Elliptic Partial Differential Equations (Courant Lecture Notes)

In this short yet concise lecture notes, almost everything about Elliptic Partial Differential Equations is touched.

Chapter 1 is about harmonic functions. The special property about harmonic function is the mean value formula and the simple for of the fundamental solution, from which we can get maximum principal and uniqueness. However, these properties cannot be extended to more general elliptic functions, even for those with constant coefficients. The maximum principle in general form and the energy method in the last two sections are more useful.

Chapter 2 is about maximum principle. The classical maximum principles are introduced from which we can derive apriori bounds on the solutions and the gradients. Alexandrov type maximum principle are very powerful for quasilinear equations like those for curvature equations, Monge-Ampere equations.