In the article: On viscosity solutions of fully non linear equations with Measurable Ingredients the following definition is made
$u \in C(\Omega)$ is a $L^p$-viscosity subsolution of $F=f$ in $\Omega$ if, for all $\varphi\in W^{2,p}_{loc}(\Omega)$ whenever $\varepsilon>0$, $O\subset\Omega$ is open and $F(x,u(x),D\varphi(x),D^2\varphi(x))-f(x)\geq\varepsilon$, then $u-\varphi$ cannot have a local maximum in $O$.
An equivalent definition is
$u \in C(\Omega)$ is a $L^p$-viscosity subsolution of $F=f$ in $\Omega$ if, for all $\varphi\in W^{2,p}_{loc}(\Omega)$ and $\hat{x}\in\Omega$ at which $u-\varphi$ has a local maximum, one has $\displaystyle \text{essliminf}_{x\to\hat{x}} F(x,u(x),D\varphi(x),D^2\varphi(x))-f(x)\leq0$.
I am not able to show this equivalence. Every help is welcome.