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AuthorRauch, Jeffrey. author
TitlePartial Differential Equations [electronic resource] / by Jeffrey Rauch
ImprintNew York, NY : Springer New York : Imprint: Springer, 1991
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Descript X, 266 p. online resource


This book is based on a course I have given five times at the University of Michigan, beginning in 1973. The aim is to present an introduction to a sampling of ideas, phenomena, and methods from the subject of partial differential equations that can be presented in one semester and requires no previous knowledge of differential equations. The problems, with hints and discussion, form an important and integral part of the course. In our department, students with a variety of specialties-notably differenยญ tial geometry, numerical analysis, mathematical physics, complex analysis, physics, and partial differential equations-have a need for such a course. The goal of a one-term course forces the omission of many topics. Everyone, including me, can find fault with the selections that I have made. One of the things that makes partial differential equations difficult to learn is that it uses a wide variety of tools. In a short course, there is no time for the leisurely development of background material. Consequently, I suppose that the reader is trained in advanced calculus, real analysis, the rudiments of complex analysis, and the language offunctional analysis. Such a background is not unusual for the students mentioned above. Students missing one of the "essentials" can usually catch up simultaneously. A more difficult problem is what to do about the Theory of Distributions


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 Cauchyโ{128}{148}Kowaleskaya Theorem -- ยง1.5. Cauchyโ{128}{148}Kowaleskaya with General Initial Surfaces -- ยง1.6. The Symbol of a Differential Operator -- ยง1.7. Holmgrenโ{128}{153}s Uniqueness Theorem -- ยง1.8. Fritz Johnโ{128}{153}s 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. Schrรถdingerโ{128}{153}s Equation -- ยง3.3. Solutions of Schrรถdingerโ{128}{153}s Equation with Data in $$\mathcal{J}({\mathbb{R}̂d})$$ -- ยง3.4. Generalized Solutions of Schrรถdingerโ{128}{153}s 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 Cauchyโ{128}{148}Riemann Operator -- ยง3.9. The Sideways Heat Equation and Null Solutions -- ยง3.10. The Hadamardโ{128}{148}Petrowsky Dichotomy -- ยง3.11. Inhomogeneous Equations, Duhamelโ{128}{153}s 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. Dirichletโ{128}{153}s 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. Hopfโ{128}{153}s Strong Maximum Principles -- APPEND -- A Crash Course in Distribution Theory -- References

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