There is a prevalent method called the "Nonlinear adjoint method" in the study of viscosity solution and Hamilton--Jacobi equation, especially equations of the form
$$
u^\varepsilon + H(x,Du^\varepsilon) - \varepsilon \Delta u^\varepsilon = 0 
$$


It usually requires taking derivative


$$
\frac{d u^\varepsilon}{d\varepsilon}
$$


and in most papers, they assume $\varepsilon \mapsto u^\varepsilon$ is smooth, once solution is uniquely defined.
My questions are:


* Why do we have such a claim?
* If solutions are unique up to additive constant, upon normalizing to get uniqueness, do we still have such a claim?



Smooth dependence of parameter of PDE - viscosity solutions
You need to justify this with finite differences. I will give only a sketch of proof. What you do is formally differentiate the equation to get an equation for $v^\epsilon = (\partial/\partial \epsilon) u^\epsilon$. You prove uniqueness of this equation, which is presumably implied by your assumptions and some regularity on $u^\epsilon$ and the function $H$. For now we consider $v^\epsilon$ to be the solution of this equation, and the fact that it actually equal to $(\partial/\partial \epsilon) u^\epsilon$ is yet to be proved. We next consider finite difference approximations
\begin{equation\*}
v\_h^\epsilon = \frac1h ( u^{\epsilon+h} - u^\epsilon),
\end{equation\*}
which also solve equations which are very close to the equation for $v^\epsilon$ as $h\to 0$. The uniqueness of the equation for $v^\epsilon$ will guarantee that these finite differences converge to $v$.
This also requires sufficient uniform (in $h$) estimates of $v\_h^\epsilon$ so that you can get convergence to something along subsequences (Arzela-Ascoli is usually what you need in the viscosity solution setting). You show that the limit of a subsequence must be a solution of the same equation as $v^\epsilon$, because the equation for $v^\epsilon\_h$ is so close to the one for $v^\epsilon$ (you again need to use some regularity of $H$ here). This requires a typical viscosity solution argument. (You touch the limiting function above by smooth function, argue that you are essentially touching the $v^\epsilon\_h$ for some small $h$, which gives a differential inequality at the touching point, send $h\to 0$.) This completes the proof.


(A similar argument appears in the context of $H^2$ estimates for elliptic equations in Chapters 5 or 6 in the Evans PDE book.)


A personal aside: I was a PhD student of Craig Evans (a long time ago now) and I asked him the exact same question. He told me "everything is completely smooth here" and he indicated in a gentle/nice way that he was not really interested in explaining further. I subsequently wrote a fifteen page set of notes proving stuff like this at the level of detail that was appropriate for me at the time.


So I completely understand the question, but twenty years later I also now understand Craig's reaction. When you do an argument like this three times, it becomes "very obvious" and you don't need to see a full proof any longer in similar situations. This is the reason that such details are not given in papers. But you do need to do it a couple of times.


