Quantifying the shape of energy landscapes and timescale separation in a few model problems

We will explore the connection between the shape of an energy landscape and the dynamics of the corresponding gradient flow for three classical model problems, the Allen-Cahn equation (in d=1), the Cahn-Hilliard equation (in d=1), and the Mullins-Sekerka evolution (in d=3). In all cases we are particularly interested in initial perturbations that are not small. Along the way we will comment on a few questions (solved and open) about stochastically perturbed versions of these models. An underlying issue of interest in the Allen-Cahn and Cahn-Hilliard equations is the phenomenon of metastability. The variational techniques developed for and explained within the context of the model problems are more generally applicable.