Chapter 1. First-Order Equations

• The Cauchy Problem for Quasilinear Equations
• Weak Solutions for Quasilinear Equations
• General Nonlinear Equations
• Concluding Remarks on First-Order Equations

• The Cauchy Problem
• Second-Order Equations in Two Variables
• Linear Equations and Generalized Solutions

• The One-Dimensional Wave Equation
• Higher Dimensions
• Energy Methods
• Lower-order Terms

• Introduction to the Laplace Equation
• Potential Theory and Green's Functions
• General Existence Theory
• Eigenvalues of the Laplacian

• The Heat Equation in a Bounded Domain
• The Pure Initial Value Problem
• Regularity and Similarity

• Function Spaces and Linear Operators
• Application to the Dirichlet Problem
• Duality and Compactness
• Sobolev Imbedding Theorems
• Generalizations and Refinements

• Calculus of Functionals and Variations
• Optimization with Constraints
• Calculus of Maps between Banach Spaces

• Elliptic Operators on a Torus
• Estimates and Regularity on Domains
• Maximum Principles
• Solvability

• Schauder Fixed Point Theory
• Semigroups and Dynamics

• Local Existence for Hyperbolic Systems
• Quasilinear Systems of Conservation Laws
• Systems of Two Conservation Laws

• Parabolic Maximum Principles
• Local Existence and Regularity
• Global Behavior
• Applications to Navier-Stokes

• Symmetric Hyperbolic Systems
• Linear Wave Dynamics
• Semilinear Wave Dynamics

• Perturbations and Bifurcations
• The Method of Sub- and Supersolutions
• The Variational Method
• Fixed Point Methods