Boundary value problem, complex-variable methods

Methods for studying boundary value problems for partial differential equations in which one uses representations of solutions in terms of analytic functions of a complex variable.

Given a second-order elliptic equation

$$\tag{1 } \Delta u + a (x, y) \frac{\partial u }{\partial x } + b (x, y) \frac{\partial u }{\partial y } + c (x, y) u = 0,$$

where $a, b, c$ are analytic functions of the real variables $x, y$ in some domain $D _ {0}$ of the $z$- plane, $z = x + iy$, consider the following boundary value problem: Find a solution of equation (1), regular in a simply-connected domain $S \subset D _ {0}$, satisfying the boundary condition

$$\tag{2 } R (u) \equiv \ \sum _ {0 \leq j + k \leq m } \left [ a _ {jk} (t) \frac{\partial ^ {j + k } u }{\partial x ^ {j} \partial y ^ {k} } + T _ {jk} \left ( \frac{\partial ^ {j + k } u }{\partial x ^ {j} \partial y ^ {k} } \right ) \right ] = f (t),$$

where $a _ {jk} (t), f (t) \in C _ \alpha ( \partial S), 0 < \alpha < 1$, and the $T _ {jk}$ are linear operators mapping $C _ \alpha ( \partial S)$ into $C _ \alpha ( \partial S)$, with $T _ {m - k, k }$ completely continuous.

This problem includes as special cases the well-known classical boundary value problems of Dirichlet, Neumann, Poincaré, etc.

Using the formula for the general representation of solutions (see Differential equation, partial, complex-variable methods),

$$u (x, y) = \ \mathop{\rm Re} \left \{ G (z, \overline{z}\; _ {0} , z, \overline{z}\; ) \Phi (z)\right . -$$

$$- \left . \int\limits _ {z _ {0} } ^ { {z } } \Phi (t) \frac \partial {\partial t } G (t, \overline{z}\; _ {0} , z, \overline{z}\; ) dt \right \} ,$$

one reduces the problem to an equivalent boundary value problem for analytic functions:

$$\tag{3 } \mathop{\rm Re} \sum _ {k = 0 } ^ { m } [a _ {k} (t) \Phi ^ {(k)} (t) + T _ {k} ( \Phi ^ {(k)} )] = f (t),$$

where $a _ {k} (t)$ are given Hölder-continuous functions, $t \in \partial S$, $T _ {m}$ is a completely-continuous operator, and the $T _ {k}$ $(k = 0 \dots m - 1 )$ are linear operators.

Suppose that the finite simply-connected domain $S$ is bounded by a closed Lyapunov contour (see Lyapunov surfaces and curves) $\partial S$ and that the $m$- th derivative, $m \geq 0$, of the function $\Phi (z)$( the latter is holomorphic in $S$), restricted to $\partial S$, is a function of class $C _ \alpha$, $0 < \alpha < 1$. Then, assuming that the point $z = 0$ is in $S$, one can express $\Phi (z)$ as follows:

$$\Phi (z) = \ \int\limits _ {\partial S } \frac{t \mu (t) dt }{t - z } + ic,\ \textrm{ if } m = 0;$$

$$\Phi (z) = \int\limits _ {\partial S } \mu (t) \left ( 1 - { \frac{z}{t} } \right ) ^ {m - 1 } \mathop{\rm ln} \left ( 1 - { \frac{z}{t} } \right ) dt +$$

$$+ \int\limits _ {\partial S } \mu (t) dt + ic,\ \textrm{ if } m \geq 1,$$

where $\mu (t)$ is a real function of class $C _ \alpha ( \partial S)$, $0 < \alpha < 1$, and $c$ is a real constant; $\mu (t)$ and $c$ are uniquely determined by $\Phi (z)$.

Substituting these expressions into the boundary condition (3), one obtains a singular integral equation, equivalent to problem (2), for the unknown function $\mu$:

$$K ( \mu ) = \ A (t _ {0} ) \mu ( t _ {0} ) + \frac{B (t _ {0} ) }{\pi i } \int\limits _ {\partial S } \frac{\mu (t) dt }{t - t _ {0} } + T ( \mu ) = f (t _ {0} ),$$

$t _ {0} \in \partial S$, where $T$ is a completely-continuous operator.

A necessary and sufficient condition for the boundary value problem (2) to be normally solvable is that

$$\tag{4 } a (t) = \ \sum _ {k = 0 } ^ { m } i ^ {k} a _ {m - k } (t) \neq 0,\ \ t \in \partial S,\ \ m \geq 0.$$

The Dirichlet problem ( $m = 0$) is always normally solvable. (Henceforth it is assumed throughout that condition (4) is satisfied.)

The index of the boundary value problem (2) is computed from the formula

$$\kappa = \ 2 (m + p),\ \ m \geq 1,$$

where $p$ is the increment of the function $(1/2 \pi ) \mathop{\rm arg} \overline{ {a (t) }}\;$ when the contour $\partial S$ is described once in the positive sense. The index of the Dirichlet problem is zero.

The homogeneous boundary value problem $R (u) = 0$ has a finite number $k \geq 0$ of linearly independent solutions, where $k \geq \kappa$; the inhomogeneous problem (2) has a solution if and only if

$$\int\limits _ {\partial S } f (t) \nu _ {j} (t) dS = 0,\ \ j = 1 \dots \overline{k}\; ,$$

where $\nu _ {j}$ is a complete system of linearly independent solutions of the associated homogeneous integral equation

$$k ^ \prime ( \nu ) = \ A (t _ {0} ) \nu (t _ {0} ) - \frac{1}{\pi i } \int\limits _ {\partial S } \frac{B (t) \nu (t) dt }{t - t _ {0} } + T ( \nu ) = 0.$$

The boundary value problem (2) has a solution, whatever the free term on the right, if and only if there exist exactly $\kappa$ linearly independent solutions of the associated homogeneous problem $R (u) = 0$. Consequently, if $\kappa > 0$ the homogeneous boundary value problem $R (u) = 0$ always has at least $\kappa$ linearly independent solutions; if $\kappa < 0$ the inhomogeneous problem (2) is not solvable for an arbitrary free term on the right, but at least $| \kappa |$ solvability conditions must be satisfied.

Necessary and sufficient conditions for the solvability of an inhomogeneous boundary value problem may be formulated in terms of the completeness of a certain system of functions. The kernel and system of functions may be constructed explicitly using the Riemann function of equation (1) and the coefficients of the boundary conditions. For example, let $\{ u _ {k} \}$ be some complete system of solutions in the basic domain $D _ {0}$ of equation (1), and let $S \subset D _ {0}$. Then a necessary and sufficient condition for problem (2) to be solvable for any free right-hand side is that the system of functions $\{ R (u _ {k} ) \}$ be complete on the boundary.

Very complete results have been obtained for the following boundary value problem (the generalized Riemann–Hilbert problem): Find a solution of the equation

$$\tag{5 } \partial _ {\overline{z}\; } w + A (z) w + B (z) \overline{w}\; = f (z),\ \ w = u + iv,$$

$$2 \partial _ {\overline{z}\; } = \partial _ {x} - i \partial _ {y} ,$$

which is continuous in $S + \partial S$ and satisfies the boundary condition

$$\tag{6 } \mathop{\rm Re} [ \overline{ {\lambda (z) }}\; , w (z)] \equiv \ \alpha u + \beta v = \gamma ,\ \ z \in \partial S,$$

where $\alpha , \beta , \gamma$ are real functions of class $C _ \alpha ( \partial S)$, $0 < \alpha < 1$, with $\alpha ^ {2} + \beta ^ {2} = 1$. The domain $S$ may be multiply connected. A problem of this type may be reduced to an equivalent singular integral equation; this yields a full qualitative analysis of the boundary value problem (6).

Suppose that the boundary $\partial S$ of $S$ is the union of a finite number of simple closed curves $\partial S _ {0} \dots \partial S _ {m}$ satisfying the Lyapunov conditions. Since the forms of the equation and the boundary condition are preserved under conformal mapping, it may be assumed without loss of generality that $\partial S _ {0}$ is the unit circle with centre at $z = 0$, the latter being a point of $S$, while $\partial S _ {1} \dots \partial S _ {m}$ are circles lying outside $\partial S _ {0}$.

The index of problem (6) is defined as the integer $n$ equal to the increment of $(1/2 \pi ) \mathop{\rm arg} [ \alpha ( \zeta ) + i \beta ( \zeta )]$ when the point $\zeta$ describes $\partial S$ once in the positive sense. The boundary condition can be reduced to the simpler form

$$\mathop{\rm Re} [z ^ {-n} e ^ {ic (z) } w (z)] = \gamma ,\ \ z \in \partial S,$$

where $c (z) = c _ {j}$ on $\partial S _ {j}$, with $c _ {0} = 0$, while $c _ {1} \dots c _ {m}$ are real constants, uniquely expressible in terms of $\alpha$ and $\beta$. The index of the adjoint problem

$$\tag{7 } \left . \begin{array}{ll} \partial _ {\overline{z}\; } w _ {*} - Aw _ {*} - B \overline{w}\; _ {*} = 0, & z \in S, \\ \mathop{\rm Re} \left \{ ( \alpha + i \beta ) \frac{d \overline{z}\; }{ds} w _ {*} (z) \right \} = 0, & z \in \partial S, \\ \end{array} \right \}$$

is calculated by the formula $n ^ \prime = - n + m - 1$.

Problem (6) has a solution if and only if

$$\int\limits _ {\partial S } ( \alpha + i \beta ) w _ {*} \gamma ds = 0,$$

where $w _ {*}$ is an arbitrary solution of the adjoint problem.

Let $e$ and $e ^ \prime$ be the numbers of linearly independent solutions of the homogeneous problems (6) and (7), respectively. Then

$$e - e ^ \prime = \ n - n ^ \prime = \ 2n + 1 - m.$$

If $n < 0$, the homogeneous problem (6) has no non-trivial solutions. If $n > m - 1$, it has exactly $e = 2n + 1 - m$ linearly independent solutions, while the inhomogeneous problem (6) is always solvable. If $n < 0$, the inhomogeneous problem (6) is solvable if and only if

$$\int\limits _ {\partial S } ( \alpha + i \beta ) w _ {* j } \gamma ds = 0,\ \ j = 1, 2 ,\dots ; \ \ e ^ \prime = m - 2n - 1,$$

where $w _ {* j }$ is a complete system of solutions of the homogeneous problem (7). If $m = 0$ and $n = 0$, then $e = 1$ and all solutions of the homogeneous problem problem (6) are given by

$$w (z) = \ ice ^ {\omega _ {0} (z) } ,$$

where $c$ is a real constant and $\omega _ {0}$ a continuous function on $S + \partial S$.

The above results completely characterize the problem in the simply-connected ( $m = 0$) and multiply-connected ( $n < 0$, $n > m - 1$) cases. The cases $0 \leq n \leq m - 1$ require special examination; they have also been worked out in considerable detail.

Boundary value problems of the type of the Poincaré problem have also been studied for equation (5).

For references see Differential equation, partial, complex-variable methods.