Most of the projects I can offer in this area take the form of so-called singular perturbation problems. A PDE, involving a certain parameter ε, describes the behaviour of a physical phenomenon. For the limit ε → 0, is it possible to derive a limiting equation that describes the behaviour of solutions to the original problem? For example, the magnetisation of a ferromagnet is often nearly constant in certain regions (called domains in this context), but there may be different such regions separated by so-called domain walls. The domain walls are not sharp transitions; there is a transition layer with a thickness of order ε. But as ε is typically very small, they are often treated as if they were sharp transitions. It is then important to derive and study an equation that describes their behaviour.
Despite the physical motivation, these are projects in rigorous analysis. They involve abstract notions of convergence in a functional analytic framework and an analysis of the underlying PDEs. A typical result will be a statement of roughly the following form. Suppose that we have a family of solutions uε of the original equation (one for each value of ε). Then uε converges to a limit u0 as ε → 0 (in a suitable sense), and u0 is the solution of a specific limiting equation.