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

Abstract

Under a suitable notion of equivalence of integral densities we prove a $Γ$-closure theorem for integral functionals: The limit of a sequence of $Γ$-convergent families of such functionals is again a $Γ$-convergent family. Its $Γ$-limit is the limit of the $Γ$-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.