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  "body": "\n## Example\n\nTake the one-dimensional one-phase [[Stefan problem|3B0HB3DHW1]] in units with thermal diffusivity and conductivity equal to $1$ and latent heat $L \u003e 0$.\nThe liquid occupies $0 \u003c x \u003c s(t)$, with $s(0) = 0$ and $s$ increasing.\nThe end $x = 0$ is heated, with a given temperature $u(0, t) \u003e 0$, and the temperature $u$ satisfies:\n$$\\partial_t u = \\partial_x^2 u \\ \\text{ for } 0 \u003c x \u003c s(t), \\qquad u(s(t), t) = 0, \\qquad L s'(t) = -\\partial_x u(s(t), t).$$\nExtend $u$ by $0$ to the solid region $x \u003e s(t)$, and define the integrated temperature\n$$w(x, t) = \\int_0^t u(x, \\tau) \\, d\\tau.$$\nThen $w$ solves the obstacle-type system\n$$w \\ge 0, \\qquad \\partial_t w - \\partial_x^2 w + L \\ge 0, \\qquad w \\, \\big(\\partial_t w - \\partial_x^2 w + L\\big) = 0.$$\n\nCheck in the liquid: fix $x \u003e 0$ and let $\\theta(x)$ be the time with $s(\\theta(x)) = x$, so $u(x, \\tau) = 0$ for $\\tau \\le \\theta(x)$.\nThen $w(x, t) = \\int_{\\theta(x)}^t u(x, \\tau) \\, d\\tau$ for $t \u003e \\theta(x)$.\nDifferentiating twice in $x$ and using $u(x, \\theta(x)) = 0$ gives $\\partial_x^2 w = \\int_{\\theta(x)}^t \\partial_x^2 u \\, d\\tau - \\partial_x u(x, \\theta(x)) \\, \\theta'(x)$.\nThe Stefan condition and $s'(\\theta(x)) \\, \\theta'(x) = 1$ give $-\\partial_x u(x, \\theta(x)) \\, \\theta'(x) = L$.\nThe heat equation turns the integral into $u(x, t) = \\partial_t w$, so $\\partial_t w - \\partial_x^2 w = -L$ where $w \u003e 0$.\nIn the solid, $w = 0$, so $\\partial_t w - \\partial_x^2 w + L = L \u003e 0$.\n\n## Remarks\n\nFor each time $t$, the system is equivalent to a [[variational inequality|6QS7X5GVMY]] over the convex set $K = \\set{v \\ge 0}$:\n$$\\int \\big(\\partial_t w - \\partial_x^2 w + L\\big) (v - w) \\, dx \\ge 0 \\quad \\text{for all } v \\in K.$$\nIndeed, where $w \u003e 0$ the first factor vanishes, and where $w = 0$ both factors are nonnegative.\nSo the free boundary $x = s(t)$ becomes the boundary of the set $\\set{w \u003e 0}$, as in the [[obstacle problem|GXN2NEHC8X]], and it no longer appears explicitly among the unknowns.\nThis change of unknown is due to Georges Duvaut, and it allows existence and regularity results for the Stefan problem through the theory of variational inequalities.\n\n## Explanations\n\n### L1 human\n\nIn a melting problem, the moving boundary between ice and water is unknown, which makes the equations hard to handle.\nThe trick is to add up the temperature over time at each place.\nThe new function is zero where the ice has not melted yet and positive in the water.\nIt solves a problem of the same type as a membrane pressed against an obstacle, where the contact set is found as part of the solution.\n",
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