In [\[CEL84, Theorem 1.1, p.489\]][1], 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 ([\[CEL84, Theorem 1.1(iii)\]][1]), which was "taken as basic" in the previous paper [\[CL83\]][2].

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

  [1]: http://www.ams.org/journals/tran/1984-282-02/S0002-9947-1984-0732102-X/S0002-9947-1984-0732102-X.pdf
  [2]: http://www.ams.org/journals/tran/1983-277-01/S0002-9947-1983-0690039-8/S0002-9947-1983-0690039-8.pdf