Partial Differential Equations

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. Schrödinger's Equation.- §3.3. Solutions of Schrödinger's Equation with Data in
$$\mathcal{J}({\mathbb{R}^d})$$.- §3.4. Generalized Solutions of Schrödinger'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 Schrödinger Propagator.- §4.5. The Wave EquationPropagator 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.