Partial Differential Equations (Paperback, Softcover Repri)

1 Power Series Methods.- 1.1. The Simplest Partial Differential Equation.- 1.2. The Initial Value Problem for Ordinary Differential Equations.- 1.3. Power Series and the Initial Value Problem for Partial Differential Equations.- 1.4. The Fully Nonlinear Cauchy-Kowaleskaya Theorem.- 1.5. Cauchy-Kowaleskaya with General Initial Surfaces.- 1.6. The Symbol of a Differential Operator.- 1.7. Holmgren's Uniqueness Theorem.- 1.8. Fritz John's Global Holmgren Theorem.- 1.9. Characteristics and Singular Solutions.- 2 Some Harmonic Analysis.- 2.1. The Schwartz Space $$\mathcal{J}({\mathbb{R}^d})$$.- 2.2. The Fourier Transform on $$\mathcal{J}({\mathbb{R}^d})$$.- 2.3. The Fourier Transform onLp$${\mathbb{R}^d}$$d):1 ?p?2.- 2.4. Tempered Distributions.- 2.5. Convolution in $$\mathcal{J}({\mathbb{R}^d})$$ and $$\mathcal{J}'({\mathbb{R}^d})$$.- 2.6. L2Derivatives and Sobolev Spaces.- 3 Solution of Initial Value Problems by Fourier Synthesis.- 3.1. Introduction.- 3.2. Schrodinger's Equation.- 3.3. Solutions of Schrodinger's Equation with Data in $$\mathcal{J}({\mathbb{R}^d})$$.- 3.4. Generalized Solutions of Schrodinger's Equation.- 3.5. Alternate Characterizations of the Generalized Solution.- 3.6. Fourier Synthesis for the Heat Equation.- 3.7. Fourier Synthesis for the Wave Equation.- 3.8. Fourier Synthesis for the Cauchy-Riemann Operator.- 3.9. The Sideways Heat Equation and Null Solutions.- 3.10. The Hadamard-Petrowsky Dichotomy.- 3.11. Inhomogeneous Equations, Duhamel's Principle.- 4 Propagators andx-Space Methods.- 4.1. Introduction.- 4.2. Solution Formulas in x Space.- 4.3. Applications of the Heat Propagator.- 4.4. Applications of the Schrodinger Propagator.- 4.5. The Wave Equation Propagator ford = 1.- 4.6. Rotation-Invariant Smooth Solutions of $${\square _{1 + 3}}\mu = 0$$.- 4.7. The Wave Equation Propagator ford =3.- 4.8. The Method of Descent.- 4.9. Radiation Problems.- 5 The Dirichlet Problem.- 5.1. Introduction.- 5.2. Dirichlet's Principle.- 5.3. The Direct Method of the Calculus of Variations.- 5.4. Variations on the Theme.- 5.5.H1 the Dirichlet Boundary Condition.- 5.6. The Fredholm Alternative.- 5.7. Eigenfunctions and the Method of Separation of Variables.- 5.8. Tangential Regularity for the Dirichlet Problem.- 5.9. Standard Elliptic Regularity Theorems.- 5.10. Maximum Principles from Potential Theory.- 5.11. E. Hopf's Strong Maximum Principles.- APPEND.- A Crash Course in Distribution Theory.- References.