Discontinuous viscosity solutions through the two envelopes

Let $n\ge 1$, let $O\subseteq {\mathbb{R}}^n$ be open, $T>0$, let $H:O\times [0,T]\times {\mathbb{R}}^n\to \mathbb{R}$ be continuous, $Z=O\times (0,T)$, and let $u_0:O\to \mathbb{R}$. A locally bounded function $u:Z\to \mathbb{R}$ is a viscosity solution of the Cauchy problem if its upper semicontinuous envelope $u^{\ast }$ is a viscosity subsolution and its lower semicontinuous envelope $u_{\ast }$ is a viscosity supersolution in the sense of Viscosity subsolutions and supersolutions of a first-order equation and of the Cauchy problem, each carrying the initial datum $u_0$ in the relaxed limsup/liminf sense. Both envelopes are taken over $Z$ as a subset of ${\mathbb{R}}^{n+1}$ (Upper and lower semicontinuous envelopes by local limsup and liminf); the subsolution inequalities are imposed at every point of $Z$ and the relaxed initial conditions at every point of $O$.

A continuous viscosity solution is a viscosity solution $u$ that is continuous on $Z$. For such a $u$ one has $u^{\ast }=u_{\ast }=u$ (apply The envelopes are the least upper and greatest lower semicontinuous functions on a small closed ball about each interior point, where continuity makes $u$ bounded). The relaxed initial conditions give a continuous extension to the initial face with value $u_0$, so the definition specializes to the one for continuous test functions. In the opposite direction, the envelope formulation is forced whenever $u$ is not continuous: it keeps the subsolution inequality attached to $u^{\ast }$ and the supersolution inequality to $u_{\ast }$, and never asks a single discontinuous function to satisfy both.

• Why the two envelopes. A merely locally bounded $u$ need not be upper or lower semicontinuous, so the test-function definition of Viscosity subsolutions and supersolutions of a first-order equation and of the Cauchy problem cannot be applied to $u$ directly. The envelopes are respectively the least upper semicontinuous majorant and the greatest lower semicontinuous minorant of $u$, so requiring $u^{\ast }$ to be a subsolution and $u_{\ast }$ a supersolution is the weakest formulation in which the two one-sided inequalities can be tested; a function is a viscosity solution exactly when both envelopes solve their respective one-sided problems.
• Where it is used. This is the notion under which the half-relaxed limits of a locally bounded family are sub- and supersolutions (Half-relaxed limits of a locally bounded family, Half-relaxed limits of sub- and supersolutions with vanishing perturbations) and under which the perron-type and comparison statements of the page are formulated when continuity of the produced object is not known in advance (The upper envelope of a locally bounded supremum of subsolutions is a subsolution). The domain conventions $Z=O\times (0,T)$ and the initial face $O\times \{0\}$ are those of The Hamilton--Jacobi Cauchy problem and its classical solutions. No choice principle is used.