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In one dimension the periodic conductivity cell problem reduces toHence $a(y)(1+w^{\prime })=C$ is constant. Periodicity requires ${\int }_0^1{w}^{\prime }dy=0$, soThe effective flux average is $a^{\ast }={\int }_0^{1}a(1+w^{\prime })dy=C$, givingOne can exhibit the corrector explicitly: $w^{\prime }=a^{\ast }/a_1-1$ in phase 1 and $w^{\prime }=a^{\ast }/a_2-1$ in phase 2. Its integrated change over a period is zero, so a continuous piecewise affine periodic $w$ exists, and an additive constant fixes its mean.

This is precisely the normal component of the laminate conductivity in part (i). In one dimension there is no tangential direction: every heat-flow path crosses the phases in sequence, so only the harmonic-mean response remains.

Take $a_1,a_2>0$ and $p_1+p_2=1$. Write $g^{(k)}=\nabla u$ in phase $k$ and $h^{(k)}=a_k{g}^{(k)}$. Here $h$ is the positive conductivity flux; physical heat flux is $-h$. Continuity of temperature on a planar interface implies continuity of its tangential derivatives, while conservation gives continuity of normal heat flux. With interface normal $e_1$, these conditions areThe mean gradient is $E=p_1{g}^{(1)}+p_2{g}^{(2)}$. ThereforeBy the definition of effective conductivity, $⟨h⟩=a^{\ast }E$. ThusThe laminate conductivity is a weighted harmonic mean across the layers and a weighted arithmetic mean along them. This also explains why isotropic constituents can yield an anisotropic thermal conductivity tensor: flux must cross each resistance in sequence in the normal direction, whereas tangential transport occurs in parallel.