More generally, viscosity solutions are stable with respect to limits. The presentation of the following result, which has more general forms, is borrowed from [\[Tou13\]][2].

> **Theorem:** Let $\mathcal{O}$ be open and $u_\epsilon$ be an LSC supersolution of $$F_\epsilon(\cdot,u_\epsilon(\cdot),Du_\epsilon(\cdot),D^2u_\epsilon(\cdot))=0 \text{ on }\mathcal{O}$$ where $(F_\epsilon)_{\epsilon > 0}$ are elliptic and continuous operators. Suppose $(\epsilon,x)\mapsto u_\epsilon(x)$ and $(\epsilon,z)\mapsto F_\epsilon(z)$ are locally bounded and let $$
\underline{u}(x)=\liminf_{(\epsilon,x^{\prime})\rightarrow(0,x)}u_{\epsilon}(x^{\prime})\text{ and }\overline{F}(z)=\limsup_{(\epsilon,z^{\prime})\rightarrow(0,z)}F_{\epsilon}(z^{\prime}).
$$ Then, $\underline{u}$ is an LSC viscosity supersolution of $$\overline{F}(\cdot,\underline{u}(\cdot),D\underline{u}(\cdot),D^{2}\underline{u}(\cdot))=0\text{ on }\mathcal{O}.$$

A similar result holds for USC subsolutions.

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Intimately related to the passage to limits above is the so-called *Barles-Souganidis framework* [\[BS91\]][1], which gives sufficient conditions for convergence of approximation schemes for equations of the form
$$F(\cdot,u(\cdot),Du(\cdot),D^2u(\cdot))=0 \text{ on } \mathcal{O}.$$
In particular, it establishes that any approximation scheme that is **monotone**, **stable**, and **consistent** with respect to a limiting equation that satisfies a *comparison principle* converges to its unique (bounded) viscosity solution. Note that the ideas in [\[BS91\]][1] are very general, applying to any nonlinear second order elliptic equation.

*Remark*: [\[BS91\]][1] use a stronger notion of comparison principle by accounting for the boundary conditions (sometimes referred to as a *strong comparison principle*). You can relax their result to more usual notions, but doing so requires that you check convergence at the boundary separately, roughly speaking.

  [1]: http://content.iospress.com/articles/asymptotic-analysis/asy4-3-05
  [2]: http://www.springer.com/la/book/9781461442851