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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?

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The hypotheses are listed directly above the theorem statement (e.g., "Recall we are assuming..."). It is common in textbooks or papers to list some of the hypotheses right before the theorem to make the theorem statements more concise. In general, you should aim to understand the proof in sufficient detail that you can deduce on your own what assumptions are necessary.

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  • $\begingroup$ Thanks and I agree with you. $\endgroup$ – user79963 Jun 9 '17 at 3:44

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