Tentative partial programme
The classical theory of elliptic and parabolic partial differential equations commences with potential methods for the Poisson equation and the heat equation. Basics tools are estimates for the Newton and heat potential in suitable Holder spaces. Using the method of continuity in combination with the a priori Schauder estimates, as well as the maximum principle, one obtains existence and uniqueness of classical solutions. For elliptic equations the basic reference for this approach is the book of Gilbarg and Trudinger, Elliptic partial differential equations of second order, Springer, 1977, ISBN/ISSN 3.540.08007.4. Although this book is very technical, it is well accessible for advanced undergraduate students in mathematics. The elliptic part of my lecture notes is taken largely from this book. It is however hard to find a parabolic counterpart of which the same can be said.
In this course we will discuss only only a few elements of this approach. Thus we begin with the fundamental solutions of the Poisson and the heat equation, some potential theory and the mean value theorems for harmonic functions and solutions of the heat equation.
Here is a problem set on harmonic functions.
We then follow the recent book of Evans, see below, which takes the modern theory of weak solutions for elliptic equations in Sobolev spaces as starting point. Thus we avoid potential theory and base the existence and uniqueness of (weak) solutions on the Lax-Milgram Theorem. This requires a proper introduction of Sobolev spaces and their basic properties (such as the Sobolev embeddings). We also establish the regularity of weak solutions which is based on testing the equations with second order difference quotients of the weak solution.
The existence, uniqueness and regularity theory is complemented with weak and strong maximum principles for (by now) classical solutions and spectral theory for selfadjoint elliptic operators. The eigenfunctions allow a natural finite dimensional approximation of evolution equations (second order parabolic and hyperbolic equations). This is the Galerkin method on which we base the theory for evolution equations involving elliptic operators. If time permits we also discuss the method for the Navier-Stokes equations. We complement this with the semigroup approach to parabolic equations.
The rest of the programme is still open.
Literature: Partial Differential Equations, Lawrence C. Evans, American Mathematical Society, Graduate Studies in Mathematics, ISBN: 0-8218-0772-2. See also the lecture notes.