Lax-type representation conjecture for state-dependent Hamilton–Jacobi equations

Let $H(x,p,t)$ be a Hamiltonian satisfying $H(x,p,t)\in C^2$ and convexity with respect to $p$, and let (A5) denote the assumption referenced in the source. Let $\varphi$ be the viscosity solution of the Hamilton–Jacobi equation and initial condition in the source, and let the Lax representation be the formula denoted by

. **Lax-type representation conjecture.** If $H(x,p,t)\in C^2$ is convex with respect to $p$ and (A5) holds, then there \exists $t_0$ such that $\varphi$ is represented by

for $t\leq t_0$. Moreover, if $\varphi(x,t)$ is differentiable with respect to $x$ in a neighbourhood of $(x,t)$ and the infimum is attained at $\tilde v$, then $\partial_x\varphi(x,t)=\tilde v$. The claim extends the rigorously proved restricted-assumption formula beyond the hypotheses used in the preceding theorem; the source reports numerical evidence and says that rigorous criteria remain to be established.