Partial Differential Equations [Paperback]

This book is a rigorous introduction to the abstract theory of partial differential equations.A rigorous introduction to the abstract theory of partial differential equations progresses from the theory of distribution and Sobolev spaces to Fredholm operations, the Schauder fixed point theorem and Bochner integrals.A rigorous introduction to the abstract theory of partial differential equations progresses from the theory of distribution and Sobolev spaces to Fredholm operations, the Schauder fixed point theorem and Bochner integrals.This book is a rigorous introduction to the abstract theory of partial differential equations. The main prerequisite is familiarity with basic functional analysis: more advanced topics such as Fredholm operators, the Schauder fixed point theorem and Bochner integrals are introduced when needed, and the book begins by introducing the necessary material from the theory of distributions and Sobolev spaces. Using such techniques, the author presents different methods available for solving elliptic, parabolic and hyperbolic equations. He also considers the difference process for the practical solution of a partial differential equation, emphasising that it is possible to solve them numerically by simple methods. Many examples and exercises are provided throughout, and care is taken to explain difficult points. Advanced undergraduates and graduate students will appreciate this self-contained and practical introduction.Preface; Part I. Sobolev Spaces: 1. Notation, basic properties, distributions; 2. Geometric assumptions for the domain; 3. Definitions and density properties for the Sobolev-Slobodeckii spaces ; 4. The transformation theorem and Sobolev spaces on differentiable manifolds; 5. Definition of Sobolev spaces by the Fourier transformation and extension theorems; 6. Continuous embeddings and Sobolev's lemma; 7. Compact embeddings; 8. The trace operator; 9. Weak sequential compactness and approximation of derivatives by difference quotienlCH