Partial Differential Equations - Jeffrey Rauch

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Sommaire:
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 dollars dollars\mathcal{J}({\mathbb{R}^d}) dollars dollars.- ?2.2. The Fourier Transform on dollars dollars\mathcal{J}({\mathbb{R}^d}) dollars dollars.- ?2.3. The Fourier Transform onLp dollars dollars{\mathbb{R}^d} dollars dollarsd):1 ?p?2.- ?2.4. Tempered Distributions.- ?2.5. Convolution in dollars dollars\mathcal{J}({\mathbb{R}^d}) dollars dollars and dollars dollars\mathcal{J}'({\mathbb{R}^d}) dollars dollars.- ?2.6. L2Derivatives and Sobolev Spaces.- 3 Solution of Initial Value Problems by Fourier Synthesis.- ?3.1. Introduction.- ?3.2. Schr?dinger's Equation.- ?3.3. Solutions of Schr?dinger's Equation with Data in dollars dollars\mathcal{J}({\mathbb{R}^d}) dollars dollars.- ?3.4. Generalized Solutions of Schr?dinger'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 Schr?dinger Propagator.- ?4.5. The Wave EquationPropagator ford = 1.- ?4.6. Rotation-Invariant Smooth Solutions of dollars dollars{\square _{1 + 3}}\mu = 0 dollars dollars.- ?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.