In [[CEL84, Theorem 1.1, p.489]](http://www.ams.org/journals/tran/1984-282-02/S0002-9947-1984-0732102-X/S0002-9947-1984-0732102-X.pdf), Crandall, Evans, and Lions give three equivalent definitions of *viscosity solution*. 


As the authors note, the first two are "more appealing in some respects and more convenient for certain purposes" than the third one ([[CEL84, Theorem 1.1(iii)]](http://www.ams.org/journals/tran/1984-282-02/S0002-9947-1984-0732102-X/S0002-9947-1984-0732102-X.pdf)), which was "taken as basic" in the previous paper [[CL83]](http://www.ams.org/journals/tran/1983-277-01/S0002-9947-1983-0690039-8/S0002-9947-1983-0690039-8.pdf).




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**Q1:** In which circumstances is the third definition actually more convenient than the other two?



On the 'usefulness' of the 'original' definition of viscosity solution
**(i).** The semijet definition [[CEL Theorem 1.1 (i)]](http://www.ams.org/journals/tran/1984-282-02/S0002-9947-1984-0732102-X/S0002-9947-1984-0732102-X.pdf) is the natural way to derive comparison principles for PDEs using, e.g., the Crandall-Ishii lemma [[CIL, Theorem 3.2]](http://www.ams.org/journals/bull/1992-27-01/S0273-0979-1992-00266-5/S0273-0979-1992-00266-5.pdf).


**(ii).** Here are some various examples in which [[CEL Theorem 1.1 (ii)]](http://www.ams.org/journals/tran/1984-282-02/S0002-9947-1984-0732102-X/S0002-9947-1984-0732102-X.pdf) is easier to work with than the above:


* Showing that the set of viscosity solutions is invariant under a particular change of variables;
* Establishing the convergence of numerical approximation schemes [[BS]](http://content.iospress.com/articles/asymptotic-analysis/asy4-3-05) (or more generally, the stability of viscosity solutions).
* Proving that the value function of an optimal control problem is a viscosity solution of a particular PDE;
* etc.


**(iii).** [[CEL Theorem 1.1 (iii)]](http://www.ams.org/journals/tran/1984-282-02/S0002-9947-1984-0732102-X/S0002-9947-1984-0732102-X.pdf) is motivated by maximums of products of a solution with a test function. Precisely, let $u$ be a classical solution and $\phi$ be continuously differentiable. Then if $\phi(y)u(y) = \max\{\phi u\}>0$, the product rule yields $$(D(\phi u))(y)=\phi(y)Du(y)+u(y)D\phi(y)=0.$$ Moving some terms around, $$Du(y) = -\frac{u(y)}{\phi(y)} D\phi(y).$$ The version of (iii) appearing in [[CEL]](http://www.ams.org/journals/tran/1984-282-02/S0002-9947-1984-0732102-X/S0002-9947-1984-0732102-X.pdf) is the same idea, except with $\phi(y)(u(y)-k)$ being a positive maximum.




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**Perhaps a disappointing answer:** I cannot think of a situation in which (iii) would be more convenient to work with than (ii). As stated in [[CEL]](http://www.ams.org/journals/tran/1984-282-02/S0002-9947-1984-0732102-X/S0002-9947-1984-0732102-X.pdf), "L. C. Evans has observed that the criterion utilizing extremals of
$u - \phi$ is more convenient in various situations." Moreover, I have not seen (iii) in *any* recent works. It is possible that the (great) authors exercised extra caution in writing "various situations," when perhaps they might have meant "all situations we can think of." However, I cannot speak for them :-)


