# Boundary value problems of analytic function theory

Problems of finding an analytic function in a certain domain from a given relation between the boundary values of its real and its imaginary part. This problem was first posed by B. Riemann in 1857 . D. Hilbert  studied the boundary value problem formulated as follows (the Riemann–Hilbert problem): To find the function $\Phi (z) = u + iv$ that is analytic in a simply-connected domain $S ^ {+}$ bounded by a contour $L$ and that is continuous in $S ^ {+} \cup L$, from the boundary condition

$$\tag{1 } \mathop{\rm Re} \{ (a + ib) \Phi \} = \ au - bv = c,$$

where $a, b$ and $c$ are given real continuous functions on $L$. Hilbert initially reduced this problem to a singular integral equation in order to give an example of the application of such an equation.

The problem (1) may be reduced to a successive solution of two Dirichlet problems. A complete study of the problem by this method may be found in .

The problem arrived at by H. Poincaré  in developing the mathematical theory of tide resembles problem (1). Poincaré's problem consists in determining a harmonic function $u(x, y)$ in a domain $S ^ {+}$ from the following condition on the boundary $L$ of this domain:

$$\tag{2 } A (s) \frac{\partial u }{\partial n } + B (s) \frac{\partial u }{\partial s } + C (s) u = f (s),$$

where $A(s), B(s), C(s)$ and $f(s)$ are real functions given on $L$, $s$ is the arc abscissa and $n$ is the normal to $L$.

The generalized Riemann–Hilbert–Poincaré problem is the following linear boundary value problem: To find an analytic function $\Phi (z)$ in $S ^ {+}$ from the boundary condition

$$\tag{3 } \mathop{\rm Re} \{ \lambda \Phi \} = f (t _ {0} ),\ \ t _ {0} \in L ,$$

where $\lambda$ is an integro-differential operator defined by the formula

$$\tag{4 } \lambda \Phi = \ \sum _ {j = 0 } ^ { m } \left \{ a _ {j} (t _ {0} ) \Phi ^ {(j)} (t _ {0} ) + \int\limits _ { L } h _ {j} (t _ {0} , t) \Phi ^ {(j)} (t) ds \right \} ,$$

where $a _ {0} (t _ {0} ), \dots, a _ {m} (t)$ are (usually complex-valued) functions of class $H$ defined on $L$ (i.e. satisfying a Hölder condition), $f(t)$ is a given real-valued function of class $H$ and $h _ {j} ( t _ {0} , t)$ are (usually complex-valued) functions on $L$ of the form

$$h _ {j} (t _ {0} , t) = \ \frac{h _ {j} ^ {0} (t _ {0} , t) }{| t - t _ {0} | ^ \alpha } ,\ \ 0 \leq \alpha < 1,$$

where $h _ {j} ^ {0} ( t _ {0} , t)$ are functions of class $H$ in both variables. The expression $\Phi ^ {(j)} (t _ {0} )$ on the right-hand side of (4) is understood to mean the boundary value on $L$ from inside the domain $S ^ {+}$ of the $j$-th order derivative of $\Phi (z)$.

A special case of the Riemann–Hilbert–Poincaré problem, in the case when $m = 0$, $h _ {j} (t _ {0} , t) = 0$, is the Riemann–Hilbert problem; Poincaré's problem is also a special formulation of the same problem. Many important boundary value problems — such as boundary value problems for partial differential equations of elliptic type with two independent variables — may be reduced to the Riemann–Hilbert–Poincaré problem.

The Riemann–Hilbert–Poincaré problem was also posed for $a _ {m} (t _ {0} ) \neq 0$, $t _ {0} \in L$, and was solved by I.N. Vekua .

An important role in the theory of boundary value problems is played by the concept of the index of the problem — an integer defined by the formula

$$\kappa = 2 (m + n),$$

where $2 \pi n$ is the increment of $\mathop{\rm arg} \overline{ {a _ {m} (t) }}$ under one complete traversal of the contour $L$ in the direction leaving the domain $S ^ {+}$ at the left.

The Riemann–Hilbert–Poincaré problem is reduced to a singular integral equation of the form

$$\tag{5 } N _ \mu \equiv A (t _ {0} ) \mu (t _ {0} ) + \int\limits _ { L } N (t _ {0} , t) \mu (t) ds =$$

$$= \ f (t _ {0} ) - c \sigma (t _ {0} ),$$

where $\mu$ is the unknown real-valued function of class $H$, $c$ is an unknown real constant, and

$$N (t _ {0} , t) = \ \frac{K (t _ {0} , t) }{t - t _ {0} } .$$

The functions $A (t _ {0} ), K ( t _ {0} , t)$ and $\sigma (t _ {0} )$ are expressed in terms of $a _ {j} (t)$ and $h _ {j} (t _ {0} , t)$, $j = 0, \dots, m$.

Let $k$ and $k ^ \prime$ be the numbers of linearly independent solutions of the homogeneous integral equation $N _ \mu = 0$ corresponding to (5) and of the homogeneous integral equation

$$\tag{6 } N _ \nu ^ { \prime } \equiv \ A (t _ {0} ) \nu (t _ {0} ) + \int\limits _ { L } N (t, t _ {0} ) \nu (t) ds = 0,$$

associated with it. The numbers $k$ and $k ^ \prime$ are connected with the index $\kappa$ of the Riemann–Hilbert–Poincaré problem by the equality

$$\kappa = k - k ^ \prime .$$

Of special interest is the case when the problem is solvable whatever the right-hand side $f(t _ {0} )$. In order for the Riemann–Hilbert–Poincaré problem to be solvable whatever the right-hand side $f (t _ {0} )$, a necessary and sufficient condition is $k ^ \prime = 0$ or $k ^ \prime = 1$, and in the latter case the solution $\nu (t)$ of equation (6) must satisfy the condition

$$\int\limits _ { L } \nu (t) \sigma (t) ds \neq 0;$$

in both cases $\kappa \geq 0$ and the homogeneous problem $\mathop{\rm Re} \{ \lambda \Phi \} = 0$ has exactly $\kappa + 1$ linearly independent solutions. If $\sigma (t) = 0$, then the Riemann–Hilbert–Poincaré problem is solvable for any right-hand side if and only if $k ^ \prime = 0$.

As regards the Riemann–Hilbert problem, the following statements are valid: 1) If $\kappa \geq 0$, then the inhomogeneous problem (1) is solvable whatever its right-hand side; and 2) if $\kappa < -2$, then the problem has a solution if and only if

$$\int\limits _ { 0 } ^ { 2 \pi } e ^ {i (k + \kappa /2) \phi } \Omega ( \phi ) c ( \phi ) d \phi = 0,\ \ k = 1, \dots, - \kappa - 1,$$

where

$$\Omega ( \phi ) = \ \frac{1}{\sqrt {a ^ {2} ( \phi ) + b ^ {2} ( \phi ) } } \mathop{\rm exp} \left \{ - { \frac{1}{4 \pi } } \int\limits _ { 0 } ^ { 2 \pi } \theta ( \phi _ {1} ) \mathop{\rm cotg} \ \frac{\phi _ {1} - \phi }{2} d \phi _ {1} \right \} ,$$

$$\theta (t) = \mathop{\rm arg} \left [ - t ^ {- \kappa } \frac{a - ib }{a + ib } \right ] ,\ a ^ {2} + b ^ {2} \neq 0.$$

The Riemann–Hilbert problem is closely connected with the so-called problem of linear conjugation. Let $L$ be a smooth or a piecewise-smooth curve consisting of closed contours enclosing some domain $S ^ {+}$ of the complex plane $z = x + iy$, which remains on the left during traversal of $L$, and let the complement of $S ^ {+} \cup L$ in the $z$-plane be denoted by $S ^ {-}$. Let a function $\Phi (z)$ be given, and let it be continuous in a neighbourhood of the curve $L$, everywhere except perhaps on $L$ itself. One says that the function $\Phi (z)$ is continuously extendable to a point $t \in L$ from the left (or from the right) if $\Phi (z)$ tends to a definite limit $\Phi ^ {+} (t)$( or $\Phi ^ {-} (t)$) as $z$ tends to $t$ along an arbitrary path, while remaining to the left (or to the right) of $L$.

The function $\Phi (z)$ is said to be piecewise analytic with jump curve $L$ if it is analytic in $S ^ {+}$ and $S ^ {-}$ and is continuously extendable to any point $t \in L$ both from the left and from the right.

The linear conjugation problem consists of determining a piecewise-analytic function $\Phi (z)$ with jump curve $L$, having finite order at infinity, from the boundary condition

$$\Phi ^ {+} (t) = \ G (t) \Phi ^ {-} (t) + g (t),\ \ t \in L,$$

where $G(t)$ and $g(t)$ are functions of class $H$ given on $L$. On the assumption that $G(t) \neq 0$ everywhere on $L$, the integer

$$\kappa = \frac{1}{2 \pi } \ [ \mathop{\rm arg} G (t)] _ {L}$$

is called the index of the linear conjugation problem.

If $\Phi (z) = ( \Phi _ {1}, \dots, \Phi _ {n} )$ is a piecewise-analytic vector, $G(t)$ is a square $(n \times n )$-matrix and $g(t) = (g _ {1}, \dots, g _ {n} )$ is a vector, and if also $\mathop{\rm det} G(t) \neq 0$, then the integer

$$\kappa = \frac{1}{2 \pi } \ [ \mathop{\rm arg} \mathop{\rm det} G (t)] _ {L}$$

is called the total index of the linear conjugation problem. The concepts of the index and the total index play an important role in the theory of the linear conjugation problem , , .

The theory of one-dimensional singular integral equations of the form (5) was constructed on the basis of the theory of the linear conjugation problem.

How to Cite This Entry:
Boundary value problems of analytic function theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boundary_value_problems_of_analytic_function_theory&oldid=51875
This article was adapted from an original article by A.V. Bitsadze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article