This book deals with elliptic differential equations, providing the analytic background necessary for the treatment of associated spectral questions, and covering important topics previously scattered throughout the literature.
Starting with the basics of elliptic operators and their naturally associated function spaces, the authors then proceed to cover various related topics of current and continuing importance. Particular attention is given to the characterisation of self-adjoint extensions of symmetric operators acting in a Hilbert space and, for elliptic operators, the realisation of such extensions in terms of boundary conditions. A good deal of material not previously available in book form, such as the treatment of the Schauder estimates, is included.
Requiring only basic knowledge of measure theory and functional analysis, the book is accessible to graduate students and will be of interest to all researchers in partial differential equations. The reader will value its self-contained, thorough and unified presentation of the modern theory of elliptic operators.
David Edmunds is an analyst who has been at the University of Sussex since 1966 where he is currently Emeritus Professor. His interests include function spaces, interpolation theory, entropy and s-numbers, and elliptic partial differential equations.
Desmond Evans is Emeritus Professor at the University of Cardiff, having been at Cardiff since 1964. His main research interests are in the spectral theory associated with differential equations and related areas of analysis and mathematical physics. These related areas include the properties of function spaces and the mappings between them, growth and asymptotic estimates for eigenvalues and s-numbers, and inequalities.
1. Preliminaries.- 2. The Laplace Operator.- 3. Second-order elliptic equations.- 4. The classical Dirichlet problem for second-order elliptic operators.- 5. Elliptic operators of arbitrary order.- 6. Operators and quadratic forms in Hilbert space.- 7. Realisations of second-order linear elliptic operators.- 8. The Lp approach to the Laplace operator.- 9. The p-Laplacian.- 10. The Rellich inequality.- 11. More properties on Sobolev embeddings.- 12. The Dirac Operator.