Heinrich Begehr - Böcker

Recent results on partial differential equations and complex analytic methods on singular integral equations are presented here with related subjects. Many of the contributions are survey articles. Topics ranging from elliptic, parabolic, hyperbolic, and mixed-type equations and systems to hyper-complex and quatern ionic analysis, M-analytic, bianalytic, polyharmonic and functions of several complex variables are covered. Applications to mathematical physics are also included. This book is aimed at specialists in partial differential equations and related topics. Those with an interest in real and complex methods and in applications to mathematical physics should also find this book useful.

Provides reviews and research on topics ranging from complex elliptic first-order systems and second-order systems with regular or singular coefficients to overdetermined systems in several complex variables and partial differential equations in Clifford analysis. Different boundary value problems are studied. Applications to crack problems in elasticity theory to cusped bars, plates, and shells are given. Wavelet transformations are constructed in Banach spaces and used to identify complex analysis from the viewpoint of geometry. Fixed-point problems, even in abstract Banach spaces, are investigated with respect to an optimal domain of existence for the solution.

This volume of the Proceedings of the congress ISAAC '97 collects the con­ tributions of the four sections 1. Function theoretic and functional analytic methods for pde, 2. Applications of function theory of several complex variables to pde, 3. Integral equations and boundary value problems, 4. Partial differential equations. Most but not all of the authors have participated in the congress. Unfortunately some from Eastern Europe and Asia have not managed to come because of lack of financial support. Nevertheless their manuscripts of the proposed talks are included in this volume. The majority of the papers deal with complex methods. Among them boundary value problems in particular the Riemann-Hilbert, the Riemann (Hilbert) and related problems are treated. Boundary behaviour of vector-valued functions are studied too. The Riemann-Hilbert problem is solved for elliptic complex equations, for mixed complex equations, and for several complex variables. It is considered in a general topological setting for mappings into q;n and related to Toeplitz operators. Convolution operators are investigated for nilpotent Lie groups leading to some consequences for the null space of the tangential Cauchy Riemann operator. Some boundary value problems for overdetermined systems in balls of q;n are solved explicitly. A survey is given for the Gauss-Manin connection associated with deformations of curve singularities. Several papers deal with generalizations of analytic functions with various applications to mathematical physics. Singular integrals in quaternionic anal­ ysis are studied which are applied to the time-harmonic Maxwell equations.

This volume is a collection of manscripts mainly originating from talks and lectures given at the Workshop on Recent Trends in Complex Methods for Par­ tial Differential Equations held from July 6 to 10, 1998 at the Middle East Technical University in Ankara, Turkey, sponsored by The Scientific and Tech­ nical Research Council of Turkey and the Middle East Technical University. This workshop is a continuation oftwo workshops from 1988 and 1993 at the In­ ternational Centre for Theoretical Physics in Trieste, Italy entitled Functional analytic Methods in Complex Analysis and Applications to Partial Differential Equations. Since classical complex analysis of one and several variables has a long tra­ dition it is of high level. But most of its basic problems are solved nowadays so that within the last few decades it has lost more and more attention. The area of complex and functional analytic methods in partial differential equations, however, is still a growing and flourishing field, in particular as these methods are not only applied. Whithin the framework of holomorphic functions but are also combined with properties of generalized analytic functions. This can be seen by the many books which recently were published in this field and also by the proceedings in this ISAAC series and the ISAAC congresses and workshops.

This work provides information for readers interested in the mathematical past and present of Berlin. It presents a comprehensive, condensed overview of mathematical activity in Berlin, beginning with the foundation of the Academy by Leibniz and carrying over almost to the present day. Many chapters are essentially concise biographies of figures in mathematical history. The presentations are held together by a few articles examining the overall development of entire periods of scientific life at Berlin. Chapters cover the foundation of the University of Berlin, the "golden age" of mathematics (spanning the second half of the 19th century), the Nazi period, the development of mathematics in East and West Berlin during the political invasion of the city, and the merging of the formerly separated communities with the reunification of Germany.