Partial differential equations First-order equations and conservation laws Theorem

Hopf–Lax formula

Builds on Hamilton–Jacobi equation Convex function Legendre transformation Lipschitz condition

Statement

Let $H \colon \R^n \to \R$ be a smooth convex function that is superlinear: $H(p)/\abs{p} \to \infty$ as $\abs{p} \to \infty$ . Let $L \colon \R^n \to \R$ be its Legendre transform, $L(q) = \sup_{p \in \R^n} \big(p \cdot q - H(p)\big)$ . Let $g \colon \R^n \to \R$ be Lipschitz continuous. For $x \in \R^n$ and $t > 0$ define $u(x, t) = \min_{y \in \R^n} \left( t\, L\!\left(\frac{x - y}{t}\right) + g(y) \right),$ and set $u(x, 0) = g(x)$ . Then the minimum is attained, and $u$ is Lipschitz continuous on $\R^n \times [0, \infty)$ . Moreover, $u$ is differentiable at almost every point of $\R^n \times (0, \infty)$ , and at each such point $\partial_t u + H(\nabla u) = 0.$ So $u$ is a solution almost everywhere of the Hamilton–Jacobi equation with initial value $g$ .

Remarks

The formula comes from the calculus of variations. The value $u(x, t)$ is the least cost $\int_0^t L(\dot w(s))\, ds + g(w(0))$ over paths $w$ with $w(t) = x$ . By convexity of $L$ , Jensen's inequality shows that straight paths are optimal, which gives the minimum over the starting point $y$ . The almost everywhere differentiability comes from the Lipschitz bound and Rademacher's theorem. The 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 $0 \le s < t$ .

Example: for $H(p) = \tfrac12 \abs{p}^2$ one has $L(q) = \tfrac12 \abs{q}^2$ , and the formula reads $u(x, t) = \min_y \big( \abs{x - y}^2/(2t) + g(y) \big)$ . For $g(y) = \abs{y}$ 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 $\abs{x} \le t$ .

Solutions almost everywhere are far from unique, so the formula alone does not settle the initial value problem. Under the hypotheses above the Hopf–Lax function is also a viscosity solution, and this is the property that singles it out.

Going deeper

For a student

The formula is the starting point of the theory of weak solutions of Hamilton–Jacobi equations in Evans' textbook. Differentiating it in $x$ in dimension $1$ gives the Lax–Oleinik formula for the entropy solution of a scalar conservation law with convex flux, since $v = \partial_x u$ solves $v_t + H(v)_x = 0$ . For $H(p) = \tfrac12 \abs{p}^2$ 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.

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Tag	`2JD72JT9FQ`
URI	`m2://2JD72JT9FQ`
Kind	statement, theorem
Writing	W0 stub
MSC	35F21, 35F25, 49L25
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About	Hamilton–Jacobi equation
Uses	Hamilton–Jacobi equation, Convex function, Legendre transformation, Lipschitz condition
Changed	2 November 2026