Theory and Applications of Partial Differential Equations

Description

This book is a product of the experience of the authors in teaching partial differential equations to students of mathematics, physics, and engineering over a period of 20 years. Our goal in writing it has been to introduce the subject with precise and rigorous analysis on the one hand, and interesting and significant applications on the other. The starting level of the book is at the first-year graduate level in a U.S. university. Previous experience with partial differential equations is not required, but the use of classical analysis to find solutions of specific problems is not emphasized. From that perspective our treatment is decidedly theoretical. We have avoided abstraction and full generality in many situations, however. Our plan has been to introduce fundamental ideas in relatively simple situations and to show their impact on relevant applications. The student is then, we feel, well prepared to fight through more specialized treatises. There are parts of the exposition that require Lebesgue integration, distributions and Fourier transforms, and Sobolev spaces. We have included a long appendix, Chapter 8, giving precise statements of all results used. This may be thought of as an introduction to these topics. The reader who is not familiar with these subjects may refer to parts of Chapter 8 as needed or become somewhat familiar with them as prerequisite and treat Chapter 8 as Chapter O. Front Matter....Pages i-ix Introduction to Partial Differential Equations....Pages 1-9 Wave Equation....Pages 11-51 Heat Equation....Pages 53-101 Laplace Equation....Pages 103-211 Elliptic Partial Differential Equations of Second Order....Pages 213-267 Abstract Evolution Equations....Pages 269-289 Hyperbolic Systems of Conservation Laws in One Space Variable....Pages 291-394 Distributions and Sobolev Spaces....Pages 395-435 Back Matter....Pages 437-444