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  "body": "\n## Example\n\nLet $\\Omega \\subseteq \\R^2$ be a bounded open set and let $\\varepsilon \u003e 0$.\nThe simplified *Ginzburg–Landau functional* acts on maps $u \\colon \\Omega \\to \\C$ in the [[Sobolev space|M4CEZ5K9HP]] $H^1(\\Omega; \\C)$ by\n$$E_\\varepsilon(u) = \\int_\\Omega \\Big( \\frac12 \\abs{\\nabla u}^2 + \\frac{1}{4 \\varepsilon^2} \\big(1 - \\abs{u}^2\\big)^2 \\Big) \\, dx.$$\nThe first term penalizes variation of $u$, and the second term pushes $\\abs{u}$ towards $1$.\n\nCheck of the zero set: $E_\\varepsilon(u) = 0$ holds exactly when $\\nabla u = 0$ and $\\abs{u} = 1$ almost everywhere.\nWhen $\\Omega$ is connected, this means that $u$ is a constant of modulus $1$.\n\nEuler–Lagrange equation: the derivative of $E_\\varepsilon$ at $u$ in a direction $v$ is\n$$\\int_\\Omega \\Big( \\operatorname{Re}\\big(\\nabla u \\cdot \\overline{\\nabla v}\\big) - \\frac{1}{\\varepsilon^2} \\big(1 - \\abs{u}^2\\big) \\operatorname{Re}\\big(u \\, \\overline{v}\\big) \\Big) \\, dx.$$\nSo critical points solve $-\\Delta u = \\varepsilon^{-2} u \\big(1 - \\abs{u}^2\\big)$ weakly, an equation of [[Ginzburg–Landau type|713MRSZ9XA]] without magnetic field.\n\n## Remarks\n\nLet $\\Omega$ be smooth and simply connected.\nImpose a boundary condition $u = g$ on $\\partial \\Omega$, where $g \\colon \\partial \\Omega \\to S^1$ is smooth with winding number $d \\ne 0$.\nNo map in $H^1$ with these boundary values has $\\abs{u} = 1$ everywhere, so minimizers must have zeros, called vortices.\nThe minimal energy grows like $\\pi \\abs{d} \\log(1/\\varepsilon)$ as $\\varepsilon \\to 0$.\nThe study of the limit $\\varepsilon \\to 0$, by direct estimates and by [[Γ-convergence|NW9MM370A5]] of $E_\\varepsilon / \\log(1/\\varepsilon)$, locates the vortices.\nThe full functional of Ginzburg–Landau theory also contains a magnetic vector potential.\n\n## Explanations\n\n### L1 human\n\nThe functional models a superconductor or a superfluid by a complex number $u$ at each point of a plane domain.\nThe material prefers $\\abs{u} = 1$, and it prefers $u$ not to change.\nIf the boundary values wind around the circle, both wishes cannot be met, and $u$ must vanish at a few points.\nNear these points, the vortices, $u$ turns around once like the hands of a clock.\nAs $\\varepsilon$ gets small, the vortices shrink to points and their positions are what the limit theory describes.\n",
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