21. Beweis der Existenz linearer Differentialgleichungen mit vorgeschriebener Monodromiegruppe

The Original Riemann-Hilbert Problem

The problem was initially raised by Hilbert in his 1900 paper as the following

The Original Riemann-Hilbert Problem (The original RHP ([2], 1900)). Proof of the existence of linear differential equations having a prescribed monodromic group.

The statement above seems to be too different from the description that we use below. The reason is that there is a misunderstanding of mathematical terms at that time, such as between regular singularities and Fuchsian singularities. Primarily, we can interpret it as the following (Organized by Thomas Bothner [1]).

Interpretation of the Original Riemann-Hilbert Problem. Given a monodromy group with encoded singularity locations, are we asked to realize it by

  • a Fuchisan linear p-th order differential equation?
  • a linear p \times p system having only regular singularities?
  • a Fuchsian system on the whole Riemann sphere \mathbb{C}\mathbb{P}^1?

The first question was proven to be false by Poincare: the number of parameters in a p-th order linear Fuchsian equation with singularities a_1, \dots,a_n is less than the dimension of the space of monodromy representations. The second interpretation was proven to be true by Plemelj in 1908 and believed to settle the third interpretation as well. However, 75 years later, Kohn and Arnold, ll’yashenko found a series gap in Plemelj’s argument. In 1989, the third argument was finally proven to be false by Bolibrukh.

The Riemann-Hilbert Boundary Value Problem

Preliminary Formulation.

Given a contour \Gamma, a normalization point z_0 \in \mathbb{C} \cup{\infty}, a normalization constant C_0 \in \mathbb{C}^{n \times m}, and jump functions G:\Gamma \to \mathbb{C}^{m \times m} and F:\Gamma \to \mathbb{C}^{n \times m}, find \Phi:\mathbb{C} \setminus \Gamma \to \mathbb{C}^{n \times m} that satisfies the following:

  • \Phi is analytic off \Gamma,
  • \Phi is bounded at \infty and has weaker than pole singularities throughout the complex plane. This means that

        \[\lim_{\tau \to z}|(\tau - z)\Phi(z)|< \infty \quad \text{for all }z \in \mathbb{C}\]

  • \Phi(z_0) = C_0, and
  • \Phi satisfies the jump condition.

    \[\Phi^{+}(s) = \Phi^{-}(s)G(s)+F(s) \quad \text{for} \ s \in \Gamma\]

Preliminary Formulation (from Trogdon & Olver [7])

Notice that in this preliminary formulation, \Phi is a sectionally analytic with two limits from both sides of \Gamma satisfying the jump condition. To make the definition of \Phi more precise, we may assume that \Gamma is a complete contour.

Definition 1. \Gamma is said to be a complete contour if \Gamma can be oriented in such a way that \mathbb{C} \setminus \Gamma can be decomposed into left and right components:

    \[\mathbb{C} \setminus \Gamma = \Omega_+ \cup \Omega_- \quad \text{and} \quad \Omega_+ \cap \Omega_- = \varnothing\]

\Omega_+(\Omega_-) lies to the left(right) of \Gamma.

We now define the left and right boundary values pointwise:

    \[\Phi^{+}(s) = \lim_{\substack{z \to s \ z \in \Omega_+}}\Phi(z) \quad \text{and} \quad \Phi^{-}(s)= \lim_{\substack{z \to s \ z \in \Omega_-}}\Phi(z)\]

To precisely define the jump condition, we need to specify \Phi^{\pm} as functions on \Gamma. Two common requirements for \Phi^{\pm} are

  • \Phi^{\pm} should exist at every interior point of the contour and be continuous functions except at endpoints of \Gamma where they should be locally integrable, or
  • \Phi^{\pm} should exist almost everywhere (with respect to Lebesgue arc length measure) and be in an appropriate L^p space.

The first case is a continuous RH problem and the second is an L^p RH problem.

The Solution of Scalar Riemann-Hilbert Problem

We need to first introduce some definitions for later use.

Definition 2. Define the space C^{0,a}(\Gamma), 0 < \alpha \leq 1, for \Gamma smooth, bounded, and closed, consisting of uniformly \alpha-Hölder continuous functions. We introduce the seminorm

    \[|f|_{0,a} =\sup_{s_1 \neq s_2,\ s_1,s_2 \in \Gamma} \frac{|f(s_1)-f(s_2)|}{|s_1-s_2|^\alpha}\]

which is finite for every function in C^{0,a}(\Gamma). C^{0,a}(\Gamma) is a Banach space when equipped with the norm

    \[\norm{f}_{0,\alpha} =\sup_{s \in \Gamma}|f(s)| + |f|_{0,a}\]

Definition 3. Given an oriented contour \Gamma and a function f:\Gamma \to \mathbb{C}, the Cauchy Integral is defined by

    \[\mathcal{C}_{\Gamma}f(z) = \frac{1}{2\pi i }\int_{\Gamma}\frac{f(s)}{s-z}\ ds\]

Note that we need to consider the existence and regularity of the Cauchy integral in the RH problem, and this is where Hölder theory is introduced. This post will not get into deep details, but there is a fairlywide class of functions, the \alpha-Hölder continuous functions, for which the limits of Cauchy integrals are well-defined and regular.

We will now consider the solution of the simplest RH problem on smooth, closed, and bounded curves.

Problem 1. Find \phi that solves the continuous RH problem

    \[\phi^+(s)-\phi^-(s) = f(s),\ s \in \Gamma,\ \phi(\infty) = 0,\ f \in C^{0,a}(\Gamma)\]

where \Gamma is smooth, bounded, and closed curve.

This problem can be solved directly by the Cauchy integral \phi(z) =\mathcal{C}_{\Gamma}f(z) from the Plemelj’s Lemma. The existence of the solution will be constructed during the precise formulation of the problem, and to prove the uniqueness, let \varphi(z) be another solution. The function

    \[D(z) = \varphi(z)-\phi(z)\]

satisfies the condition

    \[D^+(s)-D^-(s)=0,\ s \in \Gamma,\ D(\infty) = 0\]

It implies that D is continuous up to \Gamma, and is analytic at every point on \Gamma, and hence entire. By Liouville’s theorem, it must be identically zeros, which proved the uniqueness.

Now we move to the simplest case of a Riemann-Hilbert Problem with a multiplicative jump.

Problem 2. Find \phi that solves the homogeneous continuous RH problem

    \[\phi^+(s)= \phi^-(s) g(s),\ s \in \Gamma,\ \phi(\infty) = 1,\ g \in C^{0,a}(\Gamma)\]

where g(s) \neq 0, \Gamma is smooth, bounded, and closed curve.

This problem can be solved via the logarithm. Let X(z) = \log \phi(z), the problem becomes

    \[X^+(s)-X^-(s) = G(s),\ G(s) = \log g(s)\]

If \log g(s) is well-defined and Hölder continuous, the solution is given by

    \[X(z) = \mathcal{C}_{\Gamma}G(z) \quad \implies \quad \phi(z) = \exp(\mathcal{C}_{\Gamma}G(z))\]

Problem 3. Find \phi that solves the inhomogeneous continuous RH problem

    \[\phi^+(s)= \phi^-(s) g(s) +f(s),\ s \in \Gamma,\ \phi(\infty) = 0,\ g,f \in C^{0,a}(\Gamma)\]

where g(s) \neq 0, \Gamma is smooth, bounded, and closed curve.

To solve this problem, we need to find the fundamental solution of the homogeneous problem, like the method of variation of parameters.

In fact, the solution procedure for scalar RH problems is not much more difficult in practice
when the curve \Gamma is not closed. Acomplication comes from the fact that in the case of arcs, additional solutions are introduced, which can easily be seen in the continuous RH problem.

Reference and Extended Readings

[1] Thomas Bothner, On the origins of Riemann-Hilbert problems in mathematics. https://arxiv.org/pdf/2003.14374.pdf
[2] D. Hilbert, Mathematische Probleme, Gottinger Nachrichten 3 (1900), no. 1, 253–297.
[3] H. Poincare, Sur les groupes des equations lineaires, Acta Math. 4 (1884), no. 1, 201-312, DOI 10.1007/BF02418420.
[4] A. Treibich Kohn, Un resultat de Plemelj, Mathematics and physics (Paris, 1979/1982), 1983, pp. 307-312.
[5] V. I. Arnol’d and Yu. S. Il’yashenko, Ordinary differential equations, Encyclopaedia Math. Sci., vol. 1, Springer, Berlin, 1988. Translated from the Russian by E. R. Dawson and D. O’Shea.
[6] A. A. Bolibrukh, The Riemann-Hilbert problem on the complex projective line, Mat. Zametki 46 (1989), no. 3, 118-120.
[7] Trogdon, T. D., &; Olver, S. (2016). Riemann-Hilbert problems, their numerical solution, and the computation of nonlinear special functions. Philadelphia: SIAM, Society for Industrial and Applied Mathematics.
[8] Erik Koelink and Walter Van Assche. Orthogonal Polynomials and Special Functions. Springer.
[9] Yulij ll’yashenko and Sergei Yakovenko. Lectures on Analytic Differential Equations. Moscow State University.


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