Closure and Commutability Results for $\Gamma$-Limits and the Geometric Linearization and Homogenization of Multiwell Energy Functionals
Martin Jesenko, Bernd G Schmidt · SIAM Journal on Mathematical Analysis · 2014
We consider a family of integral functionals, doubly indexed by $\varepsilon > 0$ and $j \in \mathbb{N} \cup \{\infty\}$, satisfying a uniform G\r arding-type inequality. If these functionals $\Gamma$-converge for every $j \in \mathbb{N}$ as $\varepsilon \to 0$, then the same holds also for $j = \infty$ if the densities fulfill a certain equivalence condition. In that case the $\Gamma$-limit for $j = \infty$ is recovered as the limit of the $\Gamma$-limits for finite $j$ as $j \to \infty$. We thus obtain a $\Gamma$-closure theorem for such functionals and moreover find criteria for the commutability of the $\Gamma$-limits as $\varepsilon \to 0$ and $j \to \infty$. Due to our mild growth conditions from below this result not only provides a common basic principle for a number of linearization and homogenization results in elasticity theory, but it also allows for new applications which we exemplify by proving that geometric linearization and homogenization of multiwell energy functionals commute.