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Let $F(x,u,Du,D^2u)=0$ for $x\in\Omega$ be our equation of interest. One way of defining a viscosity subsolution is this:

We say $u$ is a viscosity subsolution iff for any test function $\phi\in C^2(\Omega)$ such that $u-\phi$ has a LOCAL maximum at $x_0$ then $F(x_0,D\phi(x_0),D^2\phi(x_0))\leq 0$.

However in this paper, definition 2.1 on page 6

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.52.8750&rep=rep1&type=pdf

the author replace the word local with GLOBAL. I believe this is fairly common in optimal control/stopping literature.

I think the answer is no, as long as we can solve this seemingy simple analysis question

https://math.stackexchange.com/questions/638457/finding-a-smooth-function-which-dominates-a-continuous-one

but I have struggled to do this.

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The answer is no, and how to solve the analysis question is explained in exercise 3.5 in Albert Fathi's lecture notes for example -there are other sources of course, but not freely available.

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  • $\begingroup$ That would do. If i write something on this, what should i reference? In the paper i referred to, i think this was just assumed. $\endgroup$
    – lost1
    Commented Jan 15, 2014 at 12:24
  • $\begingroup$ @lost1 you can quote Barles' book, for example, Guy Barles. Solutions de viscosit\´e des \´equations de Hamilton-Jacobi. Springer- Verlag, Paris, 1994. $\endgroup$
    – username
    Commented Jan 15, 2014 at 12:43
  • $\begingroup$ I flipped through the content page and this book seems to be interested in first order only, am I mistaken? $\endgroup$
    – lost1
    Commented Jan 15, 2014 at 12:51
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    $\begingroup$ @lost1 Viscosity solutions are introduced first for second order problem, as it is more 'natural' in this case, and then for first order. The general theory is done for both. $\endgroup$
    – username
    Commented Jan 15, 2014 at 13:00
  • $\begingroup$ @username I wonder how we can solve the analysis question by using Exercise 3.5 (and its proof) in Fathi's lecture note. Such exercise provides a smooth function dominating a given continuous function, while (in the analysis question) we are looking for a much more complicated smooth function. Thank you. $\endgroup$
    – Khoa Vu
    Commented Aug 30 at 20:48

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