv sin 2ϕ,

which implies ∫T%2 (v2 − u2) cos 2ϕ |dt|+

∫T

2 %2 uv sin 2ϕ |dt| = 0.

13

Now we have cos 2ϕ ≥ δ := cos 2γ > 0 and thus

δ ‖%u‖22 ≤ ‖%v‖2

2 + 2 ‖%u‖2 ‖%v‖2,

which finally gives the desired result

‖%u‖2 ≤1

δ(1 +

√1 + δ) ‖%v‖2 = C(γ) ‖%v‖2.

We continue with the proof of Theorem 13. In order to estimate the norm of wc let% := exp(Hϕ). It is sufficient to show that

‖H(%−1c)‖2,% ≡ ‖%H(%−1c)‖2 ≤ C ‖c‖2 ≡ C ‖%−1c‖2,%,

which is equivalent to the boundedness of H in L2%. According to Theorem 14 this

estimate holds if Hϕ admits a representation ψ+Hϕ1, where ψ, ϕ1 ∈ L∞ and ‖ϕ1‖ ≤π/4. In order to get such a representation we write the (continuous) function ϕ as thesum of a trigonometric polynomial and a function ϕ1 with sup–nom less than π/4.

Frequently the Riemann-Hilbert problem is studied together with an additional condi-tion in order to fix the value of the free constant d. Since the solution is not necessarilycontinuous on D, it makes no sense to impose a condition for values of the solution ata point on T. A typical side condition involves the value w(0) of the solution at z = 0,

αu(0) + β v(0) = γ. (4)

A straightforward computation yields the value of the homogeneous solution w0 atz = 0,

w0(0) = exp(iϕ(0)) = cos δ + i sin δ,

where

δ :=1

2π

∫ 2π

0

arg f(eiτ ) dτ.

Consequently, the Riemann-Hilbert problem Im (f w) = c with the side condition (4)is uniquely solvable if α cos δ + β sin δ 6= 0.

3.2 Problems with positive index

In this and the following section we investigate Riemann-Hilbert problems with con-tinuous coefficients and arbitrary index κ := wind f . Again we can restrict ourselvesto symbols with |f | ≡ 1.

Theorem 15. Let 1 < p < ∞, f ∈ C(T), |f | ≡ 1 on T, and κ := wind f > 0. Thenthe homogeneous Riemann-Hilbert problem Im (f w) = 0 has 2κ + 1 (real) linearlyindependent solutions and the inhomogeneous problem Im (f w) = c is solvable in Hp

for any right-hand side c ∈ LpR.

14

Proof. Let g(z) := C0 +C1z + . . .+Cκ−1zκ−1 be any polynomial of degree κ− 1. If w

is a solution of the Riemann-Hilbert problem

Im(ft−κ w

)= c− Im

(fg)

with index zero, then w defined by w(z) := zκ w(z)+g(z) is a solution of Im(fw

)= c.

For c ≡ 0 the transformed problem has a one dimensional solution space d w0. Allsolutions obtained by varying C0, . . . , Cn and d are linearly independent.Conversely, any solution w of the original problem is obtained this way by choosing gsuch that w − g has a zero of order κ− 1 at z = 0.

It follows from the proof that the solution can be made unique by imposing side con-ditions

Im ( δ w(0)) = γ, w(zj) = wj (j = 1, . . . , κ).

at 0 and pairwise distinct points zj ∈ D \ {0}. While wj ∈ C and γ ∈ R can be chosenarbitrarily, δ must satisfy a compatibility condition which depends on f and the choiceof the points zj.

3.3 Problems with negative index

Finally we consider the linear Riemann-Hilbert problem with negative index.

Theorem 16. Let 1 < p < ∞, f ∈ C(T), |f | ≡ 1 on T, and κ := wind f < 0. Thenthe homogeneous Riemann-Hilbert problem

Im (f w) = 0

has only the trivial solution w ≡ 0 in Hp. The inhomogeneous problem

Im (f w) = c

is solvable if and only if the 2|κ| − 1 solvability conditions∫ 2π

0

c(eiτ )ψ(eiτ ) cos kτ dτ = 0, k = 0, 1, . . . , |κ| − 1,∫ 2π

0

c(eiτ )ψ(eiτ ) sin kτ dτ = 0, k = 1, . . . , |κ| − 1,

are satisfied, whereψ := exp

(−H arg(t|κ|f)

).

Proof. With w(z) := z−κw(z) and g(t) := t−κf(t) we have

Im(f(t)w(t)

)= Im

(f(t)t−κt−κw(t)

)= Im

(g(t)w(t)

).

The function w is in Hp and solves the Riemann-Hilbert problem Im(fw

)= c if

and only if w is in Hp, solves Im(gw

)= c, and has a zero of order |κ| at z = 0.

15

Since wind g = 0, we have an explicit expression for w. The solvability conditions areobtained by expressing

w(0) = 0,dw

dz(0) = 0, . . . ,

d|κ|−1w

dz|κ|−1(0) = 0

in terms of the Fourier coefficients of the boundary functions. Note that the numberof conditions is reduced by one, since w involves a free constant which can be used toeliminate one condition.

Note that the solvability conditions are given by bounded linear functionals in Lp whichact on the right–hand side c.

3.4 Problems with piecewise continuous coefficients

In the final section of this chapter we suppose that f is continuous with the possibleexception of finitely many jumps at points t1, t2, . . . , tn ∈ T. We assume that the pointstj = eiτj are arranged in counterclockwise order, such that 0 ≤ τ1 < τ2 < . . . < τn < 2π,and denote the one–sided limits of f(eiτ ) at τj by f−(tj) and f+(tj). As always wenormalize the boundary condition such that |f | ≡ 1.

Lemma 2. Let f : T → T be piecewise continuous. Then f has a representation

f(t) = s(t) · tκ · f0(t) · fr(t)

where s : T → {+1,−1} is piecewise continuous, f0 : T → T is a Laurent polynomial(i.e. a finite linear combination of integer powers of t) with wind f0 = 0, fr(t) : T → Tis piecewise continuous with |arg fr| ≤ π/4, and κ ∈ Z/2. If Re

(f+(tj)/f−(tj)

)6= 0,

then sup |arg fr| < π/4 and the number κ is uniquely defined.

Note that tκ stands for any branch of this function which is continuous on T\{1}. Theproof is constructive and elementary and we leave it as an exercise.

In order to formulate the main result in the language of functional analysis, we recallthat a bounded linear operator A is said to be Fredholm if it has closed range and ifthe dimension of its kernel and the codimension of its image are finite. In this case

indA := dim kerA− codim imA

is called the (functional analytic) index of the operator A.

Theorem 17. Assume that f : T → T is piecewise continuous with jumps at t1, t2, . . . , tn.Then the Riemann-Hilbert operator

Rf : H2(T) → L2R(T), w 7→ Im (f w)

has the following properties.

16

(i) If Re f+(tj)/f−(tj) 6= 0 for all j = 1, . . . , n, then Rf is Fredholm with

dim kerRf = max{0, 2κ+ 1}, codim imRf = max{0,−2κ− 1}.

In particular Rf is invertible if and only if κ = −1/2.

(ii) If Re f+(tj)/f−(tj) = 0 for some j, then Rf is not Fredholm.

Proof. 1. Let f(t) = s(t)·tκ ·f0(t)·fr(t) be the representation of the symbol f accordingto Lemma 2. Multiplying the boundary condition by s does not change much, only cmay be replaced by −c in some arcs. So we get an equivalent problem with s ≡ 1.Moreover, we have κ = µ or κ = µ − 1/2 with µ ∈ Z, and the representation f(t) =tµ g(t), together with the techniques already used for continuous symbols, transformthe problem to one with either κ = 0 or κ = −1/2.2. Let the assumptions of (i) be satisfied with κ = 0. Then we have f = exp i(ϕ0 +ϕr),where ϕ0 := arg f0 is a trigonometric polynomial (in τ), and ϕr := arg fr has sup–normless than π/4. The rest of the proof is exactly the same like for continuous symbols,with the only difference that now the Helson-Szego theorem is applied with ϕ = ϕr

and ψ = Hϕ0.

3. If the assumptions of (i) are satisfied with κ = −1/2 we have

f(t) = t−1/2 · f0(t) · fr(t),

with a continuous branch of t−1/2 on T \ {1}. Then the functions g and d defined by

g(t) :=

{f(t2) if 0 ≤ arg t < π

−f(t2) if π ≤ arg t < 2π,d(t) :=

{c(t2) if 0 ≤ arg t < π

−c(t2) if π ≤ arg < 2π.

are odd and g has (generalized) winding number −1. If w is a solution of Im (f w) = c,then w defined by w(z) := w(z2) is a solution of Im (g w) = d. Conversely, if wsolves the second problem, then w must be even. Indeed, otherwise w− defined byw−(z) = w(−z) would be another solution of this problem, which is impossible sincewind g = −1. This ensures that the relation w(z2) = w(z) defines a function w ∈ H2.It remains to show that Im (g w) = d always has a solution. This happens if andonly if Im (tg tw) = d has a solution w with w(0) = 0. The problem has index zero,so a solution w exists. Since tg is even and d is odd, the function w− defined byw−(z) := −w(−z) is also a solution and thus (w+ w−)/2 is a solution that vanishes atz = 0.

4. It remains to prove (ii). Assume to the contrary that Rf is Fredholm. Then thereare arbitrarily small (in the sup-norm) perturbations of f which satisfy (i) and havedifferent winding numbers. This is in conflict with the stability of the Fredholm indexwith respect to small perturbations of Rf .

17

4 Nonlinear Riemann-Hilbert problems

Surprisingly, the history of boundary value problems for holomorphic functions does notstart with linear problems. Careful reading of Riemann’s thesis reveals the followingproblem, which is now frequently also referred to as the nonlinear Riemann-Hilbertproblem:

Let {Mt}t∈T be a given family of curves in the complex plane. Find all functionsw = u+ iv holomorphic in the open unit disk D and continuous in the closed unit diskD (this space of functions is called the disc algebra and we denote it by H∞ ∩C) suchthat the boundary condition

w(t) ∈Mt

is satisfied for all points t ∈ T. The curves Mt go by the name target curves orrestriction curves of the problem. The figure illustrates this definition.

For linear Riemann-Hilbert problems au + bv = c the target curves are the straightlines

Mt := {u+ iv ∈ C : a(t)u+ b(t)v = c(t)}.

A less geometrical form of writing the boundary condition of a nonlinear Riemann-Hilbert problem is

F(t, u(t), v(t)

)= 0.

Here F : T× R2 → R is a given function, called the defining function of the problem.Note that the defining function is by no means unique, only the zero set of F isimportant to define the target curves Mt.

4.1 Explicit Riemann-Hilbert problems

This section is devoted to the case where the defining function can be solved withrespect to one variable, v say, which results in boundary conditions of the type

v(t) = F (t, u(t)). (5)

Let us look for the right spaces to treat the problem. Since F : T × R → R will bedifferentiated (for technical reasons) we assume it to be continuously differentiable.

18

Further we need the Hilbert operator, which acts continuously in Lebesgue spacesLp(T), but not in C(T) and, more generally, in Ck(T). The right–hand side of (5)contains the superposition operator u 7→ F (., u). Since this operator is (in general) notwell defined in Lebesgue spaces, we try next Sobolev spaces on the unit circle. Recallthat the Sobolev space W 1

p (T) is the linear space of all functions f ∈ Lp(T) which havegeneralized derivatives ∂τf ∈ Lp(T) with respect to the polar angle τ . The norm inW 1

p (T) is given by

‖f‖W 1p

={‖f‖p

p + ‖∂τ‖pp

}1/p.

The functions in W 1p (T) are continuous and the embedding W 1

p (T) → C(T) is compact.Then, in fact, all involved operators are bounded and continuous in W 1

p .

The main result of this chapter is the existence of a unique solution under relativelymild assumptions on the defining function F .

Theorem 18. Assume that the function F : T×R → R is continuously differentiableand has bounded derivatives. Then, for each u0 ∈ R, the nonlinear Riemann-Hilbertproblem

v(t) = F (t, u(t)), u(0) = u0 (6)

has a unique solution w = u + iv in H∞ ∩ C. This solution belongs to H∞ ∩W 1p for

all p with 1 ≤ p <∞.

The proof is based on Schauder’s fixed point theorem (see [2], Sect. 7.1 [3], Sect. 2.8).Recall that a (nonlinear) operator K : E → E is said to be compact, if it is continuousand the image of each bounded subset is relatively compact. The following result isthe simplest version of Schauder’s fixed point theorem, but sufficient for our purposes.

Theorem 19. (Schauder) If the operator K : E → E is compact and has a boundedrange K(E), then K has a fixed point.

In the following we transform the nonlinear Riemann-Hilbert problem to a fixed pointequation for a compact operator K in the Sobolev space E := W 1

p (T). Note that wecan rewrite the problem as a nonlinear operator equation v = F (.,−Hv + u0), whichalready has fixed point form. However, the Hilbert operator H is not compact, and sowe need a trick to produce compactness.The idea consists in differentiating the boundary relation v(eiτ ) = F (eiτ , u(eiτ )) withrespect to the polar angle τ . This results in a quasilinear Riemann Hilbert problemfor the derivative ∂τw and integrating its solution along T reproduces u. Indeed thisworks fine, but the details are a bit more involved.For u ∈ W 1

p we define

a := ∂uF (., u− u(0) + u0), c := ∂τF (., u− u(0) + u0) (7)

and denote by w ≡ u+ i v the solution to the linear Riemann-Hilbert problem

v − au = c, u(0) = 0. (8)

19

Further we set

u(eiτ ) :=

∫ τ

0

u(eiσ) dσ. (9)

This definition is correct since the mean value of u vanishes due to the conditionu(0) = 0 imposed on u. The equations (7)–(9) define the (nonlinear) operator

K : W 1p → W 1

p , u 7→ u− u(0) + u0.

Two relevant properties of K are summarized in the next lemma.

Lemma 3. Let 1 < p, q <∞. Then the operator K has the following properties.

(i) The mapping K : W 1p → W 1

q is compact.

(ii) A function u ∈ W 1p is a fixed point of K if and only if there exists d ∈ R such

thatHu+ F (., u) = d and u(0) = u0. (10)

Condition (10) is the same as saying that w = u + i (d − Hu) is a solution of theRiemann-Hilbert problem (6).

Proof. 1. The operator K is composed as shown in the following scheme:

W 1p → C → C × C → Lq → W 1

q

u 7→ u 7→ (a, c) 7→ u 7→ u.

All mappings involved in this diagram are continuous and the first one is a compactembedding. This proves (i).

2. Let w ≡ u + iv satisfy Hu + F (., u) = const and u(0) = u0. Differentiation withrespect to the polar angle τ yields

H ∂τu+ ∂τF (., u) + ∂uF (., u) · ∂τu = 0,

H ∂τu+ c+ a · ∂τu = 0.

Since the function ∂τu− iH ∂τu is holomorphic and (∂τu)(0) = 0, it must coincide withthe unique solution w of (8), and hence ∂τu = u, H ∂τu = v. Consequently,

Ku := u− u(0) + u0 =

∫ τ

0

∂τu(eiσ) dσ + const = u+ const.

Since Ku(0) = u0 = u(0), the constant vanishes.

3. If, conversely, u is a fixed point of K, then Ku := u − u(0) + u0 = u and thusu(0) = u0. Further,

u = ∂τ u = ∂τu,

and thereforev = −Hu+ v(0) = −H ∂τu+ v(0),

20

which implies

∂τ

(F (., u) +Hu

)= ∂uF (., u) · ∂τu+H∂τu+ ∂τF (., u) = au− v + v(0) + c = const.

The mean value of the function on the left-hand side is zero, and hence the constanton the right-hand side must vanish also, so that

F (., u) +Hu = const.

This completes the proof of Lemma 3.

Once the existence of a fixed point u ofK is established, the solutions w of the Riemann-Hilbert problem is simply given by w := u+ iF (., u).

In order to apply Schauder’s fixed point theorem, we prove that K maps W 1p into a

certain ball, provided that p is sufficiently close to 1.

Lemma 4. There exists a p > 1 such that the range of K : W 1p → W 1

p is bounded.

The proof of this result requires some preparations. Above all we need a better estimateof the solutions to the linear Riemann-Hilbert problem

v − au = c, u(0) = 0.

Recall that its solutions are given by the following set of formulas:

c0 := c/|1 + ia|, ϕ := arg (1 + ia), w0 := exp (Hϕ+ iϕ)

w = Bc := w0 ·(H (c0/|w0|) + i c0/|w0|

).

We already know that B : Lp → Lp is bounded, but we get a better estimate for thenorm of B if we think of B as of a mapping B : Lq → Lp with p close to 1 and qsufficiently large. To this end we adapt the Helson-Szego theorem to pairs of weightedLebesgue spaces. With respect to the situation of interest only weights % = exp (Hϕ)are considered.

Lemma 5. For each γ with 0 ≤ γ < π/2 there exist positive numbers C, p, and q with1 < p < 2 < q <∞, such that for all weight functions % with

% := exp (H ϕ), ‖ϕ‖∞ ≤ γ < π/2,

the norm of H : Lq% → Lp

% is bounded by C,

‖%Hu‖p ≤ C ‖% u‖q.

21

Proof. We choose an ε with 0 < ε < 1 such that

δ :=1 + ε

1− εγ <

π

2

and prove that the assertion of the lemma is valid for p := 1 + ε, q := 2 + 2/ε, and acertain positive constant C. By Holder’s inequality,

‖%Hu‖pp =

∫%p |Hu|p dτ =

∫%

p2

(% |Hu|2

) p2 dτ

≤{∫

%p

2−p dτ

}1− p2

·{∫

% |Hu|2 dτ

} p2

.

Since

γp

2− p= γ

1 + ε

1− ε= δ <

π

2,

Zygmund’s lemma yields that∥∥∥% p2−p

∥∥∥1

=

∥∥∥∥ exp

(1 + ε

1− εHϕ

)∥∥∥∥1

≤ C1.

Consequently,

‖%Hu‖p ≤ C(2−p)/2p1 ‖%1/2Hu‖2 ≤ C

1/21 ‖%1/2Hu‖2. (11)

Applying now the theorem of Helson and Szego to the weight %1/2 = exp (12Hϕ), we

obtain the estimate‖%1/2Hu‖2 ≤ C2 ‖%1/2 u‖2. (12)

Again by Holder’s inequality,

‖%1/2u‖22 =

∫% |u|2 dτ =

∫%−1(%|u|)2 dτ

≤{∫

%−1−ε dτ

}1/(1+ε) {∫(%|u|)(2+2ε)/ε dτ

}ε/(1+ε)

= ‖%−1−ε‖1/(1+ε)1 ‖%u‖2

q.

Since %−1−ε = exp H((−1 − ε)ϕ

)and ‖(1 + ε)ϕ‖∞ ≤ δ < π/2, Zygmund’s inequality

yields‖%−1−ε‖1/(1+ε)

1 ≤ C1/(1+ε)1 ≤ C1,

and hence‖%1/2u‖2 ≤ C

1/21 ‖% u‖q. (13)

Combining the estimates (11), (12), and (13) we arrive at

‖%Hu‖p ≤ C1/21 ‖%1/2Hu‖2 ≤ C

1/21 C2 ‖%1/2u‖2 ≤ C1C2 ‖% u‖q. (14)

22

In order to prove Lemma 4 we remark that the uniform boundedness of the derivative∂uF implies the uniform boundedness of a := ∂τF (., u− u(0) + u0) for all u ∈ W 1

p .

Lemma 6. For each positive C0 there exist numbers p > 1 and C > 0 such that anysolution w = u+iv of the Riemann-Hilbert problem v−au = c with ‖a‖∞ ≤ C0 satisfiesthe estimate

‖w‖p ≤ C ‖c‖∞.

Proof. The symbol 1 + ia of the Riemann-Hilbert problem v − au = c satisfies

|arg (1 + ia)| ≤ δ < π/2.

The result then follows on applying the generalized Helson-Szego theorem to the for-mulas in Theorem 13 which represent the solution.

After these preparations we are in a position to prove Lemma 4. Since ‖c‖∞ is boundedby a constant that does not depend on the choice of u, Lemma 6 yields the (uniform)boundedness of ‖u‖p in Lp for all p sufficiently close to 1. Integrating along T thenfinally shows that Ku is bounded in W 1

p .

Proof of Theorem 18. 1. The existence of a fixed point of K (and hence of a solutionto the Riemann-Hilbert problem) in H∞ ∩W 1

p for p close to 1 follows from Schauder’sfixed point theorem.

2. Since, for every p > 1, K maps W 1p into W 1

q for each q with 1 ≤ q < ∞, any fixedpoint belongs to W 1

q for all finite q.

3. Finally, if w1 and w2 are two solutions of (5) in H∞ ∩ C, then their difference is asolution of a homogeneous linear Riemann-Hilbert problem

v2 − v1 = f(., u2)− f(., u1) = g · (u2 − u1)

with continuous coefficients. This problem has index zero and since u2(0)− u1(0) = 0,we have w1 = w2.

Remark: The solution w depends continuously on the parameter u0 in the side conditionu(0) = u0 and the real part of u tends uniformly to ±∞ if u0 → ±∞. Then a continuityargument shows that the Riemann-Hilbert problem v = F (., u) with the side conditionu(1) = u1 at t = 1 (or at any point on T) has a (unique) solution.

4.2 Problems with compact target manifold

This and the next section are devoted to general nonlinear Riemann-Hilbert problemsin implicit form

F (t, u(t), v(t)) = 0.

Instead of writing the boundary condition as an equation we prefer the geometricformulation

w(t) ∈Mt,

23

where {Mt : t ∈ T} is a given family of target curves in the complex plane.To write the boundary condition in a more compact form, we introduce the set

M := {(t, z) ∈ T× C : z ∈Mt},

which is referred to as the target manifold of the problem, and the trace of a solutionw as the graph of its boundary function,

trw := {(t, w(t)) : t ∈ T}.

The boundary condition can then be written simply as

trw ⊂M.

Here we consider the case where the target curves Mt are simple closed curves and thetarget manifold M is compact. A compact target manifold M is called admissible, ifit is a submanifold of T×C of smoothness class C1 which is transverse to every plane{t}×C. The figure below illustrates the construction of an admissible compact targetmanifold from its target curves and shows the trace of a solution.

In order to acquire a feeling for what may happen let us consider three examples.

Example 1. If Mt := T for all t ∈ T, the solutions w ∈ H∞ ∩C are exactly the finiteBlaschke products:

w(z) = c

n∏k=1

z − zk

1− zkz,

where zk ∈ D and c ∈ T are arbitrarily chosen numbers.

Example 2. If Mt := {w ∈ C : |w − t−1| = 1}, then w ≡ 0 is the only solution.This follows from the observation that any solution satisfies Re t w(t) ≥ 0 on T andz w(z) = 0 at z = 0, which is only possible if w vanishes throughout D.

Example 3. IfMt := {w ∈ C : |w−2 t−1| = 1}, then, by a similar reasoning (or simplyby the argument principle) there is no holomorphic function such that w(t) ∈ Mt forall t ∈ T.

24

These examples stand for three general subclasses of Riemann-Hilbert problems withcompact target manifold. We denote by intM the bounded component of (T×C) \Mand by clos intM its topological closure.

A target manifold M is said to be regularly traceable if there exists a holomorphicfunction wM ∈ H∞ ∩ C with trwM ⊂ intM ; it is called singularly traceable if it isnot regularly traceable but there still exists wM ∈ H∞ ∩ C with trwM ⊂ clos intM ;otherwise M is called nontraceable.

The classes of regularly, singularly, or nontraceable target manifolds are denoted byR, S, and N , respectively.

The first of the above examples is of class R, the second belongs to S, and third is inN , respectively. As we shall see in a moment, problems of class R have a solution setwith a rich structure. The solutions can be classified by a geometric characteristic, thewinding number windMw about the target manifold.In order to define this winding number, we fix an arbitrary complex-valued functionwM ∈ C(T) with trwM ∈ intM , and set

windM f := wind (f − wM), (15)

where the “wind” on the right-hand side refers to the usual winding number about theorigin. This definition is independent of the choice of the function wM .The figures show a target manifold with traces of solutions having winding numberszero and two, respectively.

The problem of conformally mapping D onto a simply connected C1-region G is aanother special case of a Riemann-Hilbert problem with a compact target manifold.It results from putting Mt := ∂G for all t ∈ T. Usually conformal mappings arerequired to be schlicht (i.e. one-to-one). In the context of Riemann-Hilbert problemsthis is reflected in the additional “winding condition” windMw = 1. We discuss thesolvability of Riemann-Hilbert problems with regularly traceable target manifolds. Thenext theorem shows that the solution set has the same structure as for Example 1.

Theorem 20. Let M ∈ R with wM ∈ H∞ ∩ C satisfying trwM ⊂ intM . Then thefollowing assertions hold.

25

(i) For arbitrarily given points t0 ∈ T and W0 ∈Mt0 there exists exactly one solutionw ∈ H∞ ∩ C with windM w = 0 which satisfies the boundary condition

trw ⊂M (16)

and the interpolation condition

w(t0) = W0. (17)

(ii) Given any positive integer n and points t0 ∈ T, W0 ∈ Mt0, z1, . . . , zn ∈ D, thereexists exactly one solution w ∈ H∞ ∩ C with windM w = n which satisfies theconditions (16), (17), and the interpolation conditions

w(zj) = wM(zj), j = 1, . . . , n. (18)

(iii) The target manifold M is covered in a schlicht manner by the traces of the so-lutions to (16) with winding number zero. The same result holds for the set ofsolutions with windMw = n which satisfy (18) for fixed numbers z1, z2, . . . , zn.

(iv) The solutions described in (i) and (ii) are the only solutions of (16) in H∞ ∩ C.In particular, there is no solution with a negative winding number about M .

The figure shows a target manifold and traces of solutions with winding number zero.

For a proof of this result it is not sufficient to apply a simple fixed point theorem. Allproofs known so far utilize homotopy methods. We give an outline of this approach.The reader interested in details of the proof is referred to [22].

The main idea is to imbed the given problem in a family of problems depending ona parameter λ ∈ [0, 1]. This embedding depends continuously (in a definite sense) onthe homotopy-parameter λ. The homotopy is chosen so that it coincides with the givenproblem for λ = 1, and with a simple problem (where a solution is known) for λ = 0.Since we know that solutions exist for λ = 0 we just have to show that they cannotdisappear along the homotopy when λ runs from 0 to 1. The theory of Leray-Schauderis a useful tool to settle this task. Its realization requires essentially three steps.

26

First of all one has to construct the homotopy. For Riemann-Hilbert problems withclosed target curves the idea consists in deforming these curves to unit circles for λ = 0.Secondly, the problems have to be reformulated as fixed point equations for compactoperators Kλ. This can be done in analogy to the approach used for explicit Riemann-Hilbert problems.The third step is harder and quite technical. In order to ensure that the solutions ‘donot escape to infinity’ along the homotopy, we need an a-priori estimate. This means,assuming the existence of solutions wλ corresponding to the homotopy-parameter λ,we show that all solutions are uniformly bounded, so that ‖wλ‖ < C for all λ. Thisguaranties that all solutions (if such exist) remain in a certain ball B.According to Leray and Schauder we can associate with each operator Kλ an integerfixBKλ, called its fixed-point index with respect to the ball B. If this index is differentfrom zero, then Kλ has a fixed point in B. One main property of the fixed-point indexis its homotopy invariance: If Kλ depends continuously on λ and for all λ no fixedpoint of Kλ lies on the boundary of B, which is ensured by the a-priori estimate, thenfixBKλ = const. Especially, fixBK1 = fixBK0, so that K1 has a fixed point, if we canshow that fixBK0 6= 0 for the simple operator K0.

Riemann-Hilbert problems with singularly traceable target manifolds have completelydifferent solvability properties.

Theorem 21. Let M ∈ S and suppose the function wM ∈ H∞ ∩ C satisfies trwM ⊂clos intM . Then the following assertions hold.

(i) The function wM is the only solution in H∞ ∩C of the Riemann-Hilbert problemtrw ⊂M .

(ii) The winding number of wM about M is negative.

The remaining case of nontraceable target manifolds is trivial. We formulate the resultonly for the sake of completeness.

Theorem 22. Let M ∈ N . Then there is no solution w ∈ H∞ ∩ C of the Riemann-Hilbert problem trw ⊂M .

The class M of all admissible compact target manifolds can be endowed with a naturaltopology. Then it turns out that R and N are open sets and S is their commonboundary. Thus singular problems are ‘degenerate’, and “generically” Riemann-Hilbertproblems belong either to R or to N . For details see [22].

4.3 Problems with noncompact target manifold

A prominent example of a Riemann-Hilbert problem with noncompact target manifoldis given by linear problems with continuously differentiable coefficients and right–hand

27

side. More generally, we now assume that the target manifold satisfies the followingconditions.

A non-compact orientable target manifold M is said to be admissible, if it has aparametrization

M ={(t, µ(t, s)

): t ∈ T, s ∈ R

}(19)

where µ ∈ C1(T× R) is such that

(i) for each t ∈ T the mapping s 7→ µ(t, s) is injective,

(ii) 0 < 1/C ≤ |Dsµ| ≤ C on T× R,

(iii) the limitslim

s→±∞Dsµ(t, s) := µ±(t), lim

s→±∞s−2Dtµ(t, s) = 0

exist uniformly with respect to t ∈ T.

The number indM := windµ+ is said to be the index of the problem. Geometrically,the index is the ‘twisting number’ of the target manifold M . For the time being, alladmissible target manifolds are orientable by assumption.Condition (iii) guarantees a certain regular behavior of the target curves Mt at infinity.While the condition implies that the direction of the target curves stabilizes for s →±∞, the existence of a (linear) asymptotics does not follow. Note that there is nointerrelation between the behavior at ‘both ends’ of Mt.The figure shows (part of) two target manifolds with index zero and index two

The qualitative results for nonlinear problems with noncompact orientable target man-ifold are completely analogous to the linear case.

Theorem 23. Let M be a noncompact admissible target manifold. Then the followingassertions hold.

(i) If indM = 0 the target manifold is covered by the graphs of solutions w ∈ H∞∩Cto the Riemann-Hilbert problem graphw ⊂M in a schlicht manner.

(ii) If indM < 0 there exists at most one solution w ∈ H∞ ∩ C of graphw ⊂M .

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(iii) If indM = n ≥ 0 then, for arbitrarily given t0 ∈ T, W0 ∈ Mt0 , z1, . . . , zn ∈D, W1, . . . ,Wn ∈ C, there exists exactly one solution of graphw ⊂ M whichsatisfies the interpolation conditions

w(t0) = W0, w(zk) = Wk (k = 1, . . . , n). (20)

The case of nonorientable noncompact target manifolds can be dealt with by consid-ering the doubled manifold M having the fibers Mt := Mt2 . It is remarkable thatRiemann-Hilbert problems with a unique solution are exactly those with a nonori-entable manifold with indM = −1/2. The reader interested in the details may consult[22].

5 Applications

5.1 Boundary value problems for harmonic functions

There are several boundary value problems for harmonic functions (in plane domains)which can be reformulated as Riemann-Hilbert problems. Since harmonic functionsare invariant under conformal mapping, we restrict ourselves to problems on the disk.Because any harmonic function u in the disk has a harmonic conjugate v such thatw = u+ iv is holomorphic, the Dirichlet problem

∆u(z) = 0 in D, u(t) = f(t) on T

is (more or less) equivalent to the Riemann-Hilbert problem Rew = f . Of course, theDirichlet problem can be solved explicitly by Poisson’s formula.More interesting problems result from the observation that for harmonic u the functionw := ∂xu− i∂yu is holomorphic. So any problem

∆u(z) = 0 in D, F (t, ∂xu(t), ∂yu(t)) = 0 on T

with boundary condition involving the gradient of u (but not u itself) can be reformu-lated as a Riemann-Hilbert problem. In the special case where F is linear with respectto u and v we obtain the oblique derivative problem

∆u(z) = 0 in D, a(t) ∂xu(t) + b(t)∂yu(t) = c(t) on T

which generalizes the well-known Neumann problem. The related linear RHP has indexwind(b+ia), which is always −1 for the Neumann problem (why?). In the next sectionwe give an example of a nonlinear oblique derivative problem in hydrodynamics.

5.2 A problem in hydrodynamics

It is well known that conformal mapping techniques are useful in studying flow problemsaround obstacles. Here we consider a situation where the fluid can penetrate the

29

boundary of the obstacle. More precisely, we investigate the plane steady irrotationalflow of an incompressible inviscid fluid with density % past a (not necessarily circular)cylinder with permeable (porous or perforated) surface.

Let G denote the intersection of the inner region of the cylinder with the ζ-plane(ζ = ξ+iη) orthogonal to the axis of the cylinder. We look for the pressure field p andfor the velocity field ~v = (vξ, vη). As usual the influence of gravity is neglected.In the outer region we assume that the flow at infinity is parallel to the ξ−axis andhas the speed q∞,

vξ(∞) = q∞, vη(∞) = 0. (21)

The pressure at infinity is denoted by p∞. The velocity components of the flow in thedirection of the (positively oriented) tangent and the inner normal on the boundary∂G are denoted by vσ and vν , respectively. The circulation J of the flow around ∂Gand the flux Q through ∂G are given by

J :=

∫∂G

vσ dσ, Q :=

∫∂G

vν dσ, (22)

where integration is with respect to arc length σ on ∂G. Notice that Q > 0 meanssuction of liquid out of the flow and Q < 0 injection of liquid into the flow.The assumption that the flow be irrotational in the outer region of the cylinder and thecontinuity equation guarantee that the complex velocity W := vξ− ivη is a holomorphicfunction in the outer domain C\G. Because of (21) and (22) its asymptotic expansionat infinity is

W (ζ) = q∞ −Q+ iJ

2π

1

ζ+O(ζ−2). (23)

The filtration process through theboundary of the cylinder ist de-scribed by a filtration law. Herewe assume that the filtration ve-locity is equal to the normal com-ponent vν and that is a continu-ous monotone function of the dif-ference p − pi of the pressure atthe outer and the inner surface ofthe cylinder.

30

We admit that the filtration law depends on the point s ∈ ∂G,

vν(s) = Φ(s, p(s)− pi(s)

).

The case of a linear law Φ(s, p) = K(s) p is of particular interest.The simplest model for the interior of the cylinder is the assumption of constant pres-sure p0.

From the physical (and mathematical) point of view we get three reasonable problemsif any two of the three quantities, pressure p0 inside the cylinder, total flux Q throughthe surface, and circulation J around the cylinder are prescribed and the third one isfree. Here we consider the case where Q is unknown.The Bernoulli equation in the outer region allows to eliminate the pressure p from thespeed q,

p = p∞ +%

2q2∞ −

%

2q2.

On the outer surface of the cylinder we have q2 = v2ν + v2

σ, and inserting this into thefiltration law we get the basic boundary relation

vν = Φ(., d− %

2(v2

ν + vσ)2), (24)

with the pressure constant

d :=%

2q∞ + p∞ − p0.

This is a nonlinear oblique derivative problem for the velocity potential Ψ, which is aharmonic function in the flow region with grad Ψ = (vξ, vη). In order to transform itto a Riemann-Hilbert problem in the disk we use a conformal mapping ϕ of C\G ontothe exterior C \ aD of a disk, normalized at infinity by ϕ(ζ) = ζ +O(1). The positivenumber a is uniquely determined.On account of (23) and the normalization of ϕ, the function W := (W ◦ϕ−1)/(ϕ′ ◦ϕ−1)is holomorphic in C \ aD and has the expansion

W (Z) = q∞ −Q+ iJ

2π

1

Z+O(Z−2) (25)

at infinity. The normal and tangential velocity components vν and vσ on ∂G aretransformed according to

vν + ivσ = W · (nξ + inη) = W · ϕ′ · (nξ + inη) = W · |ϕ′| · (nX + inY ),

where nξ, nη and nX , nY are the components of the inner normal vectors to ∂G and

to aT at the points ζ and Z = ϕ(ζ), respectively. Hence, putting W = U + iV andseparating the real and imaginary parts, we obtain that

vν = Λ(−U cos τ + V sin τ

), vσ = Λ

(−U sin τ − V cos τ

).

31

Here Λ := |ϕ′| is the boundary distortion of the conformal mapping ϕ, and τ denotes

the polar angle of the point Z. By the definition of W , we have on the contour

q2 = U2 + V 2 = Λ2(U2 + V 2

). (26)

By virtue of the relations (25)–(26), the boundary condition (24) takes the form

Λ(− U cos τ + V sin τ

)= Φ

(., d− Λ2%

2(U2 + V 2)

).

After the transformation z = a/Z we arrive at a Riemann-Hilbert problem for a func-

tion w(z) = u(z) + i v(z) := W (a/z), which is holomorphic in the unit disk. Theboundary condition now reads

u cos τ + v sin τ = − 1

ΛΦ

(., d− Λ2%

2(u2 + v2)

),

where u = u(eiτ ), v = v(eiτ ). The asymptotic behavior of W at infinity implies that

w(z) = q∞ −Q+ iJ

2πaz +O(z2)

in a neighborhood of the origin. The ansatz w(z) = q∞(1 + z w(z)

), with a new

holomorphic function w, and a short calculation transform the boundary conditioninto a dimension-less form,

u+ cos τ = −F(. , c− E2(u+ cos τ)2 − E2(v − sin τ)2

), (27)

and the additional condition becomes

w(0) = −Q+ iJ

2πaq∞. (28)

Here we used the abbreviations

F (t, p) := Φ(s, p% q2

∞)/(q∞Λ(s)

), E(t) := Λ(s)/

√2, c := (p∞ − p0)/(%q

2∞) + 1/2,

with s := ϕ−1(at). So the filtration problem has the following formulation as a Rie-mann -Hilbert problem in the disk:

To given real numbers c and J find a function w = u+ iv holomorphic in the unit diskand a real constant Q such that the boundary condition (27) and the side condition (28)are satisfied.

Note that only J is prescribed, while Q has to be determined, so that condition (28)fixes the imaginary part of w(0) only. After solving the problem, Q can be computedfrom the real part of w(0).The geometry of the target curves depends significantly on the growth of the filtrationlaw for p→ −∞. For laws with Φ(s, p) ∼ |p|κ and κ > 1/2 we typically obtain closed

32

target curves, while κ < 1/2 corresponds to non-closed curves. In the linear case thetarget curves are circles. If the filtration constant K is constant the problem can theneven be solved explicitly.

The Riemann-Hilbert problem under discussion has solutions with different windingnumbers. The physically relevant solutions have winding number zero. They are theonly solutions which reproduce the flow around a solid cylinder as the function Φ inthe filtration law tends to zero.It should finally be mentioned that the Riemann-Hilbert approach reduces the un-bounded exterior flow problem to a problem on the boundary of the region. In par-ticular, for computational purposes, this is an advantage over domain discretizationmethods, since it reduces a two–dimensional problem in an unbounded domain to aproblem on a bounded one–dimensional manifold.The figure shows the streamlines and the pressure distribution (color) in a flow withJ = 0 and J > 0.

5.3 Design of dynamical systems

Another question which is intimately (but not obviously) related to Riemann-Hilbertproblems was raised in the eighties by J.W.Helton. Its engineering background isdesign of dynamical systems via optimization of the frequency response function.Roughly speaking, a dynamical system takes input functions and produces output func-tions. Usually these functions live on the real line and are considered as time dependentsignals.The most important dynamical systems are causal, linear, time invariant and stablesystems. Those systems are completely characterized by their transfer function T ; theFourier transform F of the input signal and G of the output signal are related by

G(ω) = T (ω) · F (ω).

It follows from Payley–Wiener theory that the transfer function of causal systemsextends to a bounded holomorphic function in the upper half–plane (respectively to

33

the right half–plane, if one works with the Laplace transform, which is frequently donein the engineering literature).A system operating at a harmonic input exp(iωt) (here t ∈ R denotes time) withfrequency ω produces the output T (ω) · exp(iωt), which has the same frequency as theinput, but different amplitude and phase. The values |T (ω)| and arg T (ω) describe gainand phase shift of the system at frequency ω. For this reason the boundary functionf of T on the real axis is called the frequency response of the system, f(ω) := T (ω).Since T is a bounded holomorphic function it can be reconstructed from its boundaryfunction f . Conversely, any bounded holomorphic function T in the right half plane isthe transfer function of such a system.

Linear, causal, time invariant, stable systems are completely determined by its fre-quency response.

So much for the description – but equally important is the design of systems. Usuallya designer knows what he would consider to be a ‘good performance’ f(ω) of thesystem at frequency ω. However, the frequency response is the boundary function ofa holomorphic function, so it cannot be prescribed arbitrarily; design of systems is anoptimization problem.Here we assume that the performance of a (virtual) system with frequency response fis evaluated in terms of a penalty function

F : R× C → R+, (ω,w) 7→ F (ω,w),

where F (ω,w) attains small values if w = f(ω) is ‘good’ and large values if it is ‘bad’.The standard choice of F is F (ω,w) := |w − f0(ω)|, the euclidean distance from an‘ideal’ response f0. The figure shows the level sets (color) of a more general penaltyfunction.

Once the penalty function F is fixed, the over – all rating of a system, in the sense ofworst case analysis over all frequencies, is the real number

P (f) := sup {F (ω, f(ω)) : ω ∈ R}

Optimal design of a system then requires to find the optimal performance

P ∗ := inf {P (f) : f is a frequency response function}

34

and all systems (if such exist) for which the infimum is attained.In order to relate the optimization problem to Riemann-Hilbert problems we transformthe transfer functions from the right half plane to the unit disk (a related conformalmapping is the Cayley transform) and introduce the sets of bounded analytic functionswith restricted boundary values

A(p) := {w ∈ H∞(D) : F (t, w(t)) ≤ p a.e. on T}.

The set A(p) is built from the transfer functions of all systems with performance notworse than p, and the optimization problem is equivalent to find the smallest possiblevalue of p for which A(p) is not empty.The left figure shows traces of functions in A, the right figure depicts the nested levelsets

M(p) := {(t, w) ∈ T× C : F (t, w) = p}

for several values of p.

It turns out that the properties of A(p) are closely related to the Riemann-Hilbertproblem with the target manifold M(p).

Theorem 24. Let M(p) is an admissible target manifold and denote by #A(p) thenumber of elements in A(p). Then #A(p) > 1, #A(p) = 1, or #A(p) = 0, if and onlyif M(p) ∈ R, M(p) ∈ S, or M(p) ∈ N , respectively.

The result is much deeper than it looks like. For instance it implies that any A(p)which contains a bounded function w ∈ H∞ automatically also contains a function inthe disk algebra H∞ ∩ C.

Returning to the optimzation problem we consider strictly nested families of targetmanifolds M(p), which means that the assumptions (i)–(iii) are satisfied:

(i) The M(p) are admissible compact target manifolds, depending continuously onp ∈ R+.

(ii) For all p and q with 0 < p < q we have M(p) ⊂ intM(q).

(iii) M(p) ∈ R if p is sufficiently large; M(p) ∈ N if p is sufficiently small.

35

Theorem 25. For any strictly nested family of target manifolds the optimization prob-lem has a unique solution. A function w ∈ H infty is a solution of the optimizationproblem if and only if

w ∈ H∞ ∩ C, F(t, w(t)

)= p∗ = const on T, wind ∂wF (., w) ≥ 1.

Here ∂w denotes the antiholomorphic Cauchy-Riemann operator ∂w = ∂u − i∂v. To-gether with F (t, w(t)) = p∗ this yields windM(p∗)w = −∂wF (., w) = −k < 0. Forgeneric problems we have k = 1, which implies that k > 1 is ‘very unlikely’.

Assertion (ii) reflects the fact that the solution of the optimization problem is thesolution of a singular Riemann-Hilbert problem. It has the special meaning that opti-mal systems ‘flatten the penalty’, i.e., they have one and the same performance for allfrequencies.

It is an important feature of this approach that the optimization problem is transformedinto a (parameter depending) boundary value problem; we get equations instead ofinequalities. This allows, for instance, the construction of efficient numerical methodsfor solving the design problem.

6 Historical Remarks

The history of the Riemann-Hilbert problem can be traced back to the origins offunction theory. The problem enters the scene in 1851, the year of Riemann’s famousthesis “Grundlagen fur eine allgemeine Theorie der Functionen einer veranderlichencomplexen Grosse” (see [16], pp. 3–47). When studying conditions which uniquelydefine a function holomorphic in a domain, Riemann wrote: “Die Bedingungen, welchesoeben zur Bestimmung der Function hinreichend und nothwendig befunden wordensind, beziehen sich auf ihren Werth in Begrenzungspunkten . . . und zwar geben siefur jeden Begrenzungspunkt eine Bedingungsgleichung. Bezieht sich jede derselben nurauf einen Begrenzungspunkt, so werden sie durch eine Schar von Kurven reprasentirt,von denen fur jeden Begrenzungspunkt eine den geometrischen Ort bildet.”Not regarding the connection with the problem of conformal mapping, Riemann himselftreated the problem only heuristically. The first one who proposed a method appro-priate for solving the linear problem was David Hilbert. In a lecture given at the 3rdInternational Mathematical Congress in Heidelberg in 1904 he introduced the singularintegral operator with cotangent kernel and transformed the problem to a Fredholmintegral equation with a continuous kernel. Later on Hilbert published a modificationof his earlier approach and made a few remarks about the index which he disregardedformerly, as was pointed out by Fritz Noether [15] in 1921. Noether’s courageous workeven contains material on problems with piecewise continuous coefficients.Fritz Noether’s paper [15] about linear Riemann-Hilbert problems and related singularintegral equations is without doubt one of the milestones of functional analysis – it wasthis paper where the index of an operator appeared for the first time.

36

Noether’s discovery was the starting point of a rapidly developing theory which relatesgeometric properties (here: the winding number of the function f) to solvability prop-erties of an operator equation (dimension of kernel and codimension of the range). Itled subsequently to a general symbol concept, being introduced by Solomon Mikhlinin the late thirties, and culminated in Lars Hormander’s theory of pseudodifferentialoperators.

During the ‘classical period’ linear Riemann-Hilbert problems and related singularintegral equations with Holder continuous coefficients where studied by F. D. Gakhov,N. I. Muskhelishvili, I.N.Vekua, and others. The scalar linear Riemann-Hilbert problemwith measurable coefficients could finally be treated by I. B. Simonenko [19] in 1960.Beginning in the sixties, Banach algebra techniques were developed, which allowed toattack some of the remaining hard questions (cf. Gohberg and Krupnik [7]). Com-bined with so-called local principles, they form the heart of the modern theory ofRiemann-Hilbert problems, singular integral equations and Toeplitz operators. Thisis, however already another story. The reader who is willing to learn more about thosedevelopments is referred to the monograph by Bottcher and Silbermann [1].

Riemann-Hilbert problems with small nonlinearities where already studied in the fiftiesby Gekht, Gusejnov, Natalevich, and others, using the fixed point theorems of Banachand Schauder.Without assuming that the nonlinearities are “sufficiently small”, L. v.Wolfersdorf ob-tained solvability results by an application of Schauder’s fixed point theorem. Thematerial on explicit problems is directly related to [25]. The proof given there still re-quires an additional “oscillation condition”, which could be omitted in [20]. The trickof differentiating the boundary condition is taken from [26]. Explicit problems withglobally Lipschitz continuous function F have been studied by Efendiev and Wegert[24].For general classes of nonlinear Riemann-Hilbert problems the real breakthrough camein 1972 with a paper by A. I. Shnirelman [18]. He proved the solvability of regu-larly traceable Riemann-Hilbert problems with target manifold of smoothness classC2. Schnirel’man’s original proof [18] was based on a specific topological degree theoryfor so-called quasi–linearlike mappings. Later L.v.Wolfersdorf invented the ‘differentia-tion trick’ which allowed to work with the simpler concept of Leray-Schauder degree. Italso allowed to relax the smoothness assumptions to C1, see Wegert [20]. An alterna-tive C2–approach, using the implicit function theorem in Banach spaces, was proposedby Forstneric [5]. Problems with non-compact target manifold have been treated infull generality for the first time in [20].For a comprehensive treatment of nonlinear Riemann-Hilbert problems with compactand non-compact target manifolds, and a number of applications, we refer to [22]. Asurvey on later developments up to the end of the last century is in [23].

The general H∞–optimization problem in the design of dynamical systems was in-troduced by J.W.Helton in the eighties (see [9]). Though the ‘flattening property’of continuous solutions is known since 1986 (Helton and Howe [10]), the existence of

37

continuous solutions has been open for several years. It was shown independently byHelton and Marshall [11], and, in a more general setting, by Wegert [21], using theapproach sketched above.

References

[1] A.Bottcher and B. Silbermann: Analysis of Toeplitz Operators, 2nd edition.Springer 2006.

[2] W.Cheney: Analysis for Applied Mathematics. Springer 2001.

[3] K. Deimling: Nonlinear Functional Analysis. Springer 1985.

[4] P.Duren: Theory of Hp spaces. Academic Press 1970.

[5] F. Forstneric: Analytic disks with boundaries in a maximal real submanifold ofC2. Ann. Inst. Fourier 37 (1987) 1–44.

[6] J.B.Garnett: Bounded Analytic Functions. Academic Press, New York 1981.

[7] I. Gohberg and N.Krupnik: Introduction to the Theory of One-dimensional Sin-gular Integral Operators (Russian). Shtiintsa, Kishinev 1973;

[8] H.Helson and G. Szego: A problem in prediction theory. Ann.Math. Pura Appl.51 (1960) 107–138.

[9] J.W.Helton: Operator Theory, Analytic Functions, Matrices, and Electrical Engi-neering. Conference Board of the Mathematical Sciences by the AMS, ProvidenceRI 1987.

[10] J.W.Helton and R.E.Howe: A bang-bang theorem for optimization over spaces ofanalytic functions, J. Approx. Theory 47 (1986) 2, 101–121.

[11] J.W.Helton and D.E.Marshall: Frequency domain design and analytic selections,Indiana Univ. Math. J. 39 (1990) 1, 157–184.

[12] K.Hoffman: Banach Spaces of Analytic Functions. Prentice-Hall, Englewood Cliffs1962.

[13] P. Koosis: Lectures on Hp spaces. London Math. Soc. Lect. Notes Series No. 40,Cambridge Univ. Press, London 1980.

[14] W.Fischer and I. Lieb: Ausgewahlte Kaptel aus der Funktionentheorie. FriedrichVieweg, Braunschweig, Wiesbaden 1988.

[15] F.Noether: Uber eine Klasse singularer Integralgleichungen. Math.Ann. 82 (1921)42–63.

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[16] B.Riemann: Grundlagen fur eine allgemeine Theorie der Functionen einerveranderlichen complexen Grosse. In: Gesammelte mathematische Werke und wis-senschaftlicher Nachlaß (Ed. R. Dedekind and H. Weber) Leipzig 1876, pp. 3–47.

[17] M.Rosenblum and J.Rovnyak, Topics in Hardy Classes and Univalent Functions.Birkhauser 1994.

[18] A.I. Shnirel’man: The degree of quasi-linearlike mapping and the nonlinear Hilbertproblem (Russian). Mat. Sb. 89 (1972) 3, 366–389; English transl.: Math.USSRSbornik 18 (1974) 373–396.

[19] I.B. Simonenko: The Riemann boundary value problem with measurable coeffi-cients. Dokl. Akad. Nauk SSSR 135 (1960) 3, 538–541.

[20] E.Wegert: Topological methods for strongly nonlinear Riemann-Hilbert problemsfor holomorphic functions, Math. Nachr. 134 (1987) 201–230.

[21] E.Wegert: Boundary value problems and best approximation by holomorphicfunctions. J. Approx. Theory 16(1990) 3, 322–334.

[22] E.Wegert: Nonlinear Boundary Value Problems for Holomorphic Functions andSingular Integral Equations. Akademie Verlag 1992.

[23] E.Wegert: Nonlinear Riemann-Hilbert problems – History and perspectives. In:Computational Methods and Function Theory. N. Papamichael, St. Ruscheweyhand E.B. Saff (Eds.), 583–615, World Sci. Publ. 1999.

[24] E.Wegert and M.A.Efendiev: Nonlinear Riemann-Hilbert problems with Lipschitzcontinuous boundary condition. Proc. Royal Soc. Edinburgh, Ser. A. 130 (2000)793–800.

[25] L. v. Wolfersdorf: On the theory of the nonlinear Riemann-Hilbert problem forholomorphic functions. Complex Variables 3 (1984) 323–346.

[26] L. v. Wolfersdorf: A class of nonlinear Riemann-Hilbert problems for holomorphicfunctions. Math. Nachr. 116 (1984) 89–107.

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