1. Nonlinear analysis
   • Contraction mapping principle
   • Implicit function theorem in Banach space
   • Fixed point theorems
2. Existence-uniqueness theorems for initial value problems for ODE's
   • Picard-Lindelof theorem
   • Cauchy Peano theorem
   • Dependence upon initial conditions and parameters
   • Variation of constants formula
3. First-Order PDE's
   • Cauchy problem
   • Method of characteristics
   • Weak solutions
   • Conservation laws and jump conditions
   • General nonlinear equations
   • Geometric optics
4. The Cauchy Problem
   • Caushy-Kovalevski theorem
   • Lewy's example
5. Linear Equations and Generalized Solutions
   • Adjoints
   • Weak solutions
   • Distributions
   • Fundamental solutions and convolutions
6. Classical Wave, Laplace and Heat Equations
   • Seperation of variables
   • Fourier transform
   • Green's Functions
   • Spherical means
   • Energy methods
   • Mean values
   • Maximum principles
   • Eigenfunction expansions
   • Perron's method
   • Subharmonic functions
   • Scale invariance and the similarity method
7. Nonlinear Diffusion
   • Weak and strong maximum principles
   • Comparison principles
8. Hilbert Space Theory for Linear PDE
   • Sobolev spaces
   • Hahn-Banach and Riesz representation theorems
   • Lax-Milgram theorem
   • Applications of Lax-Milgram to the study of weak solutions of elliptic PDE's
   • Regularity theory
   • Mollifiers
   • Spectral theorem for elliptic operators

References:

Partial Differential Equations methods and applications, Robert McOwen, Prentice Hall
Shock Waves and Reaction-Diffusion Equations, Smoller, Springer-Verlag
Partial Differential Equations, John, Springer-Verlag
Introduction to Partial Differential Equations, Folland, Princeton University Press