Consider the following Hamilton-Jacobi (HJ) equation:
$$u\_t + H(\nabla u,x) = 0 \quad \text{ in } \mathbb{R}^n \times (0, T], $$ where $u:\mathbb{R}^n \times (0,T] \to \mathbb{R}$, and $H:\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ (*Hamiltonian*) is continuous. 
Consider the initial condition $$u = g \quad \text{ in } \mathbb{R}^n \times \{t = 0\}.$$


In his textbook on *Partial Differential Equations* (pages 586-590), Evans proves uniqueness of viscosity solutions of this initial value problem under the following assumptions:
$$\vert H(p,x) - H(q,x) \vert \le C \vert p - q \vert;$$
$$\vert H(p,x) - H(p,y)\vert \le C \vert x-y \vert (1+\vert p \vert),$$ for some $C \ge 0.$ Evans' proof is relatively simple to follow, but slightly technical and, at least to me, not quite enlightening. 




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**Question:** Where can I find an alternative proof of this result?




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**Side remark:** In [[CEL84]](http://www.ams.org/journals/tran/1984-282-02/S0002-9947-1984-0732102-X/S0002-9947-1984-0732102-X.pdf), Crandall, Lions, and Evans prove existence and uniqueness of viscosity solutions for a less general Hamiltonian.



Uniqueness of viscosity solutions of Hamilton-Jacobi equation
I would recommend the book by Bardi and Capuzzo-Dolcetta:


"Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations" Bardi, Martino, Capuzzo-Dolcetta, Italo


Read the proofs for stationary equations $H(\nabla u, x) = 0$ on bounded domains first to get the main ideas. The proof in Evans has more bells and whistles to account for the unbounded domain. 


I would also take the opportunity to plug my own notes (Section 6):


<http://math.umn.edu/~jwcalder/222BS16/viscosity_solutions.pdf>


