
Chapter 1
PDEs: definition, classification in order, type (linear, nonlinear).
Basic concepts: Classical solutions, the Cauchy problem, the boundary value problem.
Examples.
Wellposedness, weak solutions and regularity

Chapter 2: Four Important linear PDE
Transport Equation: The initial value problem, nonhomogeneous problems.
Laplace Equation: Physical interpretation, fundamental solution, Poisson's equation. Meanvalue formulas, maximum principles, uniqueness of solutions. Estimates on derivatives, Liouville's theorem. Harnack's inequality. Green's function. Uniqueness of solutions by the energy method, Dirichlet's principle.
Heat Equation: Physical interpretation, fundamental solution, the initial value problem on the whole space R^{n}, the nonhomegeneous problem. Maximum principles, uniqueness of solutions on bounded and unbounded domains. Energy methods, backward uniqueness.
Wave Equation: Physical interpretation, solutions on the real line by D'Alembert's formula, solution on the half line by reflection. Energy methods, finite propagation speed.

Chapter 3: Nonlinear First Order PDE
The method of characteristics for semilinear, quasilinear, and fully nonlinear first order equations.
HamiltonJacobi equations. The standard problem in the Calculus of Variations, necessary condition for optimality: the EulerLagrange equations. Hamilton's equations. The Legendre transform, the HamiltonJacobi PDE, the HopfLax formula.

Selected Topics From Other Chapters
(e.g., viscosity solutions)
Math 513: PDE I