Author | Rauch, Jeffrey. author |
---|---|
Title | Partial Differential Equations [electronic resource] / by Jeffrey Rauch |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1991 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-0953-9 |
Descript | X, 266 p. online resource |
1 Power Series Methods -- ยง1.1. The Simplest Partial Differential Equation -- ยง1.2. The Initial Value Problem for Ordinary Differential Equations -- ยง1.3. Power Series and the Initial Value Problem for Partial Differential Equations -- ยง1.4. The Fully Nonlinear CauchyโKowaleskaya Theorem -- ยง1.5. CauchyโKowaleskaya with General Initial Surfaces -- ยง1.6. The Symbol of a Differential Operator -- ยง1.7. Holmgrenโs Uniqueness Theorem -- ยง1.8. Fritz Johnโs Global Holmgren Theorem -- ยง1.9. Characteristics and Singular Solutions -- 2 Some Harmonic Analysis -- ยง2.1. The Schwartz Space $$\mathcal{J}({\mathbb{R}̂d})$$ -- ยง2.2. The Fourier Transform on $$\mathcal{J}({\mathbb{R}̂d})$$ -- ยง2.3. The Fourier Transform onLp$${\mathbb{R}̂d}$$d):1 ?p?2 -- ยง2.4. Tempered Distributions -- ยง2.5. Convolution in $$\mathcal{J}({\mathbb{R}̂d})$$ and $$\mathcal{J}'({\mathbb{R}̂d})$$ -- ยง2.6. L2Derivatives and Sobolev Spaces -- 3 Solution of Initial Value Problems by Fourier Synthesis -- ยง3.1. Introduction -- ยง3.2. Schrรถdingerโs Equation -- ยง3.3. Solutions of Schrรถdingerโs Equation with Data in $$\mathcal{J}({\mathbb{R}̂d})$$ -- ยง3.4. Generalized Solutions of Schrรถdingerโs Equation -- ยง3.5. Alternate Characterizations of the Generalized Solution -- ยง3.6. Fourier Synthesis for the Heat Equation -- ยง3.7. Fourier Synthesis for the Wave Equation -- ยง3.8. Fourier Synthesis for the CauchyโRiemann Operator -- ยง3.9. The Sideways Heat Equation and Null Solutions -- ยง3.10. The HadamardโPetrowsky Dichotomy -- ยง3.11. Inhomogeneous Equations, Duhamelโs Principle -- 4 Propagators andx-Space Methods -- ยง4.1. Introduction -- ยง4.2. Solution Formulas in x Space -- ยง4.3. Applications of the Heat Propagator -- ยง4.4. Applications of the Schrรถdinger Propagator -- ยง4.5. The Wave Equation Propagator ford = 1 -- ยง4.6. Rotation-Invariant Smooth Solutions of $${\square _{1 + 3}}\mu = 0$$ -- ยง4.7. The Wave Equation Propagator ford =3 -- ยง4.8. The Method of Descent -- ยง4.9. Radiation Problems -- 5 The Dirichlet Problem -- ยง5.1. Introduction -- ยง5.2. Dirichletโs Principle -- ยง5.3. The Direct Method of the Calculus of Variations -- ยง5.4. Variations on the Theme -- ยง5.5.H1 the Dirichlet Boundary Condition -- ยง5.6. The Fredholm Alternative -- ยง5.7. Eigenfunctions and the Method of Separation of Variables -- ยง5.8. Tangential Regularity for the Dirichlet Problem -- ยง5.9. Standard Elliptic Regularity Theorems -- ยง5.10. Maximum Principles from Potential Theory -- ยง5.11. E. Hopfโs Strong Maximum Principles -- APPEND -- A Crash Course in Distribution Theory -- References