"Graduate Partial Differential Equations" Webpage

Graduate Partial Differential Equations Class Notes
Partial Differential Equations, by Jeffrey Rauch,
Springer-Verlag (1991)

Copies of the classnotes are on the internet in PDF format as given below. The "Proofs of Theorem " files were prepared in Beamer and they contain proofs of the results from the class notes. The "Printout of Proofs" are printable PDF files of the Beamer slides without the pauses. These notes and supplements have not been classroom tested (and so may have some typographical errors).

The catalog description for Applied Mathematics 2 (MATH 5620) is: "Partial differential equations, Fourier series and integrals, numerical techniques." The formal prerequisite is admission to the math graduate program or permission. This description is based on the ETSU 2021-22 Graduate Catalog. These notes are based on a traditional graduate level course in partial differential equations.

Chapter 1. Power Series Methods

Section 1.1. The Simplest Partial Differential Equation.
Section 1.2. The Initial Value Problem for Ordinary Differential Equations.
Section 1.3. Power Series and Initial Value Problems for Partial Differential Equations.
Section 1.4. The Fully Nonlinear Cauchy-Kowaleskaya Theorem.
Section 1.5. Cauchy-Kowaleskaya with General Initial Surfaces.
Section 1.6. The Symbol of a Differential Operator.
Section 1.7. Holmgren's Uniqueness Theorem.
Section 1.8. Fritz John's Global Holmgren Theorem.
Section 1.9. Characteristics and Singular Solutions.

Chapter 2. Some Harmonic Analysis

Section 2.1. The Schwarz Space ℒ (ℝ^d).
Section 2.2. The Fourier Transform on ℒ (ℝ^d).
Section 2.3. The Fourier Transform on L^p (ℝ^d): 1 ≤ p ≤ 2.
Section 2.4. Tempered Distributions.
Section 2.5. Convolution in ℒ (ℝ^d) and ℒ ' (ℝ^d).
Section 2.6. L² Derivatives and Sobolev Spaces.

Chapter 3. Solution of Initial Value Problems by Fourier Synthesis

Section 3.1. Introduction.
Section 3.2. Schrödinger's Equation.
Section 3.3. Solutions of Schrödinger's Equation with Data in ℒ (ℝ^d).
Section 3.4. Generalized Solutions to Schrödinger's Equation.
Section 3.5. Alternate Characterizations of the Generalized Solution.
Section 3.6. Fourier Synthesis for the Heat Equation.
Section 3.7. Fourier Synthesis for the Wave Equation.
Section 3.8. Fourier Synthesis for the Cauchy-Riemann Operator.
Section 3.9. The Sideways Heat Equation and Null Solutions.
Section 3.10. The Hadamard-Petrovsky Dichotomy.
Section 3.11. Inhomogeneous Equations, Duhamel's Principle.

Chapter 4. Propagators and x-Space Methods

Section 4.1. Introduction.
Section 4.2. Solution Formulas in x Space.
Section 4.3. Applications of the Heat Propagator.
Section 4.4. Applications of the Schrödinger Propagator.
Section 4.5. The Wave Equation Propagator for d = 1.
Section 4.6. Rotation-Invariant Smooth Solutions of ☐₁₊₃u = 0.
Section 4.7. The Wave Equation Propagator for d = 3.
Section 4.8. The Method of Descent.
Section 4.9. Radiation Problems.

Chapter 5. The Dirichlet Problem

Section 5.1. Introduction.
Section 5.2. Dirichlet's Principle.
Section 5.3. The Direct Method of the Calculus of Variations.
Section 5.4. Variations on the Theme.
Section 5.5. H¹ and the Dirichlet Boundary Condition.
Section 5.6. The Fredholm Alternative.
Section 5.7. Eigenfunctions and the Method of Separation of Variables.
Section 5.8. Tangential Regularity of the Dirichlet Problem.
Section 5.9. Standard Elliptic Regularity Theorems.
Section 5.10. Maximum Principles from Potential Theory.
Section 5.11. E. Hopf's Strong Maximum Principles.

Appendix

A Crash Course in Distribution Theory.

Return to Bob Gardner's home page