Equivalence of alternative definitions of 'viscosity solution' Consider the first-order Hamilton-Jacobi equation (HJ):
$$H(x,u,\nabla u) = 0 \quad \text{ on } \ \Omega,$$ where  $\Omega$ is an open set of $\mathbb{R}^n$, $u:\Omega \to \mathbb{R}$, and $H:\Omega \times \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}$ (Hamiltonian) is continuous.
Definition 1 (Crandall-Lions-Evans): We say that $u \in C(\Omega)$ is a viscosity solution of (HJ) iff, $\forall \phi \in C^1(\Omega)$,
$\forall x_0$ point of local maximum of $u-\phi$, $\ H(x_0, u(x_0), \nabla\phi(x_0)) \le 0$;
$\forall x_0$ point of local minimum of $u-\phi$, $\ H(x_0, u(x_0), \nabla\phi(x_0)) \ge 0$.

Q1: Where can I find a detailed proof that we can replace 


*

*"local maximum" by "strict local maximum" or "global maximum" or "strict global maximum";

*"local minimum" by "strict local minimum" or "global minimum" or "strict global minimum"


and obtain an equivalent definition? (Or anyway how would that proof go?)
Q2: Is it true that we can replace  $C^1$ by $C^k$ or $C^k_{\text{comp}}$, with $1 < k \le \infty$, and obtain an equivalent definition? 
 A: Only the subsolution case is proven, as the supersolution case is identical.
Q1
Suppose $u$ is a subsolution under the definition with strict extremum. Let $\phi$ be a test function such that $u-\phi$ has a possibly nonstrict maximum at $x_0$. Let
$$\psi(x)=\phi(x)+|x-x_0|^2$$
so that $x_0$ is a strict maximum of $u-\psi$. Moreover,
$$H(x_0,u,\nabla \phi(x_0)) = H(x_0,u,\nabla \psi(x_0)) \leq 0.$$
Q2
Suppose $u$ is a subsolution under the definition with compact test functions and $1 < k < \infty$. Let $\phi$ be a noncompact test function such that $u-\phi$ has a local maximum at $x_0$. Let $B_r$ denote the Euclidean ball of radius $r$ around $x_0$. Without loss of generality, we assume that $B_3 \subset \Omega$ (otherwise, perform some scaling to fix things). Now, let
\begin{align*}
\psi(x)&=\phi(x) 1_{B_1}(x) + \zeta(x) \phi(\hat{x}) 1_{B_2 \setminus B_1}(x),\\
\zeta(x)&=\exp(1-1/(1-|x-\hat{x}|^{2k})),
\end{align*}
and $\hat{x}$ is the unique closest point to $x$ in $\operatorname{cl}B_1$.  $\psi$ inherits all the local properties of $\phi$ at $x_0$ and hence
$$H(x_0,u,\nabla \phi(x_0)) = H(x_0,u,\nabla \psi(x_0)) \leq 0.$$
The supersolution case is identical.
