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  "body": "\n## Definition\n\nLet $Y = [0, 1)^n$ be the unit cell and let $a = (a_{ij}) \\colon \\R^n \\to \\R^{n \\times n}$ be measurable and $Y$-periodic.\nAssume $a$ is uniformly elliptic: for constants $0 \u003c \\alpha \\le \\beta$, $a(y)\\xi \\cdot \\xi \\ge \\alpha \\abs{\\xi}^2$ and $\\abs{a(y)\\xi} \\le \\beta \\abs{\\xi}$ for all $y, \\xi \\in \\R^n$.\nLet $\\Omega \\subseteq \\R^n$ be a bounded open set and $f \\in L^2(\\Omega)$.\nFor $\\varepsilon \u003e 0$, let $u_\\varepsilon \\in H^1_0(\\Omega)$ be the [[weak solution|SJZX81A783]] of the [[elliptic|B4TTNHSTRX]] problem\n$$-\\operatorname{div}\\big(a(x/\\varepsilon) \\nabla u_\\varepsilon\\big) = f \\text{ in } \\Omega, \\qquad u_\\varepsilon = 0 \\text{ on } \\partial\\Omega.$$\nFor $j = 1, \\dots, n$, the *cell problem* asks for a $Y$-periodic $\\chi_j \\in H^1_{\\mathrm{loc}}(\\R^n)$ with mean $0$ over $Y$ and\n$$-\\operatorname{div}_y\\big(a(y)(e_j + \\nabla_y \\chi_j)\\big) = 0 \\text{ in } \\R^n,$$\nwhere $e_j$ is the $j$-th unit vector.\nThe *homogenized matrix* is the constant matrix $a^0$ with entries\n$$a^0_{ij} = \\int_Y e_i \\cdot a(y)\\big(e_j + \\nabla_y \\chi_j(y)\\big) \\, dy,$$\nand the *homogenized problem* is $-\\operatorname{div}(a^0 \\nabla u_0) = f$ in $\\Omega$, $u_0 = 0$ on $\\partial\\Omega$.\n*Homogenization* is the passage from the problems for $u_\\varepsilon$ to the homogenized problem as $\\varepsilon \\to 0$.\n\n## Remarks\n\nThe basic theorem of periodic homogenization states that $a^0$ is again uniformly elliptic and that $u_\\varepsilon \\to u_0$ weakly in the [[Sobolev space|M4CEZ5K9HP]] $H^1_0(\\Omega)$ and strongly in $L^2(\\Omega)$.\nThe formula for $a^0$ comes from the two-scale expansion $u_\\varepsilon(x) \\approx u_0(x) + \\varepsilon \\sum_j \\chi_j(x/\\varepsilon) \\partial_j u_0(x)$.\n\nExample: in one dimension, let $a$ be a $1$-periodic function with $\\alpha \\le a \\le \\beta$.\nThe cell problem gives $a(y)(1 + \\chi') = c$ for a constant $c$, and the mean-zero condition $\\int_0^1 \\chi' = 0$ gives $c = \\big(\\int_0^1 a^{-1} \\, dy\\big)^{-1}$.\nSo $a^0 = \\big(\\int_0^1 a^{-1} \\, dy\\big)^{-1}$ is the harmonic mean of $a$.\nFor $a$ equal to $1$ on half the period and $3$ on the other half, $a^0 = 3/2$, while the arithmetic mean is $2$.\n\nNon-example: the naive guess $a^0 = \\int_Y a \\, dy$, the arithmetic mean, is wrong in general, as the example shows; the weak limit of $a(x/\\varepsilon) \\nabla u_\\varepsilon$ is not the product of the weak limits.\n\n## Explanations\n\n### L1 human\n\nHomogenization replaces a finely striped material by one uniform material that behaves the same at large scale.\nThe effective coefficients come from solving one small problem on a single period of the stripes.\n",
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