Surface diffusion is one of the most important mechanisms driving crystal growth. When bulk diffusion is much faster than the surface one, the evolution of the profile of the crystal is described by the so called Einstein-Nernst relation. According to this law, the normal velocity of the profile is related to the chemical potential. This evolution equation has a variational flavour, as it can be obtained as a gradient flow of a suitable energy.
Albeit usually neglected, adatoms (atoms freely diffusing on the surface of the crystal) seem to play a fundamental role in the description of the behaviour of a solid-vapor interfaces. Some years ago Fried and Gurtin introduced a system of evolution equations to describe such a situation, where adatoms are treated as a separate variable of the problem. Also such system of evolution equations can be seen as the gradient flow of an energy.
In this talk a first step in the programme of studying the above mentioned evolution equations and the associated energy is presented. In particular, the following topics for the static problem are addressed: regular critical points, existence and uniqueness of minimisers and relaxation of the energy functional in a suitable topology, as well as a phase-field approximation.
The talk is based on a work in collaboration with Marco Caroccia (Scuola Normale Superiore-Università di Firenze), and Laurent Dietrich (Lycée Fabert).