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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].

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.

Intimately related to the passage to limits above is the so-called Barles-Souganidis framework [BS91], 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] are very general, applying to any nonlinear second order elliptic equation.

Remark: [BS91] 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.

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].

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.

Intimately related to the passage to limits above is the so-called Barles-Souganidis framework [BS91], 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] are very general, applying to any nonlinear second order elliptic equation.

Remark: [BS91] 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.

The so-called Barles-Souganidis framework [BS91] gives sufficient conditions for convergence of approximation schemes for equations of the form $$F(x,u(x),Du(x),D^2u(x))=0.$$ 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 their results are very general, applying to any nonlinear second order elliptic equation.

Caveat: Barles and Souganidis 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.

Addendum: when I have time, I will update this answer to include a full proof of the Barles-Souganidis result.

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].

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.

Intimately related to the passage to limits above is the so-called Barles-Souganidis framework [BS91], 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] are very general, applying to any nonlinear second order elliptic equation.

Remark: [BS91] 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.

The so-called Barles-Souganidis framework [BS91] gives sufficient conditions for convergence of approximation schemes for equations of the form $$F(x,u,Du,D^2u)=0.$$ 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.

Their paper is a must-read for anyone performing numerical computations in the field of viscosity solutions.

Caveat: Barles and Souganidis 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.

Addendum: when I have time, I will update this answer to include a full proof of the Barles-Souganidis result.

The so-called Barles-Souganidis framework [BS91] gives sufficient conditions for convergence of approximation schemes for equations of the form $$F(x,u(x),Du(x),D^2u(x))=0.$$ 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 their results are very general, applying to any nonlinear second order elliptic equation.

Caveat: Barles and Souganidis 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.

Addendum: when I have time, I will update this answer to include a full proof of the Barles-Souganidis result.