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**Unformatted text preview: **AMSI Jan. 14 Feb. 8, 2008 Partial Differential Equations Jerry L. Kazdan [Last revised: March 28, 2011] Copyright c circlecopyrt 2008 by Jerry L. Kazdan Contents Chapter 1. Introduction 1 1. Functions of Several Variables 2 2. Classical Partial Differential Equations 3 3. Ordinary Differential Equations, a Review 5 Chapter 2. First Order Linear Equations 11 1. Introduction 11 2. The Equation u y = f ( x,y ) 11 3. A More General Example 13 4. A Global Problem 18 5. Appendix: Fourier series 22 Chapter 3. The Wave Equation 29 1. Introduction 29 2. One space dimension 29 3. Two and three space dimensions 33 4. Energy and Causality 36 5. Variational Characterization of the Lowest Eigenvalue 41 6. Smoothness of solutions 43 7. The inhomogeneous equation. Duhamels principle. 44 Chapter 4. The Heat Equation 47 1. Introduction 47 2. Solution for R n 47 3. Initial-boundary value problems for a bounded region, part 1 50 4. Maximum Principle 51 5. Initial-boundary value problems for a bounded region, part 2 54 6. Appendix: The Fourier transform 56 Chapter 5. The Laplace Equation 59 1. Introduction 59 2. Poisson Equation in R n 60 3. Mean value property 60 4. Poisson formula for a ball 64 5. Existence and regularity for u + u = f on T n 65 6. Harmonic polynomials and spherical harmonics 67 iii iv CONTENTS 7. Dirichlets principle and existence of a solution 69 Chapter 6. The Rest 75 CHAPTER 1 Introduction Partial Differential Equations (PDEs) arise in many applications to physics, geometry, and more recently the world of finance. This will be a basic course. In real life one can find explicit solutions of very few PDEs and many of these are infinite series whose secrets are complicated to extract. For more than a century the goal is to understand the solutions even though there may not be a formula for the solution. The historic heart of the subject (and of this course) are the three fun- damental linear equations: wave equation, heat equation, and Laplace equation along with a few nonlinear equations such as the minimal sur- face equation and others that arise from problems in the calculus of variations. We seek insight and understanding rather than complicated formulas. Prerequisites: Linear algebra, calculus of several variables, and basic ordinary differential equations. In particular Ill assume some expe- rience with the Stokes and divergence theorems and a bit of Fourier analysis. Previous acquaintantance with normed linear spaces will also be assumed. Some of these topics will be reviewed a bit as needed. References: For this course, the most important among the following are the texts by Strauss and Evans. Strauss, Walter A., Partial Differential Equations: An Introduction , New York, NY: Wiley, 1992....

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