E. I. Zverovich and G. S. Litvinchuk

ON ONE-SIDED BOUNDARY-VALUE PROBLEMS IN THE THEORY OF ANALYTIC FUNCTIONS

(Presented by Academician V. I. Smirnov on February 23, 1962)

A characteristic feature of the Riemann boundary-value problem and of its various generalizations ((^{1,2})) is that on the boundary of the domain the limiting values of the sought analytic functions, taken respectively on the left and right sides of the boundary, are linearly related. A substantially new class of boundary-value problems for analytic functions is formed by the so-called one-sided problems, in which the unknown functions are sought from a linear relation between their limiting values prescribed on one side of the boundary. F. D. Gakhov and Yu. I. Cherskii ((^{3,4})) encountered problems of this kind (and partly investigated them) in connection with the study of special integral equations of convolution type, and I. Kh. Khairullin ((^5)) did so in considering one class of infinite systems of linear algebraic equations.

Let us take a Lyapunov contour (L) bounding a simply connected domain (D^+). By (D^-) we denote the complement of (D^+ + L) in the complete plane. Suppose that on (L) a function (\alpha_+(t)) is given which maps (L) one-to-one onto itself with preservation of the orientation on (L), (\alpha_+'(t) \ne 0) on (L). By the letter (t) we denote the complex coordinate of a variable point of (L). Let (\alpha_-(t)) be a function with analogous properties, but changing the orientation on (L). In the present article some results are announced which were obtained by the authors in studying the following one-sided boundary-value problems.

Find functions (\varphi(z)) and (\psi(z)), analytic in (D^{+*}), whose angular limiting values are square integrable on (L), representable by the Cauchy formula, and satisfying almost everywhere on (L) one of the boundary conditions

[
\varphi[\alpha_+(t)] = G(t)\psi(t) + g(t);
\tag{1}
]

[
\varphi[\alpha_-(t)] = G(t)\overline{\psi(t)} + g(t).
\tag{2}
]

For (\varphi(z) \equiv \psi(z)) we obtain the problems

[
\varphi[\alpha_+(t)] = G(t)\varphi(t) + g(t) \quad \text{on } L;
\tag{3}
]

[
\varphi[\alpha_-(t)] = G(t)\overline{\varphi(t)} + g(t) \quad \text{on } L.
\tag{4}
]

Find a piecewise analytic function ({\varphi^+(z), \varphi^-(z)}) from one of the boundary conditions

[
\varphi^+[\alpha_-(t)] = G(t)\varphi^-(t) + g(t);
\tag{5}
]

[
\varphi^+[\alpha_+(t)] = G(t)\overline{\varphi^-(t)} + g(t).
\tag{6}
]

The functions (G(t)) and (\alpha_\pm'(t)) are assumed to satisfy a Hölder condition on (L), and (g(t) \in L_2).

* Problems (1)—(4) may also be posed for the domain (D^-).

§ 1. Using the conditions that (\varphi(t)=G[\beta(t)]\psi[\beta(t)]+g[\beta(t)]), where (\alpha_+[\beta(t)]\equiv t), and that (\psi(t)) are the boundary values of functions analytic in the domain (D^+), from the boundary-value problem (1) we arrive at a Fredholm integral equation of the first kind with a quasi-regular kernel (((^{1}), p. 129))

[
\frac{1}{2\pi i}\int_L
\left[
\frac{G(t)}{\tau-t}
-
\frac{G(\tau)\alpha_+'(\tau)}{\alpha_+(\tau)-\alpha_+(t)}
\right]\psi(\tau)\,d\tau
=
{g[\beta(\xi)]}{\xi=\alpha+(t)} .
\tag{7}
]

By (h^{\pm}(t)) we mean the operators

[
h^{\pm}(t)=\pm\frac12 h(t)+\frac{1}{2\pi i}\int_L \frac{h(\tau)}{\tau-t}\,d\tau .
]

Further in this paragraph we shall assume that the functions (G(t)), (\alpha_\pm'(t)), as well as the angle formed by the tangent to (L) with any direction, satisfy a Hölder condition with exponent (\mu>1/2). Then the kernel of equation (7) belongs on (L) to the class (L_2), and for equation (7) the following assertion is valid, which is an extension of the well-known theorem of Picard ((^{6})) to equations with a complex nonsymmetric kernel. For the solvability of the integral equation

[
\int_L K(t,\tau)\varphi(\tau)\,d\tau=f(t),\quad t\in L,
\tag{7'}
]

it is necessary and sufficient that, in the mean,

[
\sum_k a_k\lambda_k\mu_k(s)
=
\int_0^S \overline{f[t(s_1)]}\,K(s,s_1)\,ds,
\tag{8}
]

where

[
a_k=\int_0^S f(t(s))\overline{\mu_k(s)}\,ds
=
\frac{1}{\lambda_k}\int_0^S f(t(s))\overline{\nu_k(s)}\,ds;
]

(\lambda_k) are the eigenvalues of the kernel (K[t(s),\tau(\sigma)]); (\mu_k(s)) and (\nu_k(s)) are orthonormal systems of eigenfunctions of the right and left iterated kernels, respectively,

[
K_R(s,\sigma)=\int_0^S \widetilde K(s,\xi)\overline{\widetilde K(\sigma,\xi)}\,d\xi
]

and

[
K_L(s,\sigma)=\int_0^S \overline{\widetilde K(\xi,s)}\widetilde K(\xi,\sigma)\,d\xi,\qquad
\widetilde K(s,\xi)\equiv K[t(s),\tau(\xi)].
]

The general solution of equation ((7')) is given by the formula

[
\varphi[t(s)]
=
\frac{1}{\tau'(s)}
\left[
\sum_k a_k\lambda_k\mu_k(s)
+
\sum_j c_j\widetilde\mu_j(s)
\right].
]

Here (c_j) are arbitrary constants, and the system ({\widetilde\mu_j(s)}) completes the system of eigenfunctions ({\mu_k(s)}) to a closed one. In the case of a closed kernel (K(t,\tau)), all (c_j=0), and equation ((7')) has a unique solution in (L_2).

If equation (7) is unsolvable, then problem (1) likewise has no solutions. Suppose that equation (7) is solvable, and let (\lambda(t)) be its general solution. Then on (L) the identity

[
{G[\beta(\xi)]\lambda[\beta(\xi)]+g[\beta(\xi)]}{\xi=\alpha+(t)}^{+}
\equiv
G(t)\lambda^+(t)+g(t)
]

holds.

Theorem 1. The one-sided boundary-value problem (1) is solvable if condition (8) is fulfilled, and the general solution of the problem has the form

[
\psi(z)=\lambda^+(z),\qquad
\varphi(z)=\frac{1}{2\pi i}\int_L
\frac{G[\beta(\tau)]\lambda[\beta(\tau)]+g[\beta(\tau)]}{\tau-z}\,d\tau,
\tag{9}
]

where (\lambda(t)) is the general solution of the integral equation (7).

Theorem 2. The homogeneous problem (1) has no nontrivial solutions in (D^+) if every solution (\lambda(t)) of the homogeneous equation (7) is the boundary value of a function analytic in (D^-) and vanishing at infinity, i.e. (\lambda^+(t)\equiv0) on (L). If (\lambda^+(t)\not\equiv0), then the homogeneous problem (1) is nontrivially solvable and its general solution is given by formulas (9), where (g(t)\equiv0), and (\lambda(t)) is the general solution of the homogeneous equation (7).

We note that if the general solution of the inhomogeneous equation (7) satisfies the condition (\lambda^+(t)\equiv0), then problem (1) is solvable, provided that (g^{-}[\beta(t)]=0) on (L).

Results analogous to Theorems 1 and 2 are also valid for problems (2), (5), (6). For example, the following is valid.

Theorem 3. The boundary-value problem (6) is solvable if condition (8) is satisfied. The general solution of the problem has the form

[
\varphi^{-}(z)=-\lambda^{-}(z),\qquad
\varphi^{+}(z)=\frac{1}{2\pi i}\int_L
\frac{G[\beta(\tau)]\,\overline{\lambda[\beta(\tau)]}+g[\beta(\tau)]}{\tau-z}\,d\tau,
]

where (\lambda(t)) is the general solution of the Fredholm integral equation of the first kind

[
\frac{1}{2\pi i}\int_L
\left[
\frac{\overline{G(t)}}{\tau-t}
-
\frac{\overline{G(\tau)}\,\alpha'(\tau)\tau^{2}(\sigma)}
{\alpha(\tau)-\alpha(t)}
\right]\varphi^{-}(\tau)\,d\tau
=
-\overline{{g[\beta(\xi)]}^{-}}{\xi=\alpha .}(t)
]

Let us consider boundary-value problem (3) under the condition that (\alpha_{-}[\alpha_{+}(t)]=t) on (L).

Theorem 4. If (G(t)G[\alpha_{+}(t)]\ne1), then the homogeneous problem (3) has only the trivial solution. If (G(t)G[\alpha_{+}(t)]=1), then linearly independent nontrivial solutions of problem (3) are given by the formulas (\varphi_j(z)=\lambda_j^{+}(z)), where (\lambda_j(t)) are linearly independent solutions of the homogeneous equation (7) satisfying the condition

[
{\lambda(t)-G[\alpha_{+}(t)]\lambda[\alpha_{+}(t)]}^{+}\equiv 0.
\tag{10}
]

Theorem 5. Let (G[\alpha_{+}(t)]g(t)+g[\alpha_{+}(t)]\ne0) on (L). Then the one-sided problem (3) is solvable and has a unique solution if, in addition, (G(t)G[\alpha_{+}(t)]\ne1) on (L) and the function

[
k(t)=\frac{G[\alpha_{+}(t)]g(t)+g[\alpha_{+}(t)]}
{1-G(t)G[\alpha_{+}(t)]}
]

is the boundary value of a function analytic in (D^{+}). If (G[\alpha_{+}(t)]g(t)+g[\alpha_{+}(t)]\ne0), but (G(t)G[\alpha_{+}(t)]=1), or (k^{-}(t)\ne0), then problem (3) is not solvable. If, however, (G[\alpha_{+}(t)]g(t)+g[\alpha_{+}(t)]=0), then a solution of problem (3), (\varphi(z)=\lambda^{+}(z)), exists when conditions (8), (10) and the condition ({g[\alpha_{+}(t)]}^{+}=0) on (L) are satisfied.

Analogous assertions hold for problem (4).

§ 2. We give several simple results clarifying the influence of the properties of (G(t)) and (\alpha_{\pm}(t)) on the solvability of one-sided problems and on the character of their general solutions. Consider the case where, in problems (1), (3), (5), the functions (\alpha(t)), and in problems (2), (4), (6) the functions (\overline{\alpha(t)}), are analytically continuable into (D^{+}) with a finite number of poles. Then, from the properties of the transformation (\alpha(t)) indicated above, it follows that the function (\alpha(z)) ((\overline{\alpha(z)})) is single-valued in (D^{+}) and maps (D^{+}) one-to-one and conformally onto the domain bounded by the curve (\xi=\alpha(t)) ((\xi=\overline{\alpha(t)})), (t\in L). Consequently, boundary-value problem (1) in the case under consideration is equivalent to a problem of the form:

Find two functions (\varphi_1(z)), (\psi_1(z)) analytic in (D^{+}) from the boundary condition

[
\varphi_1(t)=G(t)\psi_1(t)+g(t)\quad \text{on } L.
\tag{1′}
]

For problem (1′) the following obvious assertions are valid:

1) If (G(t)) is not the boundary value of a function meromorphic in (D^{+}), then the homogeneous problem (1′) has only the trivial solution, while the nonhomogeneous problem (1′) can have at most one solution.

2) If (G(t)) has the form (G(t)=g(t)/h(t)), where (g(z)) and (h(z)) are functions analytic in (D^{+}), representable by the Cauchy formula, and whose boundary values belong to (L_2) on (L), then, when condition (8) is satisfied, the general solution of problem (1′) is given by the formulas

[
\psi_1(z)=\Omega(z)h(z)+\psi_0(z),\qquad
\varphi_1(z)=\Omega(z)g(z)+\varphi_0(z),
]

where (\varphi_0(z)), (\psi_0(t)) is any particular solution of problem ((1')), and (\Omega(z)) is an arbitrary analytic function in (D^+) with boundary values from (L_2).

Analogous results are obtained for the boundary-value problems (2), (5), and (6).

Let us note the case when, in the boundary condition (6), (\alpha_+(t)\equiv t). If (t) is a boundary value of a function analytic in (D^-), then problem (6) has the same character as problem ((1')). If, moreover, (G(t)\equiv 1), (g(t)\equiv 0), and (L) is a circle or a lemniscate, then we obtain a problem that was considered by another method by A. I. Markushevich ({}^{7}). The result for the case when (D^+) is a disk, (\alpha_+(t)\equiv t), and (G(t)) and (g(t)) are boundary values of functions analytic in (D^+), was formulated in a paper of L. G. Mikhailov ({}^{8}).

Let us consider the boundary-value problems (3) and (4). Without loss of generality, we shall assume that (D^+) is the disk (|z|<1). Then, introducing as before the condition of analytic continuability of (\alpha_+(t)), we obtain that (\alpha_+(t)=e^{i\theta}(t-a)/(1-\bar a t)), (|a|<1). If the homogeneous problem corresponding to the solvable problem (3) has nontrivial solutions, then the general solution of problem (3) has the form

[
\varphi(z)=\varphi_0(z)+\varphi_1(z)\Omega(z),
]

where (\varphi_0(z)) is a particular solution of the nonhomogeneous problem (3); (\varphi_1(z)) is any nontrivial solution of the homogeneous problem (3); (\Omega(z)) is an arbitrary meromorphic function in (|z|<1) satisfying the conditions: a) the boundary values (\Omega^+(t)\in L_2) on (|t|=1); b) (\Omega(z)) is an automorphic function with respect to the group ({t,\alpha(t),\alpha[\alpha(t)],\ldots}); c) (\Omega(z)\varphi_1(z)) is an analytic function in (|z|<1).

Let us consider problem (4), assuming that (\alpha_-(t)) is a real function on (|t|=1) (i.e., (\alpha(t)) is real for (t=\pm1)), satisfying the condition (\alpha_-[\alpha_-(t)]=t). Assuming the analytic continuability of (\alpha_-(t)) inside (|z|<1), we obtain that (\alpha_-(t)=(a-t)/(1-\bar a t)), (\operatorname{Im}a=0), (|a|>1). Introduce the functions
[
u(z)=\frac12[\Psi(z)+\overline{\Psi}(z)],\qquad
v(z)=\frac{1}{2i}[\Psi(z)-\overline{\Psi}(z)],
]
where (\Psi(z)) is a meromorphic function in (|z|<1). The functions (u(z)), (v(z)) are real meromorphic functions.

Let problem (4) be solvable, and let (\varphi_0(z)) be its particular solution. Let (\varphi_1(z)) be some nontrivial solution of the homogeneous problem (4). Then the general solution of problem (4) has the form

[
\varphi(z)=\varphi_0(z)+\varphi_1(z)\left{u(z)+i\left[f_0(z)-f_0\left(\frac{1-az}{a-z}\right)\right]w(z)\right},
\tag{11}
]

where
[
\left[f_0(z)-f_0\left(\frac{1-az}{a-z}\right)\right]w(z)=v(z);
]
(u(z)) and (w(z)) are real meromorphic functions automorphic with respect to the group ({t,(1-at)/(a-t)}) and such that the second term in (11) is an analytic function in (|z|<1); (f_0(z)) is a fixed real meromorphic function, nonautomorphic with respect to ({t,(1-at)/(a-t)}); (u(t)), (w(t)), (f_0(t)\in L_2) on (|t|=1).

The question of the character of the general solution of the boundary-value problems (1)—(6) for the case of functions (\alpha(t)) (or (\overline{\alpha(t)})) that are not analytically continuable inside the domain (D^+) remains open.

Rostov-on-Don
State University

Received
6 II 1962

CITED LITERATURE

({}^{1}) F. D. Gakhov, Boundary-Value Problems, Moscow, 1958.
({}^{2}) D. A. Kveselava, Tr. Tbilisi Math. Inst., 16, 39 (1948).
({}^{3}) F. D. Gakhov, Yu. I. Cherskii, Izv. AN SSSR, ser. matem., 20, No. 1, 33 (1956).
({}^{4}) Yu. I. Cherskii, Izv. AN SSSR, ser. matem., 22, No. 3, 361 (1958).
({}^{5}) I. Kh. Khairullin, Tr. All-Union Conf. on Differential Equations, Yerevan, 1960.
({}^{6}) F. Tricomi, Integral Equations, Moscow, 1960.
({}^{7}) A. I. Markushevich, Uchen. zap. Moscow Univ., no. 100, 20 (1946).
({}^{8}) L. G. Mikhailov, DAN, 139, No. 2 (1961).