There is no textbook for this course, suggested books are listed below.
501, 502, 503, 517, some very rudimentary Functional analysis
This will be a basic PDE course with a strong focus on basic estimates that lead to regularity and existence of solutions. I plan to cover Schauder theory of elliptic PDE, and attendant regularity theory. I hope to cover basic Sobolev spaces if it has not been covered in Math 517. I shall then also lecture on basic properties of pseudo-differential operators. The important theory of L^p regularity of solutions that is fundamental for the study of Nonlinear problems can only be covered if students have understood some Harmonic Analysis and the Calderon-Zygmund theory. If students have had some exposure to Harmonic Analysis, I will deal with the L^p theory, or else restrict myself to Schauder theory. No one book covers all this material. However, a good combination of books that has all this material is:
 Elliptic Partial Differential equations of Second Order, David Gilbarg and Neil S. Trudinger
 Introduction to Pseudodifferential and Fourier Integral Operators, Volume I, J. Francois Treves.