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In [CL84], Crandall and Lions considered an initial value problem for an Hamilton-Jacobi equation and proved [CL84, Theorem 5.1] a result on the convergence of the vanishing viscosity approximation to the (unique) viscosity solution of the problem.

Q1: Where can I find a more recent exposition of this result (possibly with an improved proof)?

Q2: Are there sharper/significantly more general results of this kind?

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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.

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  • $\begingroup$ @Kei: was my answer useful? I'm interested to see if this is what you were looking for. (just following up here) $\endgroup$
    – parsiad
    Commented Oct 26, 2016 at 23:05
  • $\begingroup$ Yes, thanks (and thank you very much for following up): the works you have pointed out look very interesting; I hope to get around to read them more carefully shortly. Do you also know some expositions of the result (or slightly sharper results) obtained by Crandall and Lions for first-order equations? $\endgroup$
    – user99249
    Commented Oct 28, 2016 at 21:18
  • $\begingroup$ In terms of first-order equations, you can use the "passage to limits" result above and simply remove the $D^2$ terms. In terms of making the results sharper, you should be able to relax the above so that each $F_\epsilon$ is only USC instead of continuous (and LSC for the subsolution case). As for other results on approximation schemes, I recommend looking at the first paragraph of the Barles-Souganidis paper, in which the authors list approximation results for special cases (including Hamilton-Jacobi, if my memory serves me correctly). $\endgroup$
    – parsiad
    Commented Oct 28, 2016 at 22:54

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