Let $u: \Omega \to \mathbf{R}$ be a $C^{0,\alpha}$ function, with $\alpha \in (0,1]$, defined on a bounded, open domain $\Omega$. Suppose that $u$ is a viscosity subsolution of the equation $\Delta U = A$ in $\Omega$, for some constant $A > 0$: $u$ is '$A$-strictly subharmonic' in the viscosity sense.
How well can $u$ be approximated by a regular subsolution $v$, say with $\Delta v \geq (1-\epsilon)A$ in the classical sense? Can one find such a $v \in C^2(\Omega)$ with $\lvert u - v \rvert_{C^{0,\alpha}} \leq \epsilon$?
- I suspect this kind of thing is well-known (maybe via an inf-convolution or similar), but unfortunately I know very little viscosity theory.
- If there is a trade-off between ensuring $\Delta v \geq (1 - \epsilon) A$ and the smallness of $\lvert u - v \rvert_{C^{0,\alpha}}$, then I would prioritize the latter—say something like $\Delta v \geq 2^{-N}A$ and $\lvert u - v \rvert_{C^{0,\alpha}} \leq 2^{-N}$ would be fine.
