Boundary Value Problems for Elliptic Systems

This book examines the theory of boundary value problems for elliptic systems of partial differential equations, a theory which has many applications in mathematics and the physical sciences. The aim is to 'algebraize' the index theory by means of pseudo-differential operators and methods in the spectral theory of matrix polynomials. This latter theory provides important tools that will enable the student to work efficiently with the principal symbols of the elliptic and boundary operators on the boundary. This book is ideal for use in graduate-level courses on partial differential equations, elliptic systems, pseudo-differential operators and matrix analysis. Since many new methods and results are introduced and used throughout the book, all the theorems are proved in detail, and the methods are well illustrated through numerous examples and exercises.

Table of Contents:
Part I. A Spectral Theory of Matrix Polynormials: 1. Matrix polynomials; 2. Spectral triples for matrix polynomials; 3. Monic matrix polynomials; 4. Further results; Part II. Manifolds and Vector Bundles: 5. Manifolds and vector bundles; 6. Differential forms; Part III. Pseudo-Differential Operators and Elliptic Boundary Value Problems: 7. Pseudo-differential operators on Rn; 8. Pseudo-differential operators on a compact manifold; 9. Elliptic systems on bounded domains in Rn; Part IV. Reduction Of A Boundary Value Problem To An Elliptic System On The Boundary: 10. Understanding the L-condition; 11. Applications to the index; 12. BVPs for ordinary differential operators and the connection with spectral triples; 13. Behaviour of a pseudo-differential operator near a boundary; 14. The Main Theorem revisited; Part V. An Index Formula For Elliptic Boundary Problems In The Plane: 15. Further results on the Lopatinskii Condition; 16. The index in the plane; 17. Elliptic systems with 2 x 2 real coefficients.

Review :
'It is the strength of this book that, for the first time, the theory of (elliptic) systems is presented on the level of recent research theory of (scalar) pseudodifferential operators … the authors put new life into the classical method of shifting boundary value problems in a domain to its boundary.' Hans Triebel, Bulletin of the London Mathematical Society 'The book can be recommended both as a textbook for graduate students and as a handbook for researchers.' T. Weidl, Proceedings of the Edinburgh Mathematical Society '… certainly of great interest for specialists and can be used for advanced lectures or seminars in this field.' Monatshefte für Mathematik