CorollaryStatement: 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.

Uniqueness and sup-norm contraction for the Cauchy problem

Statement

Let $n \geq 1$ , $T > 0$ , and let $H : R^{n} \times [0, T] \times R^{n} \to R$ satisfy the Lipschitz conditions of part (a) of Comparison for first-order Hamilton--Jacobi equations. (1) If $u, v$ are bounded viscosity solutions of the Cauchy problem in $Z = R^{n} \times (0, T)$ with the same bounded continuous initial datum $u_{0}$ , then $u = v$ on $Z$ ; in particular the classical and the Hopf--Lax solutions of later sections are the unique ones in the bounded class whenever $H$ satisfies those conditions. (2) More generally, if $u, v$ are bounded viscosity solutions with initial data $u_{0}, v_{0}$ , then for every $T > 0$ $\sup_{z \in Z} (u - v) \leq \max (0, \sup_{x \in R^{n}} (u_{0} (x) - v_{0} (x))),$ and applying the same bound to $v - u$ gives $\sup_{Z} ∣ u - v ∣ \leq \sup_{R^{n}} ∣ u_{0} - v_{0} ∣$ when the initial difference is bounded. In particular the solution operator is a contraction in the supremum norm on the bounded initial data. No choice principle is used.

Facts & Assumptions

Given: A Hamiltonian $H$ satisfying the Lipschitz conditions of comparison case (a), $T > 0$ , bounded viscosity solutions $u, v$ of the Cauchy problem in $Z = R^{n} \times (0, T)$ with bounded continuous data $u_{0}, v_{0}$ .

[F1]

Comparison, case (a): if $U$ is a bounded upper semicontinuous subsolution and $V$ a bounded lower semicontinuous supersolution on the closed slab with $U (x, 0) \leq V (x, 0)$ pointwise, then $U \leq V$ on $Z$ (Comparison for first-order Hamilton--Jacobi equations).

[F2]

For a bounded viscosity solution $u$ , its upper envelope $u^{*}$ is a bounded upper semicontinuous subsolution and its lower envelope $u_{*}$ is a bounded lower semicontinuous supersolution, each satisfying the corresponding relaxed initial inequality (Discontinuous viscosity solutions through the two envelopes). Adding a constant $K$ shifts both envelopes by $K$ and preserves their one-sided viscosity inequalities because $H$ is independent of the unknown (Discontinuous viscosity solutions through the two envelopes, Comparison for first-order Hamilton--Jacobi equations for the equation class).

[F3]

Boundedness of $u, v$ and of their continuous initial data is assumed in the statement and Given, so the displayed suprema are finite. The relaxed joint initial limsup/liminf conditions are part of Discontinuous viscosity solutions through the two envelopes, as recorded in [F2]. No uniform-continuity hypothesis is needed for this comparison consequence.

Proof

technique · comparison applied twice, plus the constant-shift invariance of the equation

1.1F1F2F3

Uniqueness. Let $u, v$ be bounded viscosity solutions with the same datum $u_{0}$ . By [F2], $u^{*}$ is a bounded upper semicontinuous subsolution and $v_{*}$ is a bounded lower semicontinuous supersolution. Extend them to the initial face by $U (x, 0) = u_{0} (x)$ and $V (x, 0) = u_{0} (x)$ . Their relaxed initial inequalities and continuity of $u_{0}$ make $U$ upper semicontinuous and $V$ lower semicontinuous on the closed slab, with ordered pointwise initial values. Comparison [F1] gives $u^{*} \leq v_{*}$ on $Z$ . Since $u \leq u^{*}$ and $v_{*} \leq v$ , this yields $u \leq v$ . Applying the same argument to $(v, u)$ gives $v \leq u$ , hence $u = v$ on $Z$ ; in fact all four envelopes and functions coincide. In particular, whenever a classical or Hopf--Lax solution is known to be a bounded viscosity solution of the same Cauchy problem, it is the unique bounded solution.

1.2F1F2F3algebra

The one-sided bound for general data. Put $K : = \max {0, \sup_{R^{n}} (u_{0} - v_{0})} < \infty$ . By [F2], $u^{*}$ is a bounded upper semicontinuous subsolution and $(v + K)_{*} = v_{*} + K$ is a bounded lower semicontinuous supersolution. Extend these envelopes to $t = 0$ by $u_{0}$ and $v_{0} + K$ , respectively; their relaxed initial inequalities and continuity of the data make the extensions semicontinuous on the closed slab with ordered pointwise initial values. Comparison [F1] gives $u^{*} \leq v_{*} + K$ on $Z$ . Since $u \leq u^{*}$ and $v_{*} \leq v$ , this implies $u - v \leq K$ ; taking the supremum over $Z$ gives $\sup_{Z} (u - v) \leq K$ .

2.1step 1.1step 1.2∎

Conclusion. Applying step 1.2 to the pair $(u, v)$ and to the exchanged pair $(v, u)$ gives $\sup_{Z} (u - v) \leq \max (0, \sup (u_{0} - v_{0}))$ and $\sup_{Z} (v - u) \leq \max (0, \sup (v_{0} - u_{0}))$ ; when the initial difference is bounded, both right-hand sides are at most $\sup_{R^{n}} ∣ u_{0} - v_{0} ∣$ , hence $\sup_{Z} ∣ u - v ∣ \leq \sup_{R^{n}} ∣ u_{0} - v_{0} ∣$ and the solution operator is a contraction in the supremum norm.

Remarks

Domain. The corollary is stated on $O = R^{n}$ because comparison case (a) is; on a bounded domain without lateral data uniqueness fails, as the companion counterexample shows.
Choice. Only comparison and the constant shift are used, both choice-free.

Depends on

Used by

Dependency tree · two levels

17 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)