PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Classical solutions are viscosity solutions and differentiable viscosity solutions solve the equation pointwise

Statement

Let $O \subseteq R^{n}$ be open, $T > 0$ , let $H : O \times [0, T] \times R^{n} \to R$ be continuous, let $u_{0} : O \to R$ be continuous, and let $Z = O \times (0, T)$ . (1) If $u \in C^{1} (Z) \cap C^{0} (\overline{Z})$ satisfies $u_{t} + H (x, t, D u) = 0$ pointwise on $Z$ and $u (x, 0) = u_{0} (x)$ on $O$ (a classical solution in the sense of The Hamilton--Jacobi Cauchy problem and its classical solutions), then $u$ is a viscosity solution of the Cauchy problem. (2) Conversely, if $u : Z \to R$ is a continuous viscosity solution and $u$ is differentiable at a point $z_{0} \in Z$ , then $u_{t} (z_{0}) + H (z_{0}, D u (z_{0})) = 0.$ In particular, a viscosity solution of class $C^{1} (Z)$ is a classical solution of the equation on $Z$ (its initial trace being part of the Cauchy-problem notion). No choice principle is used.

Facts & Assumptions

Given: Open $O \subseteq R^{n}$ , $T > 0$ , continuous $H : O \times [0, T] \times R^{n} \to R$ , continuous $u_{0} : O \to R$ , $Z = O \times (0, T)$ .

[F1]

$w$ is a viscosity subsolution of $u_{t} + H (x, t, D u) = 0$ in $Z$ when $ϕ_{t} (z_{0}) + H (z_{0}, D ϕ (z_{0})) \leq 0$ at every local maximum of $w - ϕ$ , $ϕ \in C^{1} (Z)$ ; a supersolution satisfies the reverse inequality at every local minimum; a subsolution of the Cauchy problem also satisfies ${lim sup}_{(y, s) \to (x, 0), s > 0, y \in O} w (y, s) \leq u_{0} (x)$ and a supersolution ${lim inf}_{(y, s) \to (x, 0), s > 0, y \in O} w (y, s) \geq u_{0} (x)$ at every $x \in O$ (Viscosity subsolutions and supersolutions of a first-order equation and of the Cauchy problem).

[F2]

If a real-valued function on an open set is differentiable at a point where it has a local maximum or a local minimum, then its total derivative vanishes there (Fermat's theorem: an interior differentiable local extremum has zero gradient).

[F3]

For an upper semicontinuous $u$ : $u$ is a viscosity subsolution of the equation in $Z$ if and only if $p_{t} + H (z_{0}, p_{x}) \leq 0$ for every $z_{0} \in Z$ and every $p \in D^{+} u (z_{0})$ ; for a lower semicontinuous $v$ : $v$ is a supersolution if and only if $p_{t} + H (z_{0}, p_{x}) \geq 0$ for every $p \in D^{-} v (z_{0})$ (Viscosity testing by first-order jets, and closure of the jet inequality).

[F4]

Total differentiability of $u$ at $z_{0}$ with derivative $D u (z_{0})$ means $u (z_{0} + h) = u (z_{0}) + D u (z_{0}) h + o (∣ h ∣)$ ; consequently $D u (z_{0}) \in D^{+} u (z_{0}) \cap D^{-} u (z_{0})$ (The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ).

Proof

technique · Fermat's theorem at a $C^1$ contact for the classical direction, and the jet characterisation for the converse

1.1F1F2algebra

Classical solutions are viscosity solutions. Let $u \in C^{1} (Z) \cap C^{0} (\overline{Z})$ solve the equation pointwise, and let $ϕ \in C^{1} (Z)$ with $u - ϕ$ having a local maximum at $z_{0} \in Z$ . Then $u - ϕ$ is differentiable at $z_{0}$ and has a local extremum there, so $D ϕ (z_{0}) = D u (z_{0})$ by [F2]; hence $ϕ_{t} (z_{0}) + H (z_{0}, D ϕ (z_{0})) = u_{t} (z_{0}) + H (z_{0}, D u (z_{0})) = 0 \leq 0$ . At a local minimum the same computation gives $ϕ_{t} (z_{0}) + H (z_{0}, D ϕ (z_{0})) = 0 \geq 0$ . Since $u$ extends continuously to $\overline{Z}$ with $u (x, 0) = u_{0} (x)$ , the relaxed initial conditions of [F1] hold; so $u$ is both a subsolution and a supersolution of the Cauchy problem.

1.2F1F3F4

Differentiable viscosity solutions solve the equation pointwise. Let $u : Z \to R$ be continuous and differentiable at $z_{0} \in Z$ . By [F4], $D u (z_{0}) \in D^{+} u (z_{0}) \cap D^{-} u (z_{0})$ ; since $u$ is a continuous viscosity solution, it equals its envelopes (Discontinuous viscosity solutions through the two envelopes), so $u$ is both an upper semicontinuous subsolution and a lower semicontinuous supersolution in the sense of [F1]. Applying [F3] at $z_{0}$ with $p = D u (z_{0})$ gives both $u_{t} (z_{0}) + H (z_{0}, D u (z_{0})) \leq 0$ and $u_{t} (z_{0}) + H (z_{0}, D u (z_{0})) \geq 0$ , hence equality.

2.1step 1.1step 1.2F1∎

Conclusion. If in addition $u \in C^{1} (Z)$ , then the pointwise equation holds at every $z_{0} \in Z$ by step 1.2, and by step 1.1 the classical solution is a viscosity solution; the initial trace of a Cauchy-problem viscosity solution is the datum by [F1], so a $C^{1}$ viscosity solution solves the equation classically on $Z$ and extends continuously to the initial face. To be a classical solution of the Cauchy problem in the stronger sense of The Hamilton--Jacobi Cauchy problem and its classical solutions, it must additionally extend continuously to all of $\overline{Z}$ .

Remarks

What is not claimed. Part (2) presupposes that the viscosity solution is differentiable at the point; viscosity solutions of Hamilton--Jacobi equations are typically not differentiable everywhere, and the proposition says nothing about the nondifferentiable set.
Choice. Both directions are pointwise computations with the definitions; no selection principle occurs.

Depends on

Used by

Dependency tree · two levels

25 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Hung Vinh Tran, Hamilton--Jacobi Equations: Theory and Applications, 2020 preliminary author manuscript of AMS Graduate Studies in Mathematics 213 (complete text) (standard reference, not scraped)
Michael G. Crandall, Hitoshi Ishii and Pierre-Louis Lions, User's guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society 27 (1992), 1--67 (complete article) (standard reference, not scraped)
Alberto Bressan, Viscosity Solutions of Hamilton--Jacobi Equations and Optimal Control Problems, complete author lecture notes, Penn State University (PDF records Fall 2019 revision) (standard reference, not scraped)