What I would most like you to gain from this class is a sufficient understanding of the ideas and language of PDE so that you can begin to read the literature. The literature not only of theoretical PDE (studying PDE mostly for its own sake), but also of applied math and a good bit of geometry as well. And, to a greater or lesser extent, other fields. I feel that within the shortness of one quarter, this is a reasonable goal.
I will give a broad survey of the diverse areas of PDE, so you at least have some understanding of the field as a whole, and give a closer look at a few selected areas. We will study distributions and Sobolev spaces, for though they are more properly part of functional analysis, they are the language in which most of modern PDE is written.
There will be no official textbook, although the textbook of record (or so I discovered) is Lawrence C. Evans's "Partial Differential Equations." This is, in fact, a beautiful text, and will influence much of what I have to say, but if you don't want to buy it, you need not. Evans avoids the use of distributions, however, sticking with Sobolev spaces.
A reasonable prerequisite for this class is a good understanding of measure theory, as one would gain from the analysis sequence here at UCR.
I haven't entirely worked out the details of the grading system yet, but I can say this: There will be no exams. Regular attendance in the class will give you an A-. I will also probably assign recommended exercises, and working these exercise will increase your grade.