Abstract Generated abstract
The paper studies a Riemann type boundary value problem for two piecewise analytic functions on domains separated by a smooth closed contour, with boundary conditions involving derivatives and Fredholm integral terms. It introduces auxiliary integral representations for analytic functions under index conditions, then applies them to the highest derivatives of the unknown functions. These representations express the solutions through a Hölder continuous contour density and a finite set of constants, with separate forms according to the relation between the derivative order and the index. Substitution into the boundary condition and use of the Sokhotski formulas reduce the original problem directly to a Fredholm integral equation, avoiding the more cumbersome regularization required in earlier approaches based on singular integral equations.
Full Text
V. S. Rogozhin
A New Integral Representation of a Piecewise-Analytic Function and Its Application
(Presented by Academician V. I. Smirnov on 28 VI 1960)
Consider the following boundary-value problem of the Riemann-problem type. A simple smooth closed contour \(L\) is given, dividing the plane into two domains \(D^+\), \(D^-\). Determine two functions \(\Phi^+(z)\), \(\Phi^-(z)\) \((\Phi^-(\infty)=0)\), analytic respectively in the domains \(D^+\), \(D^-\), and satisfying on the contour the boundary condition
\[ \sum_{k=0}^{n}\left[ a_k(t)\frac{d^k\Phi^+(t)}{dt^k} + \int_L A_k(t,\tau)\frac{d^k\Phi^+(\tau)}{d\tau^k}\,d\tau \right] - \sum_{k=0}^{p}\left[ b_k(t)\frac{d^k\Phi^-(t)}{dt^k} + \int_L B_k(t,\tau)\frac{d^k\Phi^-(\tau)}{d\tau^k}\,d\tau \right] =f(t), \tag{1} \]
where \(a_k(t)\), \(b_k(t)\), \(f(t)\) are given continuous functions, with \(a_n(t)\), \(b_p(t)\) satisfying a Hölder condition (condition H) and not vanishing; \(A_k(t,\tau)\), \(B_k(t,\tau)\) are Fredholm kernels.
This problem was first considered by L. G. Magnaradze in [1]. A simpler investigation was later carried out by Yu. M. Krikunov [2, 3], who, with the aid of an integral representation of the required functions, reduced the problem to a singular integral equation with Cauchy kernel.
In the present paper a new integral representation is indicated for the required functions in the boundary-value problem (1), which makes it possible to reduce it directly to a Fredholm-type integral equation. This introduces considerable simplifications into the solution, since the equation obtained by Yu. M. Krikunov requires a regularization involving cumbersome calculations.
We begin with the formulation of auxiliary propositions.
Lemma 1. Let \(c(t)\) satisfy condition H on \(L\), be different from zero, and let \(\operatorname{ind} c(t)=\varkappa \ge 0\); let \(f^+(z)\) be analytic in \(D^+\) and continuous in the closed domain \(\overline{D^+}\); let \(f^-(z)\) be analytic in \(D^-\) and have a zero of order \(p+1\) at infinity. Under these conditions the representation holds
\[ f^+(z)=\frac{1}{2\pi i}\int_L \frac{\nu(\tau)}{c(\tau)}\,\frac{d\tau}{\tau-z}, \qquad z\in D^+; \tag{2a} \]
\[ f^-(z)=-\frac{1}{z^p\cdot 2\pi i}\int_L \nu(\tau)\,\frac{d\tau}{\tau-z}, \qquad z\in D^-, \tag{2b} \]
where \(\nu(\tau)\) is a complex function of points of the contour \(L\), satisfying condition H, determined from the given functions \(f^+(z)\) and \(f^-(z)\) up to \(\varkappa\) arbitrary constants.
Lemma 2. Suppose that, under the conditions of the preceding lemma, \(\operatorname{ind} c(t)=\varkappa<0\). Then the representation holds
\[ f^{+}(z)=\frac{1}{2\pi i}\int_L \frac{v(\tau)}{c(\tau)}\frac{d\tau}{\tau-z} +p_0+p_1z+\ldots+p_{-\varkappa-1}z^{-\varkappa-1}, \tag{3a} \]
\[ f^{-}(z)=-\frac{1}{z^p\cdot 2\pi i}\int_L v(\tau)\frac{d\tau}{\tau-z}, \tag{3b} \]
where \(p_k\) are constants determined by the functions \(f^{+}(z)\) and \(f^{-}(z)\).
Applying Lemmas 1 and 2 to the functions \(\Phi^{+(n)}(z)\) and \(\Phi^{-(p)}(z)\), one can prove the following theorems.
Theorem 1. Suppose \(c(t)\) satisfies condition H and \(\operatorname{ind} c(t)=\varkappa\). Then, for \(n>\varkappa\), the integral representation holds
\[ \Phi^{+}(z)= \frac{(-1)^n}{(n-1)!}\frac{1}{2\pi i} \int_L \frac{v(\tau)}{c(\tau)}(\tau-z)^{n-1} \ln\left(1-\frac{z}{\tau}\right)d\tau +\sum_{k=1}^{n-\varkappa}\frac{c_k}{(k-1)!}z^{k-1}, \tag{4a} \]
\[ \Phi^{-}(z)= \frac{(-1)^p}{(p-1)!}\frac{1}{2\pi i} \int_L \frac{v(\tau)}{\tau^p}(\tau-z)^{p-1} \ln\left(1-\frac{\tau}{z}\right)d\tau +\sum_{k=1}^{p-1} d_{k-1}z^{k-1}, \tag{4b} \]
where \(v(\tau)\) is a function satisfying condition H; \(c_k\) are complex constants, with \(v(\tau)\) and \(c_k\) determined by \(\Phi^{+}(z)\) and \(\Phi^{-}(z)\) uniquely, while the constants \(d_k\) have the form
\[ d_0=\frac{(-1)^p}{(p-1)!}\frac{\beta_0}{2\pi i} \int_L v(\tau)\tau^{-1}\,d\tau+\Phi^{-}(\infty), \]
\[ d_k=\frac{(-1)^p}{(p-1)!}\frac{\beta_k}{2\pi i} \int_L v(\tau)\tau^{-(k+1)}\,d\tau,\qquad k=1,2,\ldots,p-2; \tag{5} \]
here
\[ \beta_k=\sum_{q=k+1}^{p-1}\frac{(-1)^q}{q-k}c_{p-1}^{q}. \]
Theorem 2. If, under the conditions of the preceding theorem, \(n\leqslant\varkappa\), then the integral representation holds
\[ \Phi^{+}(z)= \frac{(-1)^n}{(n-1)!}\frac{1}{2\pi i} \int_L \frac{v(\tau)}{c(\tau)}(\tau-z)^{n-1} \ln\left(1-\frac{z}{\tau}\right)d\tau+c_1, \tag{6a} \]
\[ \Phi^{-}(z)= \frac{(-1)^p}{(p-1)!}\frac{1}{2\pi i} \int_L \frac{v(\tau)}{\tau^p}(\tau-z)^{p-1} \ln\left(1-\frac{\tau}{z}\right)d\tau +\sum_{k=1}^{p-1} d_{k-1}z^{k-1}, \tag{6b} \]
where \(c_1=\Phi^{+}(0)\); \(d_k\) are determined by formulas (5).
We now proceed to reducing the boundary-value problem (1) to an integral equation. For this purpose we set \(c(t)=a_n(t)/b_p(t)t^p\) and use the integral representations (4a), (4b) or (6a), (6b), depending on whether \(n>\varkappa\) or \(n\leqslant\varkappa\). By means of differentiation we can determine all derivatives of the sought functions \(\Phi^{+}(z)\) and \(\Phi^{-}(z)\) entering into the boundary condition (1). The derivatives of order \(n\) of \(\Phi^{+}(z)\) and of order \(p\) of \(\Phi^{-}(z)\) will then be represented by Cauchy-type integrals. Passing to the limit as \(z\to t\) \((t\in L)\) and using the Sokhotski formulas
for the limiting values of the Cauchy-type integral, we obtain the integral equation
\[ \frac{a_n(t)+t^{-p}b_p(t)}{2}\,\nu(t) +\frac{1}{2\pi i}\int_L a_n(t) \left[ \frac{b_p(\tau)}{a_n(\tau)\tau^p} - \frac{b_p(t)}{a_n(t)t^p} \right] \frac{\nu(\tau)}{\tau-t}\,d\tau + \tag{7} \]
\[ +\int_L K(t,\tau)\nu(\tau)\,d\tau + \sum_{k=1}^{n-\varkappa} h_k g_k(t) = f(t), \]
where \(h_k\) are arbitrary constants; \(g_k(t)\) are known functions; \(K(t,\tau)\) is a Fredholm kernel. For \(n \leqslant \varkappa\), only the first term should be retained from the sum. Since \(a_n(t)\) and \(b_p(t)\) satisfy condition H, the kernel entering the first integral term will also be a Fredholm kernel. Thus problem (1) has been reduced to a Fredholm integral equation.
Rostov-on-Don State
University
Received
24 VI 1960
CITED LITERATURE
¹ L. G. Magnaradze, Reports of the Academy of Sciences of the Georgian SSR, 4, No. 2 (1943). ² Yu. M. Krikunov, Scientific Notes of Kazan State University named after V. I. Ulyanov-Lenin, 112, book 10 (1952). ³ Yu. M. Krikunov, Scientific Notes of Kazan State University named after V. I. Ulyanov-Lenin, 116, book 4 (1956).