1.Important linear PDE: Laplace’s equation, heat equation and wave equation.
2.Nonlinear first order equations: characteristics, Hamilton-Jacobi equations, conservation laws
3.Similarity solutions, Transform methods
4.Sobolev spaces: weak derivative, Sobolev inequalities, compactness
5.Second-Order elliptic equations: existence of weak solutions, regularity, maximum principles and Harnack’s inequality