Perron's method for the Cauchy problem: existence between two barriers

Well-definedness and the upper envelope. The barrier ${\varphi }_{-}$ is itself an admissible subsolution by [F1], so $\mathcal{W}$ is nonempty, and every $w\in \mathcal{W}$ satisfies $w\le {\varphi }_{+}$, so $W$ is real-valued and bounded above on compact subsets of $Z$. By [F2] the envelope $W^{\ast }$ is a viscosity subsolution; moreover $W\le {\varphi }_{+}$ gives $W^{\ast }\le {\varphi }_{+}$ because ${\varphi }_{+}$ is continuous (the limsup defining the envelope of a function bounded above by the continuous ${\varphi }_{+}$ is at most ${\varphi }_{+}$), and $W^{\ast }\ge W\ge {\varphi }_{-}$. Hence $W^{\ast }$ is itself an admissible member of $\mathcal{W}$, so $W^{\ast }\le W$ by maximality and therefore $W=W^{\ast }$ is upper semicontinuous.

The lower envelope is a supersolution. Suppose $W_{\ast }$ failed the supersolution test strictly at some $\widehat{z}\in Z$: there is $\varphi \in C^1$ with $W_{\ast }-\varphi$ having a local minimum at $\widehat{z}$ and ${\varphi }_t(\widehat{z})+H(\widehat{z},D\varphi (\widehat{z}))<0$. First, $W_{\ast }(\widehat{z})<{\varphi }_{+}(\widehat{z})$: otherwise $W_{\ast }(\widehat{z})={\varphi }_{+}(\widehat{z})$ and, since $W_{\ast }\le {\varphi }_{+}$, the function ${\varphi }_{+}-\varphi$ would have a local minimum at $\widehat{z}$, so the supersolution inequality for the classical supersolution ${\varphi }_{+}$ would give ${\varphi }_t(\widehat{z})+H(\widehat{z},D\varphi (\widehat{z}))\ge 0$, a contradiction. Choose a small bump supported in $B(\widehat{z},\kappa )$; by [F2] it gives a viscosity subsolution $W_{\kappa }\ge W$ that exceeds $W$ at some point in that ball and equals $W$ outside it. It is constructed as $\max (W,\chi )$ on a smaller ball, where $\chi$ is a classical subsolution and is below $W$ on the surrounding annulus. In the construction of the bump lemma, the unshifted smooth part $\tilde{\varphi }+m$ has value $W_{\ast }(\widehat{z})<{\varphi }_{+}(\widehat{z})$. First choose its ball radius $r$ small enough that $\tilde{\varphi }+m$ lies strictly below ${\varphi }_{+}$ throughout the closed ball. Then choose the offset $\delta$ smaller than both the positive minimum of ${\varphi }_{+}-(\tilde{\varphi }+m)$ on that ball and the annular allowance $\gamma (r\mathrm{/}2{)}^4$. The resulting $\chi =\tilde{\varphi }+m+\delta$ stays below ${\varphi }_{+}$ while all annular gluing inequalities hold; together with $W\le {\varphi }_{+}$ this gives $W_{\kappa }\le {\varphi }_{+}$, while $W_{\kappa }\ge W\ge {\varphi }_{-}$ always holds. Hence $W_{\kappa }$ is squeezed between the barriers, and by the two-sided initial control [F1] it satisfies the relaxed initial condition; so $W_{\kappa }\in \mathcal{W}$, contradicting maximality because $W_{\kappa }$ exceeds $W$ at the point supplied by the bump. Therefore $W_{\ast }$ is a viscosity supersolution.

Comparison, continuity and uniqueness. The upper envelope $W^{\ast }=W$ is a bounded upper semicontinuous subsolution and $W_{\ast }$ is a bounded lower semicontinuous supersolution; both carry the datum $u_0$ in the relaxed sense by [F1] applied to $W$, which lies between the barriers. Comparison [F3] gives $W^{\ast }\le W_{\ast }$; since always $W_{\ast }\le W\le W^{\ast }$, all three coincide, so $W$ is continuous and is a viscosity solution of the Cauchy problem with datum $u_0$. For uniqueness, let $V$ be any, possibly discontinuous, viscosity solution with ${\varphi }_{-}\le V\le {\varphi }_{+}$. By definition $V^{\ast }$ is a bounded upper semicontinuous subsolution and $V_{\ast }$ a bounded lower semicontinuous supersolution, both with datum $u_0$. Continuity of the barriers and ${\varphi }_{-}\le V\le {\varphi }_{+}$ give ${\varphi }_{-}\le V_{\ast }\le V\le V^{\ast }\le {\varphi }_{+}$, so $V^{\ast }$ belongs to $\mathcal{W}$ and hence $V^{\ast }\le W$ by maximality. Comparison between the subsolution $W$ and supersolution $V_{\ast }$ gives $W\le V_{\ast }$. Thus $W\le V_{\ast }\le V\le V^{\ast }\le W$, so all are equal.