# Uniqueness of viscosity solutions of Hamilton-Jacobi equation

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.

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

Side remark: In [CEL84], Crandall, Lions, and Evans prove existence and uniqueness of viscosity solutions for a less general Hamiltonian.

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

• Thanks; I'll have a look at that and then I'll let you know.
– user99249
Oct 5, 2016 at 19:18

The result (for slightly more general $$H$$) is Theorem 3.15 p. 158 in the mentioned book by Bardi and Capuzzo-Dolcetta (it is a consequence of the Comparison Principle).