= Solution
Write $R=\Lambda/\lambda$. The coefficient is measurable, so interpret the <divergence-form elliptic operator> weakly: $\int A u'\phi'=0$ for compactly supported $H^1$ tests. Testing with $\phi=\zeta^2u$ gives
$$
\lambda\int\zeta^2|u'|^2\le2\Lambda\int\zeta|u'||u||\zeta'|\le2\Lambda\left(\int\zeta^2|u'|^2\right)^{1/2}\left(\int u^2|\zeta'|^2\right)^{1/2}.
$$
Divide if the first integral is positive; the zero case is immediate. Since $\zeta=1$ on the inner interval,
$$
\boxed{\int_{-1/2}^{1/2}|u'|^2\le4R^2\int_{-1}^1u^2|\zeta'|^2.}
$$
This <Caccioppoli inequality> applies in particular to the smooth solutions in the question. No derivative of $A$ is taken.
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