My general question is given in the title. Let us consider Hopf-Lax formula - Theorem 4 of Section 3.3.2 of the PDE book by Evans, Click here.
[Theorem 4] If $x\in \mathbb R^n$ and $t>0$, then the solution $u= u(x, t)$ of the minimization problem (17) is $$u(x, t) = \ldots$$
The assumption given for Hopf-Lax formula above is "If $x\in \mathbb R^n$ and $t>0$". But obviously, one shall impose some conditions for the functions $L$ and $g$, at least measurability condition. Therefore, I wonder what exactly the conditions needed to have Hopf-Lax formula valid.
From the context, $L$ is Lagrangian which may be required strongly convex as of (19) on the same page. In addition, $g$ is mentioned to be Lipschitz in (20). So my understanding for Hopf-Lax formula is
[Restatement of Hopf-Lax] $L$ is strongly convex, $g$ is Lipschitz. If $x\in \mathbb R^n$ and $t>0$, then ....
In fact, I have many similar questions on other Theorems. Is there any unified approach to figure out the exact conditions for theorems? Or otherwise, we have to read the proof back and forth and extract the conditions by ourselves?