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  "body": "\n## Definition\n\nLet $d \\ge 1$ and let $W = W^{1,\\infty}(\\R^d; \\R^d)$ be the Banach space of bounded Lipschitz vector fields $\\theta \\colon \\R^d \\to \\R^d$, with norm $\\norm{\\theta}_W = \\norm{\\theta}_{L^\\infty} + \\norm{\\nabla \\theta}_{L^\\infty}$.\nLet $\\mathcal{O}$ be a class of open subsets of $\\R^d$, and let $J \\colon \\mathcal{O} \\to \\R$ be a shape [[functional|1EQM0DJ4NC]].\nLet $\\Omega \\in \\mathcal{O}$ be such that $(\\mathrm{Id} + \\theta)(\\Omega) \\in \\mathcal{O}$ whenever $\\norm{\\theta}_W$ is small enough.\nThen $J$ is *shape differentiable* at $\\Omega$ if the map $\\theta \\mapsto J((\\mathrm{Id} + \\theta)(\\Omega))$ is [[Fréchet differentiable|08EVNYZF42]] at $\\theta = 0$.\nIts Fréchet derivative, a bounded linear map $J'(\\Omega) \\colon W \\to \\R$, is the *shape derivative* of $J$ at $\\Omega$.\n\n## Remarks\n\nFor $\\norm{\\theta}_W \u003c 1$ the map $\\mathrm{Id} + \\theta$ is a bi-Lipschitz homeomorphism of $\\R^d$, so $(\\mathrm{Id} + \\theta)(\\Omega)$ is again open.\nExample: the volume $J(\\Omega) = \\abs{\\Omega}$ of a bounded open set.\nBy the change of variables formula $J((\\mathrm{Id} + \\theta)(\\Omega)) = \\int_\\Omega \\det(I + \\nabla \\theta) \\, dx$.\nSince $\\det(I + \\nabla \\theta) = 1 + \\operatorname{div} \\theta + O(\\norm{\\nabla\\theta}^2)$, the shape derivative is $J'(\\Omega)(\\theta) = \\int_\\Omega \\operatorname{div} \\theta \\, dx$.\nIf $\\Omega$ has a $C^1$ boundary with outward unit [[normal vector|0XYP2KKSPC]] $n$, the divergence theorem gives $J'(\\Omega)(\\theta) = \\int_{\\partial \\Omega} \\theta \\cdot n \\, ds$.\nNon-example: $J(\\Omega) = \\abs{\\abs{\\Omega} - 1}$ at a set $\\Omega$ with $\\abs{\\Omega} = 1$ is not shape differentiable.\nTake $\\Omega$ bounded and a smooth function $\\chi$ with compact support that equals $1$ on a neighbourhood of $\\overline{\\Omega}$.\nAlong $\\theta_s(x) = s \\chi(x) x$ one gets $(\\mathrm{Id} + \\theta_s)(\\Omega) = (1 + s) \\Omega$ and $J = \\abs{(1 + s)^d - 1}$, which has a corner at $s = 0$.\nFor smooth $\\Omega$ and smooth $J$, the Hadamard structure theorem gives $J'(\\Omega)(\\theta) = \\int_{\\partial \\Omega} g \\, (\\theta \\cdot n) \\, ds$ for a function $g$ on $\\partial \\Omega$, the shape gradient.\n",
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