Por Marco Caroccia (Scuola Normale Superiore & Università di Firenze).

Abstract: A perimeter type energy leading an epitaxial growth process is treated. We consider an energy of the type
\[
\mathcal{F}(E,u)= \int_{\partial E} \psi(u)\d \H^{n-1}
\]
where a density $u: \partial E \rightarrow \mathbb{R}$ living on the boundary of $E$ is considered as a variable of the problem. Such energy has been used to model an epitaxial growth process with the presence of adatoms: free atoms of the crystal allowed to move on the surface and which impacts the chemical potential of the evolution. In this context the set $E$ represents the bulk, $\partial E$ is the surface that is evolving in the process and the density $u$ cathces the presence of the so-called adatoms. Such energy lacks of lower-semicontinuity in the natural topology under which the problem can be considered. In a contribution of 2016 jointly with Riccardo Cristoferi and Laurent Dietrich we provided a lower-semicontinuous envelope of $\mathcal{F}$ and, subsequently, a phase-field approximation is developed. I will briefly describe the models involving $\mathcal{F}$, the state of art and the lower-semicontinuous envelope calculation.