Differential Analysis
Elliptic and parabolic linear partial differential equations form the core of this MIT graduate course, the second half of a two-semester Differential Analysis sequence. Lectures build the theoretical machinery mathematicians use to prove existence, uniqueness, and regularity of solutions: Sobolev spaces, weak derivatives, maximum principles, and energy estimates. Elliptic equations get treated first, with boundary value problems and their solution spaces, before the course turns to parabolic equations and the behavior of solutions over time, connecting the two through the heat equation as a running example. Materials on MIT OpenCourseWare include lecture notes, problem sets, and readings, giving students a full record of the course even without an instructor present. The course assumes real analysis and measure theory as background and is aimed at doctoral-level mathematics students rather than beginners. No certificate is offered, since MIT OpenCourseWare simply publishes the materials for independent study.