= Periodic homogenization of a diffusion equation
Consider $u_t=\partial_x[a(x/\varepsilon)u_x]$, where $a$ has period one and $0<a_-\leq a(y)\leq a_+<\infty$. At scales large compared with $\varepsilon$, the one-dimensional effective <diffusion coefficient> is
$$
a_{\rm eff}=\left(\int_0^1\frac{dy}{a(y)}\right)^{-1}.
$$
The <harmonic mean> comes from flux continuity. Impose a unit mean <gradient> across one microscopic cell and let $v'(y)$ be its local gradient. A constant conservative flux requires $a(y)v'(y)=q$. The unit total change gives $1=\int_0^1v'=q\int_0^1a^{-1}$, so $q=a_{\rm eff}$. The associated cell correction is periodic because $v(y)-y$ has zero change across a period.
For two equally thick layers with coefficients $a_1,a_2>0$, this yields $a_{\rm eff}=2a_1a_2/(a_1+a_2)$. Taking $a_1=1$ and $a_2=9$ gives $1.8$, rather than the arithmetic mean $5$. The less diffusive layer requires a larger local <gradient> to carry the same flux. This explicit harmonic-mean formula is one-dimensional; a multidimensional periodic medium generally has an effective diffusion <tensor>.
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