Calculus of Variations and Geometric Measure Theory

S. Neukamm - P. E. Stelzig

On the interplay of two-scale convergence and translation

created by stelzig on 07 Apr 2010
modified on 24 Jun 2010


Accepted Paper

Inserted: 7 apr 2010
Last Updated: 24 jun 2010

Journal: Asymptot. Anal.
Pages: 19
Year: 2010


We study the effects of translation on two-scale convergence. Given a two-scale convergent sequence $(u_\varepsilon(x))_\varepsilon$ with two-scale limit $u(x,y)$, then in general the translated sequence $(u_\varepsilon(x+t))_\varepsilon$ is no longer two-scale convergent, even though it remains two-scale convergent along suitable subsequences. We prove that any two-scale cluster point of the translated sequence is a translation of the original limit and has the form $u(x+t,y+r)$ where the microscopic translation $r$ belongs to a set that is determined solely by $t$ and the vanishing sequence $(\varepsilon)$. Finally, we apply this result to a novel homogenization problem that involves two different coordinate frames and yields a limiting behavior governed by emerging microscopic translations.

Keywords: Homogenization, Gamma-convergence, Two-scale convergence, Translation