of Theorem \[Theorem:K1\] and Theorem \[Theorem:K2\]. The upper bounds are proven in Section \[sectUB\], the lower bounds in Section \[seclowerbound\].

Upper bounds {#sectUB}
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The upper bounds are proven by explicit constructions. Whereas the basic construction in the proof of Theorem \[Theorem:K1\] is a direct generalization of the one by Kohn and Müller, the different scaling in Theorem \[Theorem:K2\] arises from a somewhat different construction, which exploits the vectorial nature of the problem and the invariance under rotations of the energy density. Therefore, we start with a discussion of the latter, and postpone the proof of Theorem \[Theorem:K1\] to Section \[secubtheo1\] below. In what follows, $c$ denotes a positive constant which does not depend on $\alpha$, $\varepsilon$, $L$ and $H$ and may be different from line to line. In this section we write $(x,y)$ for generic points in $\R^2$.

Proof of the upper bound in Theorem \[Theorem:K2\] {#secUBtheo2}
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The key idea is to use a number of period-doubling steps which join fine-scale oscillations close to the boundary with coarse-scale oscillations in the interior of the sample. The difference between the two cases considered and the main novelty in the present construction reside in the treatment of the period-doubling step, 