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  "body": "\n## Summary\n\nHomogenization is the study of partial differential equations whose coefficients oscillate on a small scale $\\varepsilon$, in the limit $\\varepsilon \\to 0$.\nThe solutions converge to the solution of a *homogenized* equation with effective, slowly varying or constant coefficients.\nFor periodic media the effective coefficients are computed from a cell problem on one period.\nIt explains, for example, the effective conductivity of a composite material made of a fine mixture of two materials.\n\n## Explanations\n\n### L1 human\n\nLook at a material made of many thin alternating layers of copper and plastic.\nAt the scale of the whole piece, heat flows as if through one uniform material with some average conductivity.\nHomogenization computes this average, which is in general not the plain arithmetic mean.\nAcross the layers, heat must cross both materials in series, and the effective conductivity is the harmonic mean, as for resistors in series.\nAlong the layers, heat flows through both in parallel, and the effective conductivity is the arithmetic mean.\n",
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