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  "body": "\n## Statement\n\nLet $H \\colon \\R^n \\to \\R$ be a smooth [[convex function|07Q7CNM5WF]] that is superlinear: $H(p)/\\abs{p} \\to \\infty$ as $\\abs{p} \\to \\infty$.\nLet $L \\colon \\R^n \\to \\R$ be its [[Legendre transform|ZQYMWVT1TT]], $L(q) = \\sup_{p \\in \\R^n} \\big(p \\cdot q - H(p)\\big)$.\nLet $g \\colon \\R^n \\to \\R$ be [[Lipschitz continuous|0AW1E6YTJY]].\nFor $x \\in \\R^n$ and $t \u003e 0$ define\n$$u(x, t) = \\min_{y \\in \\R^n} \\left( t\\, L\\!\\left(\\frac{x - y}{t}\\right) + g(y) \\right),$$\nand set $u(x, 0) = g(x)$.\nThen the minimum is attained, and $u$ is Lipschitz continuous on $\\R^n \\times [0, \\infty)$.\nMoreover, $u$ is differentiable at almost every point of $\\R^n \\times (0, \\infty)$, and at each such point\n$$\\partial_t u + H(\\nabla u) = 0.$$\nSo $u$ is a solution almost everywhere of the [[Hamilton–Jacobi equation|5TRXQKP7DY]] with initial value $g$.\n\n## Remarks\n\nThe formula comes from the calculus of variations.\nThe value $u(x, t)$ is the least cost $\\int_0^t L(\\dot w(s))\\, ds + g(w(0))$ over paths $w$ with $w(t) = x$.\nBy convexity of $L$, Jensen's inequality shows that straight paths are optimal, which gives the minimum over the starting point $y$.\nThe almost everywhere differentiability comes from the Lipschitz bound and Rademacher's theorem.\nThe equation itself follows from the semigroup property $u(x, t) = \\min_y \\big( (t - s) L((x - y)/(t - s)) + u(y, s) \\big)$ for $0 \\le s \u003c t$.\n\nExample: for $H(p) = \\tfrac12 \\abs{p}^2$ one has $L(q) = \\tfrac12 \\abs{q}^2$, and the formula reads $u(x, t) = \\min_y \\big( \\abs{x - y}^2/(2t) + g(y) \\big)$.\nFor $g(y) = \\abs{y}$ in dimension $1$ this gives $u(x, t) = \\abs{x} - t/2$ for $\\abs{x} \\ge t$ and $u(x, t) = x^2/(2t)$ for $\\abs{x} \\le t$.\n\nSolutions almost everywhere are far from unique, so the formula alone does not settle the initial value problem.\nUnder the hypotheses above the Hopf–Lax function is also a [[viscosity solution|H1ERJ3T75T]], and this is the property that singles it out.\n\n## Explanations\n\n### L1 human\n\nThe Hopf–Lax formula writes down a solution of a nonlinear first-order equation with no differential equation solving at all: one only takes a minimum.\nTo get $u$ at a point $x$ and time $t$, one looks at all starting points $y$.\nFor each, one adds the start value $g(y)$ to the cost of travelling from $y$ to $x$ in time $t$, and keeps the cheapest.\nThe solution stays meaningful even after smooth solutions break down and corners appear.\n\n### L2 human\n\nThe formula is the starting point of the theory of weak solutions of Hamilton–Jacobi equations in Evans' textbook.\nDifferentiating it in $x$ in dimension $1$ gives the Lax–Oleinik formula for the entropy solution of a scalar [[conservation law|8RA7AMWSAF]] with convex flux, since $v = \\partial_x u$ solves $v_t + H(v)_x = 0$.\nFor $H(p) = \\tfrac12 \\abs{p}^2$ the formula is a Moreau envelope of $g$, and it is also the limit as $\\nu \\to 0$ of the solutions of the viscous equation obtained by the [[Cole–Hopf transformation|VBMQ9SJ3DH]].\n",
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