Synopses & Reviews
The book contains review chapters as well as state-of-the-art research chapters 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. Audience: Researchers working in the field as well as scientists interested in the applications.
Synopsis
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.
Table of Contents
Preface.
1. A reflection principle and its applications;
J. Witte. 2. On some problems for first order elliptic systems in the plane;
D.Q. Dai. 3. Differential-operator solutions for complex partial differential equations;
O. Celebi, S. Sengül. 4. On a generalized Riemann-Hilbert Boundary value problem for second order elliptic systems in the plane;
M. Akal. 5. Boundary value problems of the theory of generalized analytic functions;
G. Manjavidze, G. Akhalaia. 6. On well-posedness of problems for nonclassical systems of equations;
D.Kh. Safarov. 7. An application of the periodic Riemann boundary value problem to a periodic crack problem;
X. Li. 8. Initial and boundary value problems for singular differential equations and applications to the theory of cusped bars and plates;
G. Jaiani. 9. Multidimensional logarithmic residues and their applications;
L.A. Aizenberg. 10. The Neumann problem for the inhomogeneous pluriharmonic system in polydiscs;
A. Mohammed. 11. Second order Cauchy-Pompeiu representations;
H. Begehr. 12. On a class of second order elliptic overdetermined systems;
A. Dzhuraev. 13. Boundary spinors and values of holomorphic functions;
J. Cnops. 14. Two approaches to non-commutative geometry;
V.V. Kisil. 15. Some partial differential equations in Clifford analysis;
E. Obolashvili. 16. Generalized monogenic functions satisfying differential equations with anti-monogenic right-hand sides;
W. Tutschke, U. Yüksel. 17. Complex analytic method for hyperbolic equations of second order;
G.-C. Wen. 18. Remarks on the solvability of Dirichlet problems in different function spaces;
F. Rihawi. 19. Complex methods in the theory of initial value problems;
W. Tutschke. 20. Optimal balls for solving fixed-point problems in Banach spaces;
T. Tutschke. 21. Wavelet transform of operators and functional calculus;
V.V. Kisil.