I. Various ways of representing solutions to various PDEs: Fundamental solutions. Mean value formula (also for heat equations). Maximal principle. Liouville theorem. Analyticity. Self-similar solutions. Hopf-Cole transformations. Potential functions. Hodograph, Legendre, and Lewy transformations. Power series, Cauchy-Kovalevskaya theorem. Lewy's counterexample.

II. Qualitative theory for (linear elliptic and parabolic) PDEs: Sobolev spaces. Existence of weak solutions. Regularity. Maximum principle. Harnack inequality. Eigenvalues and eigenfunctions.

III. Variational and non-variational techniques for nonlinear PDEs: Existence of minimizers. Regularity. Constraints. Critical points. Monotonicity method. Fixed point methods. Sub/super solutions. Nonexistence. Moving plane method. Viscosity solutions.

Textbook: Partial differential equations, L. C. Evans.

Prerequisites: Math~557 or equivalent