Perturbation and Variational Methods for Partial Differential Equations

In the course two different approaches to analyse PDEs will be presented. The first method will focus on approximating solutions to PDEs. In the lectures we elaborate on asymptotic approximations, matched asymptotic expansions, multiple scales, WKB method, bifurcation and stability analysis.

In the second approach, a modern introduction to the field of elliptic PDEs is given. In the lectures we consider weak solutions to elliptic equations (such as Laplace equation), Sobolev spaces. Modern PDEs uses tools from functional analysis and measure theory. The precise regularity and behavior of solutions of elliptic equations is one of the main features in the lectures. For non-linear PDEs such results are fundamental. The course might be useful for student in analysis, PDE and numerics, or anyone who wants to know more about Sobolev spaces and second order equations. (Key words: elliptic, weak derivatives, Sobolev spaces extension theorem, trace theorem, Sobolev inequalities, Hilbert techniques, compactness theorems, second order elliptic equations, Weak solution, energy estimates, regularity, eigenvalue techniques).