Abstract Generated abstract
This note studies the Carleman boundary value problem with orientation reversing shift on the boundary of a finite multiply connected domain. It constructs a special conformal welding function that maps the problem to a Riemann boundary value problem on a contour consisting of open arcs, using integral representations and solvability properties for auxiliary shifted problems. The reduction yields a complete solution of the homogeneous and nonhomogeneous Carleman problems in the multiply connected case, including formulas for the index and for the number of solutions in terms of the index of the coefficient and the fixed points where the coefficient equals minus one.
Full Text
Reports of the Academy of Sciences of the USSR
1970. Volume 190, No. 1
UDC 517.544
MATHEMATICS
V. A. CHERNETSKII
ON THE CONFORMAL EQUIVALENCE OF THE CARLEMAN BOUNDARY-VALUE PROBLEM TO THE RIEMANN BOUNDARY-VALUE PROBLEM ON AN OPEN CONTOUR
(Presented by Academician N. I. Muskhelishvili on 26 V 1969)
1. Let \(D\) be a finite \((m+1)\)-connected domain bounded by a Lyapunov contour \(L\), consisting of a curve \(L_0\) enclosing the curves \(L_1,\ldots,L_m\). We assume that the origin belongs to the domain \(D\).
By \(\omega(z,L_k)\), \(k=0,\ldots,m\), we denote the harmonic measures of the boundary curves \(L_k\) with respect to the domain \(D\) (\(^{1}\)), and by \(D_0^{-},\ldots,D_m^{-}\) the complement of the domain \(D+L\) to the complete plane.
Let
\[ \alpha(t)=\sum_{k=0}^{m}\omega(t,L_k)\alpha_k(t), \]
where \(\alpha_k(t)\) is an orientation-changing homeomorphism of the contour \(L_k\) onto itself, satisfying the following two conditions: 1) \(\alpha_k[\alpha_k(t)]\equiv t\) (Carleman condition); 2) the function \(\alpha'_k(t)\in H(L)^*\), \(\alpha'_k(t)\ne 0,\ t\in L\). Let \(G(t)\in H(L)\), \(g(t)\in H(L)\), \(G(t)\ne 0,\ t\in L\).
Consider the following problem.
Find a function \(\Phi(z)\), single-valued and analytic in \(D\), \(H\)-continuous in \(\bar D\), satisfying one of the following conditions on \(L\):
\[ \Phi^+[\alpha(t)]=G(t)\Phi^+(t), \tag{1} \]
\[ \Phi^+[\alpha(t)]=G(t)\Phi^+(t)+g(t). \tag{2} \]
Only the case in which the conditions (\(^{2,3}\)) are fulfilled is of interest:
\[ G[\alpha(t)]G(t)=1,\qquad G[\alpha(t)]g(t)+g[\alpha(t)]=0. \tag{3} \]
The boundary-value problem (1) was first posed by T. Carleman (\(^{4}\)). A complete solution of problems (1), (2) for a bounded simply connected domain was given by D. A. Kveselava (\(^{2}\)). The case of a simply connected unbounded exterior domain was investigated by T. S. Litvinchuk (\(^{3}\)). Problem (1) for a multiply connected domain was first considered by E. I. Zverovich (\(^{5,6}\)), who computed the number of solutions of problem (1) for the cases \(\chi>0\) and \(\chi<-2(m-1)\), where \(\chi=\operatorname{Ind}G(t)|_L\). However, the method applied in (\(^{5}\)) did not provide an algorithm for finding the solutions of problem (1) and proved insufficient for finding the number of solutions of the problem in the case \(-2(m-1)\le \chi \le 0\).
In the present note we establish the fact of the conformal equivalence of the Carleman problem to the Riemann problem on a contour consisting of \(m+1\) open arcs. This is achieved by constructing a special gluing function. Up to now, what was known was the conformal equivalence, established by G. F. Mandzhavidze and B. V. Khvedelidze (\(^{7}\)) and by I. B. Simonenko (\(^{8}\)), of the Haseman problem to the Riemann problem (on a closed contour); a problem of the type of the Haseman problem was included by E. I. Zverovich (\(^{5}\)) in the general
* That is, satisfying the Hölder condition (\(H\)-continuous) on \(L\).
theory of the Riemann problem on abstract Riemann surfaces; the equivalence of a problem of Carleman type to the Hilbert problem was proved by G. S. Litvinchuk and E. G. Khasabov [9]. The Carleman problem (2) remained the only one of the principal boundary-value problems with shift*, for which such a fact had not hitherto been known.
The connection established between the Carleman problem and the Riemann problem on an open contour makes it possible to give a complete solution of the Carleman problem for a multiply connected domain.
2. Lemma 1. A function \(\Phi(z)\), single-valued and analytic in \(D\), \(H\)-continuous in \(\overline{D}\), can be represented in the form:
\[ \Phi(z)=\frac{1}{\pi i}\int_L \frac{\varphi(\tau)}{\tau-z}\,d\tau+ \int_L \omega(\tau,L_m)\varphi(\tau)[1+|\alpha'(\tau)|]\,d\sigma, \tag{4} \]
where the density \(\varphi(t)\) satisfies the condition
\[ \varphi(t)+\varphi[\alpha(t)]=\sum_{k=1}^{m}c_k\omega(t,L_k); \tag{5} \]
here \(c_k,\ k=1,\ldots,m-1\), are arbitrary constants, while \(c_m\) is determined by \(\Phi(z)\) uniquely; moreover the density \(\varphi(t)\) is determined up to a term of the form \(\sum_{k=1}^{m-1}\mu_k\omega(t,L_k)\), where \(\mu_k\) are arbitrary constants.
Lemma 2. The general solution of the problem
\[ \Phi^+[\alpha(t)]=\Phi^+(t) \tag{6} \]
is an arbitrary constant.
Using representation (4), we reduce problem (6) to the integral equation
\[ \varphi(t)+\frac{1}{2\pi i}\int_L\left[\frac{1}{\tau-t} -\frac{\alpha'(\tau)}{\alpha(\tau)-\alpha(t)}\right]\varphi(\tau)\,d\tau=0. \tag{7} \]
It is easy to show that every solution of equation (7) satisfies condition (5); the functions \(\omega(t,L_1),\ldots,\omega(t,L_m)\) are eigenfunctions of equation (7), and to the functions \(\omega(t,L_k)\), \(k=1,\ldots,m-1\), there corresponds the trivial solution of problem (7), while the function \(\omega(t,L_m)\) corresponds to a complex constant.
Consider the equation
\[ \psi(t)-\frac{1}{2\pi i}\int_L\left[\frac{1}{\tau-t} -\frac{\alpha'(t)}{\alpha(\tau)-\alpha(t)}\right]\psi(\tau)\,d\tau=0, \tag{8} \]
conjugate to equation (7). For a fundamental system of solutions \(\psi_1(t),\ldots,\psi_m(t)\) of equation (8), the conditions
\[ \psi_k(t)+\psi_k[\alpha(t)]\alpha'(t)=0,\qquad k=1,\ldots,m. \tag{9} \]
are fulfilled. Hence follows the unconditional solvability of the problem**
\[ \Phi^+[\alpha(t)]=\Phi^+(t)+g(t)\qquad (g(t)+g[\alpha(t)]=0). \tag{10} \]
3. The following theorem on conformal welding is based on the last assertion, under the assumption that \(g(t)=1/t-1/\alpha(t)\).
Theorem 1. In the domain \(D\) there exists a univalent function \(\omega^+(z)\), analytic in \(D\), except for the point \(z=0\), where \(\omega^+(z)\) has a simple
\[ \text{* Here and above we use the terminology of the survey article [6].} \]
\[ \text{** It is easy to see that the solvability condition for problem (10) } \int g(t)\psi_k(t)\,dt=0,\ k=1,\ldots,m,\ \text{is satisfied.} \]
strip satisfying on \(L\) the gluing condition
\[ \omega^{+}[\alpha(t)]=\omega^{+}(t). \tag{11} \]
Moreover, the gluing line \(\Gamma\) consists of \(m+1\) simple open Lyapunov contours \(\Gamma_0,\ldots,\Gamma_m\), given by the equation
\[ w=\omega^{+}(t),\qquad t\in L. \tag{12} \]
The function \(\omega^{+}(z)\) has the form \(\omega^{+}(z)=1/z+\Phi^{+}(z)\). It is not difficult to show that \(\omega^{+}(z)\) is univalent.
- Let us introduce a new unknown function
\[ \Psi(\omega)=\Phi[z(\omega)], \tag{13} \]
where \(z(\omega)\) is the function inverse to \(\omega^{+}(z)\). Then problem (2) reduces to two Riemann problems on the open contour \(\Gamma\), consisting of \(m+1\) arcs. Both problems, by virtue of conditions (3), turn out to be identical. Thus, the Riemann problem corresponding to the Carleman problem has the form
\[ \Psi^{+}(w)=G[z^{-}(w)]\Psi^{-}(w)+g[z^{-}(w)],\qquad w\in\Gamma. \tag{14} \]
The index of problem (14) is represented by formula (10)
\[ \nu=-(\chi+m_-)/2, \tag{15} \]
where \(m_-\) is the number of fixed points of the shift \(\alpha(t)\) at which \(G(t)=-1\), \(\chi=\operatorname{Ind} G(t)|_L\).
Relying on the theory of the Riemann problem on an open contour \((^{10})\), we obtain the result for the Carleman problem in the case of an \((m+1)\)-connected domain.
Theorem 2. The number of solutions of the homogeneous Carleman problem (1) is \(l=1-(\chi+m_-)/2\) for \(\chi\le -m_-\), where \(\chi=\operatorname{Ind} G(t)|_L\), and \(m_-\) is the number of fixed points of the shift \(\alpha(t)\) at which \(G(t)=-1\).
For \(\chi>-m_-\), the homogeneous problem (1) has no nontrivial solutions.
For \(\chi>-m_-\), the nonhomogeneous problem (2) is solvable and has a unique solution if \(p=(\chi+m_-)/2-1\) conditions are fulfilled.
In the case \(m=0\), the known theorem of D. A. Kveselava is obtained \((^{2,6})\).
I thank É. I. Zverovich, G. S. Litvinchuk, and A. P. Nechaev for valuable discussions and assistance in the work.
Odessa State University
named after I. I. Mechnikov
Received
5 V 1969
CITED LITERATURE
\(^{1}\) R. Nevanlinna, Single-Valued Analytic Functions, Moscow, 1941.
\(^{2}\) D. A. Kveselava, Tr. Tbilissk. matem. inst., 16, 39 (1948).
\(^{3}\) G. S. Litvinchuk, Izv. vyssh. uchebn. zaved., Matem., No. 6 (25) (1961).
\(^{4}\) T. Carleman, Verhandl. internat. mathem. congr., Zurich, 1 B, 1932.
\(^{5}\) É. I. Zverovich, Sibirsk. matem. zhurn., 7, No. 4 (1966).
\(^{6}\) É. I. Zverovich, G. S. Litvinchuk, UMN, 23, 3 (141) (1968).
\(^{7}\) T. F. Madzhavidze, B. V. Khvedelidze, DAN, 123, No. 5 (1958).
\(^{8}\) F. D. Gakhov, Boundary Value Problems, Moscow, 1963.
\(^{9}\) G. S. Litvinchuk, É. G. Khasanov, Sibirsk. matem. zhurn., 5, No. 3 (1964).
\(^{10}\) N. I. Muskhelishvili, Singular Integral Equations, Moscow, 1962.