Layered medium: harmonic normal and arithmetic tangential means

Set $c=(\rho \mathrm{/}{a}_1+(1-\rho )\mathrm{/}{a}_2{)}^{-1}$. Apply [F4] to the one-dimensional layered coefficient, with $\lambda =\min (a_1,a_2)$, $\mathrm{\Lambda }=\max (a_1,a_2)$, phase zero and ratio one. Its mean-zero solution $w\in V_{\mathrm{\#}}(0,1)$ satisfies ${\int }_0^{1}a(t)(1+w^{\mathrm{\prime }}(t))v^{\mathrm{\prime }}(t)dt=0$ for every $v\in V_{\mathrm{\#}}(0,1)$ and has mean flux $c$. By the defining periodic closure, take smooth periodic $w_{\mathrm{\ell }}\to w$ in $H^1(0,1)$ and subtract their means. Their lifts $W_{\mathrm{\ell }}(y)=w_{\mathrm{\ell }}(y_1)$ are smooth periodic, and iterated integration gives $\mathrm{\parallel }{W}_{\mathrm{\ell }}-W_k{\mathrm{\parallel }}_{H^1(Y)}=\mathrm{\parallel }{w}_{\mathrm{\ell }}-w_k{\mathrm{\parallel }}_{H^1(0,1)}$. Thus their $H^1$ limit is ${\chi }_1(y)=w(y_1)\in V_{\mathrm{\#}}(Y)$, with gradient $w^{\mathrm{\prime }}(y_1)e_1$. For every smooth periodic $v(y)$, each slice $v(\cdot ,y^{\mathrm{\prime }})$ is an admissible one-dimensional cell test after subtracting its mean, so iterated integration gives ${\int }_{Y}a(y_1)(1+w^{\mathrm{\prime }}(y_1)){\mathrm{\partial }}_{1}v(y)dy=0$. Density extends this identity to $V_{\mathrm{\#}}(Y)$; uniqueness [F2] identifies ${\chi }_1$ as the first corrector. Its mean flux is $c{e}_1$ by [F4] and iterated integration.

For each $j\ge 2$, ${\chi }_j=0$ solves the cell equation: for smooth periodic $v$, iterated integration [F3] in $y_j$ gives ${\int }_{Y}a(y_1){\mathrm{\partial }}_{j}v=0$ by equality of the opposite-face values, and density extends the identity to $V_{\mathrm{\#}}$. Its mean flux is ${\int }_{Y}a(y_1)e_j=(\rho a_1+(1-\rho )a_2)e_j$. These columns yield the displayed diagonal matrix.