On the Gamma convergence of functionals defined over pairs of measures and energy-measures

TL;DR

This paper introduces a new framework for analyzing the Gamma convergence of functionals over pairs of measures and energy-measures, with applications to epitaxial growth, revealing how non-continuous energy limits affect relaxation.

Contribution

It develops a general theory for Gamma convergence of measure-based functionals, linking their limits to underlying energies and accounting for non-continuous energy effects.

Findings

The framework identifies Gamma limits from underlying energies.

Non-continuous energy limits influence the relaxation process.

Applications to epitaxial growth demonstrate the theory's utility.

Abstract

A novel general framework for the study of $\Gamma$-convergence of functionals defined over pairs of measures and energy-measures is introduced. This theory allows us to identify the $\Gamma$-limit of these kind of functionals by knowing the $\Gamma$-limit of the underlining energies. In particular, the interaction between the functionals and the underlining energies results, in the case these latter converge to a non continuous energy, in an additional effect in the relaxation process. This study was motivated by a question in the context of epitaxial growth evolution with adatoms. Interesting cases of application of the general theory are also presented.