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 CauchyKowaleskaya Theorem. - 1. 5. CauchyKowaleskaya 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 CauchyRiemann Operator. - 3. 9. The Sideways Heat Equation and Null Solutions. - 3. 10. The HadamardPetrowsky 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. Language: English