Boundary Value Problems for Elliptic Systems
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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 'algebraise' the index theory by means of pseudo-differential operators and new 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 equatio...
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 'algebraise' the index theory by means of pseudo-differential operators and new 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.
This book examines the theory of boundary value problems for elliptic systems of partial differential equations. The aim is to 'algebraise' the index theory by means of pseudo-differential operators and new methods in the spectral theory of matrix polynomials.
This book examines the theory of boundary value problems for elliptic systems of partial differential equations.
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.
This book examines the theory of boundary value problems for elliptic systems of partial differential equations. The aim is to 'algebraise' the index theory by means of pseudo-differential operators and new methods in the spectral theory of matrix polynomials.
This book examines the theory of boundary value problems for elliptic systems of partial differential equations.