After a review of Fourier transform and related real analysis, we begin by studying the most ubiquitous elliptic operator, the laplacian. We study harmonic functions and solutions to the Dirichlet problem in various domains. In order to extend the regularity and existence results we observed for the laplacian to more general elliptic operators, we next study Sobolev spaces, proving the embedding theorems, the Rellich compactness theorem, and trace theorems.
We will apply the Sobolev theory to prove basic regularity results for solutions to elliptic equations and to study existence and uniqueness of solutions to elliptic boundary value problems.
We will give alternate existence proofs based on variational techniques. We will show how to apply variational techniques to mildly nonlinear problems.
Applications to advanced topics will be considered if time permits.