Closure and commutability results for Gamma-limits and the geometric linearization and homogenization of multi-well energy functionals

TL;DR

This paper establishes a $ ext{Gamma}$-closure theorem for integral functionals, showing that limits of $ ext{Gamma}$-convergent families remain $ ext{Gamma}$-convergent, and demonstrates the commutativity of geometric linearization and homogenization in multi-well energy functionals.

Contribution

It introduces a $ ext{Gamma}$-closure theorem for integral functionals and proves the commutativity of geometric linearization and homogenization for multi-well energies.

Findings

The limit of $ ext{Gamma}$-convergent families is again $ ext{Gamma}$-convergent.

The $ ext{Gamma}$-limit equals the limit of the $ ext{Gamma}$-limits of the original problems.

Geometric linearization and homogenization commute for multi-well energy functionals.

Abstract

Under a suitable notion of equivalence of integral densities we prove a $\Gamma$-closure theorem for integral functionals: The limit of a sequence of $\Gamma$-convergent families of such functionals is again a $\Gamma$-convergent family. Its $\Gamma$-limit is the limit of the $\Gamma$-limits of the original problems. This result not only provides a common basic principle for a number of linearization and homogenization results in elasticity theory. It also allows for new applications as we exemplify by proving that geometric linearization and homogenization of multi-well energy functionals commute.