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On the 'usefulness' of the 'original' definition of viscosity solution

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

QQ1: In which circumstances is the third definition actually more convenient than the other two?

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

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

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

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

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user99249
user99249

On the 'original' definition of viscosity solution

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

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