This text on partial differential equations is intended for readers who want to understand the theoretical underpinnings of modern PDEs in settings that are important for the applications without using extensive analytic tools required by most advanced texts. The assumed mathematical background is at the level of multivariable calculus and basic metric space material, but the latter is recalled as relevant as the text progresses. The key goal of this book is to be mathematically complete without overwhelming the reader, and to develop PDE theory in a manner that reflects how researchers would think about the material. A concrete example is that distribution theory and the concept of weak solutions are introduced early because while these ideas take some time for the students to get used to, they are fundamentally easy and, on the other hand, play a central role in the field. Then, Hilbert spaces that are quite important in the later development are introduced via completions which give essentially all the features one wants without the overhead of measure theory.
There is additional material provided for readers who would like to learn more than the core material, and there are numerous exercises to help solidify one's understanding. The text should be suitable for advanced undergraduates or for beginning graduate students including those in engineering or the sciences.
András Vasy, Stanford University, Stanford, CA
• Preface 10 • Chapter 1. Introduction 12 • Chapter 2. Where do PDE come from? 30 • Chapter 3. First order scalar semilinear equations 40 • Chapter 4. First order scalar quasilinear equations 56 • Chapter 5. Distributions and weak derivatives 66 • Chapter 6. Second order constant coefficient PDE: Types and d’Alembert’s solution of the wave equation 92 • Chapter 7. Properties of solutions of second order PDE: Propagation, energy estimates and the maximum principle 104 • Chapter 8. The Fourier transform: Basic properties, the inversion formula and the heat equation 124 • Chapter 9. The Fourier transform: Tempered distributions, the wave equation and Laplace’s equation 144 • Chapter 10. PDE and boundaries 158 • Chapter 11. Duhamel’s principle 170 • Chapter 12. Separation of variables 180 • Chapter 13. Inner product spaces, symmetric operators, orthogonality 190 • Chapter 14. Convergence of the Fourier series and the Poisson formula on disks 212 • Chapter 15. Bessel functions 232 • Chapter 16. The method of stationary phase 246 • Chapter 17. Solvability via duality 256 • Chapter 18. Variational problems 274 • Bibliography 288 • Index 290