Viscosity subsolutions and supersolutions of a first-order equation and of the Cauchy problem

Let $n\ge 1$, let $U\subseteq {\mathbb{R}}^{n+1}$ be nonempty open with coordinates $(x,t)\in {\mathbb{R}}^n\times \mathbb{R}$, and let $H:U\times {\mathbb{R}}^n\to \mathbb{R}$ be continuous. A function $u:U\to \mathbb{R}$ that is upper semicontinuous on $U$ (Upper and lower semicontinuity on subsets of ${\mathbb{R}}^n$) is a viscosity subsolution of $u_t+H(x,t,Du)=0$ in $U$ if for every $\varphi \in C^1(U)$ ($C^k$ maps and multi-index derivative notation in Euclidean space) and every point $z_0\in U$ at which $u-\varphi$ has a local maximum, ${\varphi }_t(z_0)+H(z_0,D\varphi (z_0))\le 0.$ A function $v:U\to \mathbb{R}$ that is lower semicontinuous on $U$ is a viscosity supersolution if for every $\varphi \in C^1(U)$ and every point $z_0\in U$ at which $v-\varphi$ has a local minimum, ${\varphi }_t(z_0)+H(z_0,D\varphi (z_0))\ge 0.$ A viscosity solution of the equation in $U$ is a function that is both a viscosity subsolution and a viscosity supersolution.

Cauchy problem. Let $Z=O\times (0,T)$ and $H:O\times [0,T]\times {\mathbb{R}}^n\to \mathbb{R}$ be as in The Hamilton--Jacobi Cauchy problem and its classical solutions, and let $u_0:O\to \mathbb{R}$. A viscosity subsolution of the Cauchy problem is an upper semicontinuous $u:Z\to \mathbb{R}$ satisfying the subsolution inequality at every $z_0\in Z$ together with the relaxed initial condition ${\mathrm{lim\hspace{0.17em}sup}}_{\begin{array}{c}(y,s)\to (x,0)\\ s>0,y\in O\end{array}}u(y,s)\le u_0(x)\text{for every }x\in O;$ a viscosity supersolution $v$ satisfies the supersolution inequality at every $z_0\in Z$ together with ${\mathrm{lim\hspace{0.17em}inf}}_{\begin{array}{c}(y,s)\to (x,0)\\ s>0,y\in O\end{array}}v(y,s)\ge u_0(x)\text{for every }x\in O\mathrm{.}$ A viscosity solution of the Cauchy problem is a function that is both a subsolution and a supersolution of the Cauchy problem; a continuous solution of the Cauchy problem satisfies the initial data pointwise, $u(x,0)=u_0(x)$ for every $x\in O$.

• Direction of the contact. The subsolution test is taken at a local maximum of $u-\varphi$, equivalently where $\varphi$ touches $u$ from above, and the supersolution test at a local minimum; reversing the extremum reverses the direction of the required inequality. This is exactly the sign asymmetry that A strict subsolution can fail the supersolution lower-test condition ↗ records.
• Semicontinuity is part of the definition. The subsolution must be upper semicontinuous and the supersolution lower semicontinuous: those are the classes in which the tested extrema behave well under localisation and limits. No continuity of $u$ on $U$ and no almost-everywhere class is assumed; a merely locally bounded function is handled through the two envelopes in Discontinuous viscosity solutions through the two envelopes.
• The initial condition is relaxed. The limsup/liminf condition is imposed at points of the initial face through sequences with $s>0$; it does not require $u$ to extend to $t=0$. For a continuous $u$ the two relaxed conditions force $u(x,s)\to u_0(x)$ as $(y,s)\to (x,0)$, hence the pointwise equality. No choice principle is used.