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. 



Does replacing 'local' with 'global' in the definition of viscosity sub(super)solution change the definition?
The answer is no, and how to solve the analysis question is explained in exercise 3.5 in [Albert Fathi's lecture notes](http://www.google.fr/url?sa=t&rct=j&q=&esrc=s&source=web&cd=3&cad=rja&ved=0CEcQFjAC&url=http://www.ceremade.dauphine.fr/~pbernard/enseignement/m2/fathi.pdf&ei=_HzWUuzuG86r0gWLl4GADQ&usg=AFQjCNF6bx0iQFYFSBuEa6n_KA8RemSELw&sig2=VL9hcdctrCfU7_wuBEpABA&bvm=bv.59378465,d.d2k) for example -there are other sources of course, but not freely available.


