Stability of viscosity sub-, super- and solutions under locally uniform convergence

The strict-contact case. Let $\varphi \in C^1(Z)$ and suppose $u-\varphi$ has a strict local maximum at $z_0\in Z$. Choose $r>0$ with $K:=\bar{B}(z_0,r)\subseteq Z$ and strict inequality at every point of $K\setminus \{z_0\}$. For each $k$, let $M_k$ be the nonempty compact set of maximisers of $u_k-\varphi$ on $K$; it is compact because $u_k-\varphi$ is continuous. For every $r_0\in (0,r)$, the upper semicontinuous function $u-\varphi$ has a strict gap below its value at $z_0$ on the compact annulus $K\setminus B(z_0,r_0)$. Uniform convergence on $K$ therefore puts every point of $M_k$ inside $B(z_0,r_0)$ for all sufficiently large $k$. Since $r_0$ is arbitrary, ${\sup }_{z\in M_k}\mathrm{\mid }z-z_0\mathrm{\mid }\to 0$, without choosing a maximiser for each $k$. Every point $z\in M_k$ is then an interior local maximum of $u_k-\varphi$ and satisfies [F1]. If the desired residual ${\varphi }_t(z_0)+H(z_0,D\varphi (z_0))$ were positive, continuity would make ${\varphi }_t(z)+H(z,D\varphi (z))$ positive on a neighbourhood of $z_0$; uniform convergence of $H_k$ to $H$ on the compact set $K\times D\varphi (K)$ would make ${\varphi }_t(z)+H_k(z,D\varphi (z))>0$ there for all large $k$. This contradicts [F1] at every point of the nonempty set $M_k$. Hence the subsolution inequality holds at $z_0$.

The initial trace and the supersolution case. The local uniform convergence on $Z_0$ makes $u$ continuous on $Z_0$ with $u(x,0)={\lim }_k{u}_k(x,0)={\lim }_k{u}_0^{(k)}(x)=u_0(x)$ for every $x\in O$, the convergence on compact subsets of $O$ being uniform; hence $u$ has the continuous initial trace $u_0$ and satisfies the relaxed initial condition ${\mathrm{lim\hspace{0.17em}sup}}_{(y,s)\to (x,0),s>0}u(y,s)=u_0(x)$. The same argument as in step 1.1 with local minima in place of local maxima, and the supersolution inequality of [F1] in place of the subsolution inequality, shows that a locally uniform limit of supersolutions is a supersolution with the same initial trace; no sign of the convergence is used, only that the test function is fixed.

Removal of strictness and conclusion. If $u-\varphi$ merely has a (nonstrict) local maximum at $z_0$, fix $r>0$ with $\bar{B}(z_0,r)\subseteq Z$ on which the maximum inequality holds and strictify: by [F2], ${\varphi }_{\epsilon }=\varphi +\epsilon \mathrm{\mid }z-z_0{\mathrm{\mid }}^4$ satisfies $D{\varphi }_{\epsilon }(z_0)=D\varphi (z_0)$, ${\mathrm{\partial }}_t{\varphi }_{\epsilon }(z_0)={\varphi }_t(z_0)$ and makes $u-{\varphi }_{\epsilon }$ strictly maximised at $z_0$ over $\bar{B}(z_0,r)$; step 1.1 applied to ${\varphi }_{\epsilon }$ gives ${\varphi }_t(z_0)+H(z_0,D\varphi (z_0))\le 0$. Hence $u$ is a viscosity subsolution of the limit equation in $Z$, and by step 2.1 it carries the datum $u_0$; the supersolution statement and the solution statement follow by step 2.1 and by combining the two one-sided conclusions.