# Equilibria configurations for epitaxial crystal growth with adatoms

## TL;DR

This paper analyzes a surface energy model for epitaxial crystal growth incorporating adatoms, discussing critical points, minimizers, and relaxation properties to advance understanding of morphological evolution in materials science.

## Contribution

It introduces a new analytical framework for the surface energy involving adatoms, including existence, uniqueness, and relaxation of minimizers in crystal growth models.

## Key findings

- Existence and uniqueness of regular critical points established.
- Characterization of the relaxation of the energy functional.
- Insights into the morphological evolution driven by adatoms.

## Abstract

The behavior of a surface energy $\mathcal F(E,u)$, where $E$ is a set of finite perimeter and $u\in L^1(\partial^* E, \mathbb R_+)$ is studied. These energies have been recently considered in the context of materials science to derive a new model in crystal growth that takes into account the effect of atoms freely diffusing on the surface (called adatoms), which are responsible for morphological evolution through an attachment and detachment process. Regular critical points, existence and uniqueness of minimizers are discussed and the relaxation of $\mathcal F$ in a general setting under the $L^1$ convergence of sets and the vague convergence of measures is characterized. This is part of an ongoing project aimed at an analytical study of diffuse interface approximations of the associated evolution equations.