Consider a linear partial differential equation
$$ (L u)(x)=f(x) \quad \forall x \text{ in } \Omega\\ u=g \quad\text{on }\partial\Omega. $$ Assuming that $f$ and $L$ depend on a parameter $y\in\mathbb{R}$, $f=f(x,y)$, $L=\sum_{\alpha} a_\alpha(x,y)\partial^\alpha$ how is the regularity of $u$ with respect to $y$ related to $L$? Just by differentiating formally, I would guess that $u$ is approximately as regular w.r.t. $y$ as $f$ is, whenever there is some sort of regularity result for $L(y)$, because $$ (L \frac{d}{dy} u)(x)=\frac{d}{dy}f(x,y)-\sum_{\alpha}\frac{d}{dy} a(x,y)\partial^{\alpha} u(x) $$ In particular, I expect infinite differentiability w.r.t. $y$ of $f$ and $L$ imply the same for $u$ at points where there are regularity results for $L(y)$ in a neighborhood of $y$.
If what I guess is wrong, can you hint at the problems with my arguments/their rigorousification?