Partial Differential Equations
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Partial Differential Equations 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 uy = 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. Duhamel?s principle. 44
Chapter 4. The Heat Equation 47
1. Introduction 47
2. Solution for Rn 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 Rn 60
3. Mean value property 60
4. Poisson formula for a ball 64
5. Existence and regularity for −∆u + u = f on Tn 65
6. Harmonic polynomials and spherical harmonics 67
7. Dirichlet?s principle and existence of a solution 69
Chapter 6. The Rest 75
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