Analysis I (18.100C)
The idea of the course was to give a solid introduction to PDE for advanced undergraduate students. We required only advanced calculus. The course went quite rapidly through a lot of material, but our focus was linear second order uniformly elliptic and parabolic equations. Some of the topics included the Laplace equation, harmonic functions, second order elliptic equations in divergence for, L-harmonic functions, heat equations, Green's function and heat kernels, maximum principles, Hopf's maximum principle, Harnack inequalities and gradient estimates for L-harmonic functions and more generally for solutions of heat equations. Morrey's and Capanato's lemmas, regularity of general solutions of second order elliptic equations in divergence form, the De Giorgi-Nash-Moser iteration argument, boundary regularity were also covered.
The course consisted mostly of lectures given by the instructor but there were also sessions where the students presented solutions to the homework. The class was small (4 honors students) which allowed for plenty of interaction and discussion of the course material.
The instructor graded each student according to class attendance, participation in class, and homework assignments. No final was given.