Smooth test functions suffice for the first- and second-order HJB viscosity notions

Start with an upper contact at $z_0=(x_0,t_0)$ for a $C^1$ or $C^{1,2}$ test $\varphi$. On a closed Euclidean space-time ball $K$ contained in the interior cylinder, define $\tilde{\varphi }(z)=\varphi (z)+\epsilon (\mathrm{\mid }x-x_0{\mathrm{\mid }}^4+\mathrm{\mid }t-t_0{\mathrm{\mid }}^4)$ for $\epsilon >0$. The added term and all derivatives relevant to the chosen test class vanish at $z_0$, while $u-\tilde{\varphi }$ is strictly smaller than $u(z_0)-\varphi (z_0)$ at every other point of the contact neighborhood. Thus the original and strictified tests have the same PDE residual at $z_0$.

Choose a nonnegative unit-mass mollifier ${\rho }_{\delta }$ supported in the ball of radius $\delta$ from [F1], with $\delta$ small enough that its convolution samples only points in the contact neighborhood. Choose a smooth cutoff equal to one on a neighborhood of $K$ and supported inside that contact neighborhood using [F1], multiply $\tilde{\varphi }$ by it, and extend the product by zero. Convolve this compactly supported function with ${\rho }_{\delta }$ to obtain ${\varphi }_{\delta }$. Rewriting the convolution with the smooth kernel as $\int g(y){\rho }_{\delta }(z-y)dy$ shows by uniform difference quotients on the compact support that ${\varphi }_{\delta }$ is $C^{\mathrm{\infty }}$. On $K$ it coincides with convolution of $\tilde{\varphi }$ alone. For a $C^1$ function $g$, [F5] gives $\frac{g(z+h{e}_j)-g(z)}{h}={\int }_0^1{\mathrm{\partial }}_{j}g(z+\theta h{e}_j)d\theta$ when the indicated derivative exists. Uniform continuity of ${\mathrm{\partial }}_{j}g$ on the compact neighborhood makes these difference quotients converge uniformly as $h\to 0$. Integrating this uniform limit against ${\rho }_{\delta }$ proves ${\mathrm{\partial }}_j({\rho }_{\delta }\ast g)={\rho }_{\delta }\ast {\mathrm{\partial }}_{j}g$. For a second spatial derivative, apply this identity first to $g=\tilde{\varphi }$ and then to $g={\mathrm{\partial }}_j\tilde{\varphi }$; this uses only the continuous second derivative, not a third derivative. Apply it once for time and once for each first spatial derivative. Thus $D^{\alpha }{\varphi }_{\delta }={\rho }_{\delta }\ast D^{\alpha }\tilde{\varphi }$ for exactly the derivatives required in the $C^1$ or $C^{1,2}$ class. By [F2], ${\sup }_{z\in K^{\mathrm{\prime }}}\mathrm{\mid }{D}^{\alpha }{\varphi }_{\delta }(z)-D^{\alpha }\tilde{\varphi }(z)\mathrm{\mid }\le {\sup }_{z\in K^{\mathrm{\prime }},\mathrm{\mid }y\mathrm{\mid }\le \delta }\mathrm{\mid }{D}^{\alpha }\tilde{\varphi }(z-y)-D^{\alpha }\tilde{\varphi }(z)\mathrm{\mid }⟶0$ on each smaller compact ball $K^{\mathrm{\prime }}$. Hence the approximants converge in $C^1$ or $C^{1,2}$, respectively.

The upper semicontinuous function $u-{\varphi }_{\delta }$ attains a maximum on $K$ by [F3]. The strict gap from step 1.1 and uniform convergence from the preceding approximation step imply that, for every fixed neighborhood of $z_0$, all maximizing points lie in that neighborhood once $\delta$ is sufficiently small; the boundary of $K$ is excluded for the same reason. At any such maximizing point $z_{\delta }$, adding the constant $u(z_{\delta })-{\varphi }_{\delta }(z_{\delta })$ makes ${\varphi }_{\delta }$ a smooth upper test. Apply the assumed smooth-test inequality there.

The maximizing points in step 3.1 approach $z_0$ uniformly over the maximizing set, and the required derivatives of ${\varphi }_{\delta }$ converge uniformly to those of $\tilde{\varphi }$ on $K^{\mathrm{\prime }}$ by step 2.1. Continuity of the PDE residual [F4] therefore lets the smooth-test inequalities pass to the residual of $\tilde{\varphi }$ at $z_0$, which equals that of $\varphi$ by step 1.1. This proves the upper-test implication. For a lower contact, replace maxima by minima and reverse the inequalities; the same uniform gap and derivative convergence apply. Since smooth tests are themselves in both larger test classes, the reverse implications are immediate.