MATH 513: Partial Differential Equations

Outline of the course (subject to minor modifications)

• Chapter 1
  
  PDEs: definition, classification in order, type (linear, nonlinear).
  Basic concepts: Classical solutions, the Cauchy problem, the boundary value problem.
  Examples. Well-posedness, weak solutions and regularity
  
  
• Chapter 2: Four Important linear PDE
  
  Transport Equation: The initial value problem, non-homogeneous problems.
  
  Laplace Equation: Physical interpretation, fundamental solution, Poisson's equation. Mean-value 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 Rⁿ, the non-homegeneous 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: Non-linear First Order PDE
  
  The method of characteristics for semilinear, quasilinear, and fully nonlinear first order equations.
  
  Hamilton-Jacobi equations. The standard problem in the Calculus of Variations, necessary condition for optimality: the Euler-Lagrange equations. Hamilton's equations. The Legendre transform, the Hamilton-Jacobi PDE, the Hopf-Lax formula.
  
  
• Selected Topics From Other Chapters (e.g., viscosity solutions)