Local minima and energy barriers in the d-dimensional Cahn-Hilliard energy landscape

For mean values close to -1, it is easy to see that the constant state is a local energy minimizer of the Cahn-Hilliard energy. As already described in the seminal work of Cahn and Hilliard, stochastic fluctuations lead to nucleation of small, droplet-shaped regions of +1, which may then grow and coalesce. Moreover, whether the regions of +1 grow or shrink should depend on whether their mass is large enough to form a so-called critical nucleus. We describe recent (deterministic) work on the Cahn-Hilliard energy landscape in the regime of mean value close to -1 and large system size, which leads to quantitative bounds on the volume and approximate "droplet-shape" of a candidate for the critical nucleus. Our methods involve Gamma-limits, quantitative isoperimetric inequalities, and Steiner symmetrization.