Let $n \in \mathbb{N}$, let $Q \subset \mathbb{R} \times \mathbb{R}^n$ be open, let $F: Q \times \mathbb{R} \times \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}$ be a [continuous function](/page/Continuous%20Function), and let $u: Q \to \mathbb{R}$ be a $C^1$ function.
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Use the following first-order viscosity convention for the equation $F(t,x,r,a,p)=0$ on $Q$.
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The function $u$ is a viscosity subsolution if, for every $(t_0,x_0) \in Q$ and every open neighbourhood $V \subset \mathbb{R} \times \mathbb{R}^n$ of $(t_0,x_0)$ with $\phi: V \to \mathbb{R}$ of class $C^1$, whenever $u-\phi$ has a local maximum at $(t_0,x_0)$ relative to $Q \cap V$, one has
The function $u$ is a viscosity supersolution if, under the same hypotheses on $(t_0,x_0)$, $V$, and $\phi$, whenever $u-\phi$ has a local minimum at $(t_0,x_0)$ relative to $Q \cap V$, one has