Boundary value problems for analytic functions.

*(English)*Zbl 0818.30027
Series in Pure Mathematics. 16. Singapore: World Scientific. xiii, 466 p. (1993).

From the two basic boundary value problems for analytic functions in this monograph the central topic is the Riemann or linear conjugacy boundary value problem. It demands to find a sectionally analytic function with prescribed jump-behaviour on a given system of (closed or open) curves. The other problem is the Riemann-Hilbert boundary value problem, a generalization of the Dirichlet problem, where for a given functional relation of the real and the imaginary parts on the boundary of the domain under consideration an analytic function is looked for. In the simplest case the functional relation is just a linear combination and hence the problem is a linear one. This problem is related to the Riemann problem when one is involved with the unit disc and in this case this last problem is considered too. For good reasons the author sticks to the case of linear problems. Nonlinear (Riemann-Hilbert) problems for analytic functions are treated in E. Wegert [Nonlinear boundary value problems for holomorphic functions and singular integral equations (1992; Zbl 0745.30040)]. From the two classical books on the subject N. I. Muskhelishvili [Singular integral equations. Fizmatgiz, Moscow, 1946 (Russian); Noordhoff, Groningen 1953 (English); Akademie-Verlag, Berlin, 1965 (German)] and F. D. Gakhov [Boundary value problems. Fizmatgiz, Moscow, 1963 (Russian); Pergamon, Oxford, 1966 (English)] the present one differs by incorporating many results of the author on subjects not touched or only just mentioned there. These include periodic, doubly periodic, doubly quasi-periodic, automorphic functions. Moreover, besides systems of closed curves open curves are considered as are curves with even multiple nodes. Closely related to Riemann boundary value problems are singular integral equations. In a natural way they are included here for the different situations of curve systems. Many exercises are completing the text but there are no hints or solutions given. On the other hand occasionally there are worked-out examples.

The introductory Chapter I (Cauchy-type integrals) familiarizes with the basic concepts and results: sectionally analytic functions, boundary values of Cauchy-type integrals, Plemelj formulas, Privalov theorem, PoincarĂ©-Bertrand formula, Cauchy principle value integral, singular integrals. The classical situation of boundary value problems is explained in Chapter II (Fundamental boundary value problems for closed contours). Besides the classical Riemann, Riemann-Hilbert and compound boundary value problems, mainly the Riemann problem is studied for periodic, doubly periodic and (as well additive as multiplicative) doubly quasi-periodic functions.

While in the classical formulation of the jump condition \(\phi^ += G\phi^ -+ g\) on the curve system \(\Gamma\) the coefficient \(G\) is assumed not to vanish the author considers also the non-normal type of problems with \(G\) having zeroes. The Hilbert problem is just studied in the unit disc and then transformed to a related Riemann problem as was done already in the Gakhov and Muskhelishvili books. The restriction to this particular domain is allowed as far as simply connected domains are concerned because of the necessary smoothness assumptions on the boundary of the domain. Multiply connected domains need other treatments. Besides the references on p. 106 here A. Dzhuraev [Methods of singular integral equations. Nauka, Moscow, 1987 (Russian); Longman, Harlow, 1992 (English)] should be mentioned where Bergman kernel functions are effectfully used. Another detailed discussion of the Riemann-Hilbert problem for analytic functions in multiply connected domains can be found in G. C. Wen and the referee [Boundary value problems for elliptic equations and systems (1990; Zbl 0711.35038)]. For the Riemann problem particular attention is paid to the special case where \(\Gamma\) is the real line. Natural generalization of this situation are mentioned. The results for periodic functions are related to those for non-periodic ones by letting the period tend to infinity. While the singularity of the Cauchy-type integral is just the critical one the author has considered singular integrals of higher fractional order. His results for the case of higher entire order are included here (section 7).

“Singular integral equations in case of closed contours” (Chapter III) are closely related to the classical Riemann problem. Here this relation together with the knowledge about the Riemann problem is used to solve these integral equations. The Noether theorem is developed. By the way an Appendix (Some results on Fredholm integral equations) collects necessary background in Fredholm theory for integral equations.

Moreover, singular integral equations with periodic kernels such as the Hilbert kernel and doubly periodic kernels are treated. In section 5 a direct method from the author to solve \[ a(t) \phi(t)+ {1\over \pi i} \int_ \Gamma {K(t, \tau)\over \tau- t} \phi(\tau) d\tau= f(t),\quad t\in \Gamma, \] for a finite system \(\Gamma\) of non-intersecting closed contours is developed. It does not use the connection to the Riemann problem, just the Plemelj formula and gives the solution as well as the solvability conditions explicitly. Here \(a\) and \(K\) are particularly assumed to be analytic in the closure of the domain bounded by \(\Gamma\). Different cases occur depending on the location of the zeroes of \(a(z)+ K(z, z)\).

The following two chapters enlighten the situation in the case where open curves are occurring. At first in Chapter IV (Boundary value problems in general case) the behaviour of Cauchy integrals at the end-points of an open curve is discussed in a similar way as they are in Muskhelishvili’s book. But here also nodes are considered where several curves originate or terminate. The Riemann problem for systems of these curves in particular include the case of a simple closed arcwise smooth curve with piecewise continuous coefficients in the jump condition. The Riemann- Hilbert problem is studied under the same conditions for the unit disc and the upper half plane. Also the compound problem, the periodic, doubly periodic, doubly quasi-periodic Riemann problems are explained.

Similarly in Chapter V (Singular integral equations in general case) the problems studied in Chapter III for systems of closed curves are now considered for open arcs. As a particular case the problem of inverting Cauchy principle value integrals is handled. In cases where the problem is unsolvable a modified inversion problem is introduced, see section 3.4. While in general the singularities as well of the coefficients of the integral equations as of their solutions are restricted to have order less than one in the last section 4 some results of the author allowing singularities of order one are presented.

In Chapter VI (Boundary value problems for systems of functions and systems of singular integral equations) the Riemann, the Riemann-Hilbert and the compound problem and systems of singular integral equations including the inversion problem are treated.

The final Chapter VII (Miscellaneous problems) contains the Riemann boundary value problem and singular integral equations for automorphic functions, boundary value problems (Haseman, Carleman) and singular integral equations with shift including the conformal glue theorem, and integral equations of convolution type. At last some simple methods and results on approximating Cauchy principal value integrals are included.

The book is self-contained and clearly written. The referee has realized only a few missprints. It can well be used for advanced courses in complex analysis and for seminars, and is readable by graduate students themselves. The list of references is far from being complete for the subject which is under intensive research since about six decades. Many results of the author are presented here for the first time in English language.

The introductory Chapter I (Cauchy-type integrals) familiarizes with the basic concepts and results: sectionally analytic functions, boundary values of Cauchy-type integrals, Plemelj formulas, Privalov theorem, PoincarĂ©-Bertrand formula, Cauchy principle value integral, singular integrals. The classical situation of boundary value problems is explained in Chapter II (Fundamental boundary value problems for closed contours). Besides the classical Riemann, Riemann-Hilbert and compound boundary value problems, mainly the Riemann problem is studied for periodic, doubly periodic and (as well additive as multiplicative) doubly quasi-periodic functions.

While in the classical formulation of the jump condition \(\phi^ += G\phi^ -+ g\) on the curve system \(\Gamma\) the coefficient \(G\) is assumed not to vanish the author considers also the non-normal type of problems with \(G\) having zeroes. The Hilbert problem is just studied in the unit disc and then transformed to a related Riemann problem as was done already in the Gakhov and Muskhelishvili books. The restriction to this particular domain is allowed as far as simply connected domains are concerned because of the necessary smoothness assumptions on the boundary of the domain. Multiply connected domains need other treatments. Besides the references on p. 106 here A. Dzhuraev [Methods of singular integral equations. Nauka, Moscow, 1987 (Russian); Longman, Harlow, 1992 (English)] should be mentioned where Bergman kernel functions are effectfully used. Another detailed discussion of the Riemann-Hilbert problem for analytic functions in multiply connected domains can be found in G. C. Wen and the referee [Boundary value problems for elliptic equations and systems (1990; Zbl 0711.35038)]. For the Riemann problem particular attention is paid to the special case where \(\Gamma\) is the real line. Natural generalization of this situation are mentioned. The results for periodic functions are related to those for non-periodic ones by letting the period tend to infinity. While the singularity of the Cauchy-type integral is just the critical one the author has considered singular integrals of higher fractional order. His results for the case of higher entire order are included here (section 7).

“Singular integral equations in case of closed contours” (Chapter III) are closely related to the classical Riemann problem. Here this relation together with the knowledge about the Riemann problem is used to solve these integral equations. The Noether theorem is developed. By the way an Appendix (Some results on Fredholm integral equations) collects necessary background in Fredholm theory for integral equations.

Moreover, singular integral equations with periodic kernels such as the Hilbert kernel and doubly periodic kernels are treated. In section 5 a direct method from the author to solve \[ a(t) \phi(t)+ {1\over \pi i} \int_ \Gamma {K(t, \tau)\over \tau- t} \phi(\tau) d\tau= f(t),\quad t\in \Gamma, \] for a finite system \(\Gamma\) of non-intersecting closed contours is developed. It does not use the connection to the Riemann problem, just the Plemelj formula and gives the solution as well as the solvability conditions explicitly. Here \(a\) and \(K\) are particularly assumed to be analytic in the closure of the domain bounded by \(\Gamma\). Different cases occur depending on the location of the zeroes of \(a(z)+ K(z, z)\).

The following two chapters enlighten the situation in the case where open curves are occurring. At first in Chapter IV (Boundary value problems in general case) the behaviour of Cauchy integrals at the end-points of an open curve is discussed in a similar way as they are in Muskhelishvili’s book. But here also nodes are considered where several curves originate or terminate. The Riemann problem for systems of these curves in particular include the case of a simple closed arcwise smooth curve with piecewise continuous coefficients in the jump condition. The Riemann- Hilbert problem is studied under the same conditions for the unit disc and the upper half plane. Also the compound problem, the periodic, doubly periodic, doubly quasi-periodic Riemann problems are explained.

Similarly in Chapter V (Singular integral equations in general case) the problems studied in Chapter III for systems of closed curves are now considered for open arcs. As a particular case the problem of inverting Cauchy principle value integrals is handled. In cases where the problem is unsolvable a modified inversion problem is introduced, see section 3.4. While in general the singularities as well of the coefficients of the integral equations as of their solutions are restricted to have order less than one in the last section 4 some results of the author allowing singularities of order one are presented.

In Chapter VI (Boundary value problems for systems of functions and systems of singular integral equations) the Riemann, the Riemann-Hilbert and the compound problem and systems of singular integral equations including the inversion problem are treated.

The final Chapter VII (Miscellaneous problems) contains the Riemann boundary value problem and singular integral equations for automorphic functions, boundary value problems (Haseman, Carleman) and singular integral equations with shift including the conformal glue theorem, and integral equations of convolution type. At last some simple methods and results on approximating Cauchy principal value integrals are included.

The book is self-contained and clearly written. The referee has realized only a few missprints. It can well be used for advanced courses in complex analysis and for seminars, and is readable by graduate students themselves. The list of references is far from being complete for the subject which is under intensive research since about six decades. Many results of the author are presented here for the first time in English language.

Reviewer: H.Begehr (Berlin)

##### MSC:

30E25 | Boundary value problems in the complex plane |

45E05 | Integral equations with kernels of Cauchy type |

30-02 | Research exposition (monographs, survey articles) pertaining to functions of a complex variable |