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  "body": "\n## Statement\n\nLet $\\Omega \\subseteq \\R^n$ be a bounded open set with Lipschitz boundary, and let $W(t) = t^2 (1 - t)^2$ for $t \\in \\R$.\nFor $\\varepsilon \u003e 0$ define $F_\\varepsilon \\colon L^1(\\Omega) \\to [0, +\\infty]$ by\n$$F_\\varepsilon(u) = \\int_\\Omega \\Big( \\varepsilon \\abs{\\nabla u}^2 + \\frac{1}{\\varepsilon} W(u) \\Big) \\, dx$$\nif $u$ lies in the [[Sobolev space|M4CEZ5K9HP]] $W^{1,2}(\\Omega)$, and $F_\\varepsilon(u) = +\\infty$ otherwise.\nDefine $F \\colon L^1(\\Omega) \\to [0, +\\infty]$ by\n$$F(u) = c_W \\operatorname{Per}(E; \\Omega), \\qquad c_W = 2 \\int_0^1 \\sqrt{W(s)} \\, ds = \\frac13,$$\nif $u = \\mathbf{1}_E$ is the characteristic function of a set $E \\subseteq \\Omega$ of finite [[perimeter|Y68BHXZJBV]] $\\operatorname{Per}(E; \\Omega)$ in $\\Omega$, and $F(u) = +\\infty$ otherwise.\nThen $F_\\varepsilon$ [[Γ-converges|NW9MM370A5]] to $F$ in the [[Lp space|SASP14MA1Z]] $L^1(\\Omega)$ as $\\varepsilon \\to 0$.\n\n## Remarks\n\nThe lower bound comes from the inequality $a^2 + b^2 \\ge 2ab$, which gives $\\varepsilon \\abs{\\nabla u}^2 + W(u)/\\varepsilon \\ge 2 \\sqrt{W(u)} \\abs{\\nabla u} = \\abs{\\nabla (\\phi \\circ u)}$ with $\\phi(t) = 2 \\int_0^t \\sqrt{W(s)} \\, ds$.\nThe lower semicontinuity of the total variation then bounds the limit from below by the perimeter of $E$ times $\\phi(1) - \\phi(0) = c_W$.\nFor the recovery sequence, one puts the optimal one-dimensional profile across $\\partial E$.\nHere $q(s) = 1/(1 + e^{-s})$ solves $q' = \\sqrt{W(q)}$.\nWhen $\\partial E$ is smooth, $u_\\varepsilon(x) = q(d(x)/\\varepsilon)$ works, with $d$ the signed distance to $\\partial E$.\nThe same result holds for other continuous double-well potentials $W \\ge 0$ that vanish exactly at two points and grow suitably at infinity.\nThen $c_W$ is the integral of $2\\sqrt{W}$ between the two wells.\n\nThe theorem is often combined with a volume constraint $\\int_\\Omega u \\, dx = m$.\nThen the [[fundamental theorem of Γ-convergence|23RK8EZ20P]] shows that minimizers of $F_\\varepsilon$ converge, along subsequences, to characteristic functions of sets of least perimeter with the given volume.\n\n## Explanations\n\n### L1 human\n\nA mixture of two fluids, such as oil and water, can be described by a function $u$ that is near $0$ in one phase and near $1$ in the other.\nThe energy $F_\\varepsilon$ punishes values between the phases through $W$ and punishes fast changes through the gradient term.\nWhen the parameter $\\varepsilon$ becomes small, the transition layer gets thin and its energy concentrates on the interface.\nThe theorem says that in the limit the energy is a constant times the area of the interface, so minimizing the energy means minimizing the area of the interface.\n\n### L2 human\n\nThis is the model example of [[Γ-convergence|NW9MM370A5]] and the reason it became popular.\nIt connects the diffuse-interface energy behind the [[Allen–Cahn equation|7S813PEV69]] with the sharp-interface perimeter functional of geometric measure theory.\nThe constant $c_W$ is the energy per unit area of the optimal one-dimensional transition profile.\nThe result is due to Luciano Modica and Stefano Mortola, and Modica used it to study phase transitions with a volume constraint.\n",
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