Text: Partial Differential Equations I and II, 2nd edition, by Michael E.Taylors.  You don't want to buy a hardcopy of the textbook, here is the the PDF files of the book:
                 http://www.springerlink.com/content/978-1-4419-7055-8#section=802405&page=1
                 http://www.springerlink.com/content/978-1-4419-7052-7#section=805524&page=1 
                 http://www.springerlink.com/content/978-1-4419-7049-7#section=806206&page=1       

Reference: A Guide to Distribution Theory and Fourier Transforms,  by Robert S. Strichartz
                        Fourier Analysis and Nonlinear Partial Differential Equations,  by Hajer Bahouri, Jean-Yves Chemin and Raphaël Danchin 
                        Elliptic Equations: An Introductory Course, by Michel Chipot

Prerequisites:   The theory of measure and integration, Complex Analysis. You should have taken Math 8100 and 8150 at UGA or equivalent courses.

Objectives of the Course: This is a course on the theory of PDEs and Fourier analysis.  We will introduces basic examples of PDEs arising in continuum mechanics, electromagnetism, complex analysis and other areas, and develops a number of tools for their solution, in particular Fourier analysis, distribution theory, and Sobolev spaces. These tools are then applied to the treatment of basic problems in linear PDE, including the Laplace equation, heat equation, and wave equation, as well as more general elliptic, parabolic, and hyperbolic equations.