Regularity estimates for parametrized family of elliptic operators Let $\Omega\subset\mathbb{R}^3$ be a bounded Lipschitz domain. Suppose we have a parametrized family of linear operators $\{A_\varepsilon\ :\ \varepsilon\in\mathbb{R}_{\geq0}\}$ such that $A_\varepsilon:H_0^1(\Omega)\to H^{-1}(\Omega)$ and such that for all $\varepsilon\in\mathbb{R}_{\geq0}$ and $m\in\mathbb{N}$ we have that $A_\varepsilon:H^{m+2}(\Omega)\cap H_0^1(\Omega)\to H^m(\Omega)$ and that $A_\varepsilon u\in H^m(\Omega)$ implies $u\in H^{m+2}(\Omega)\cap H_0^1(\Omega)$, i.e., $A_\varepsilon$ is regularizing. 
Furthermore, the $A_\varepsilon$ are uniformly elliptic, that is there exists two constants $C_1,\ C_2$ such that 
$$\langle A_\varepsilon u,u\rangle_{H^{-1}(\Omega)H_0^1(\Omega)}\geq C_1\|u\|^2_{H_0^1(\Omega)},$$
$$\langle A_\varepsilon u,v\rangle_{H^{-1}(\Omega)H_0^1(\Omega)}\leq C_2\|u\|^2_{H_0^1(\Omega)}\|v\|^2_{H_0^1(\Omega)}$$   
for all $u,\ v\in H_0^1(\Omega)$ and for all $\varepsilon\in\mathbb{R}_{\geq0}$. Note, that the constants $C_i$ are independent of the parameter $\varepsilon$.
Now, suppose we are given a $f\in H^m(\Omega)$ such that $A_\varepsilon u_\varepsilon=f$ in $H^{-1}(\Omega)$ for all $\varepsilon\in\mathbb{R}_{\geq0}$. We then have $u_\varepsilon\in H^{m+2}(\Omega)\cap H_0^1(\Omega)$.
Is it possible to conclude, maybe after posing some additional smoothness assumption on $\Omega$ and $A_\varepsilon$, that
$$\|u_\varepsilon\|^2_{H^{m+2}(\Omega)}\leq C_3\|f\|_{H^m(\Omega)}?$$ 
It is true for $m=-1$, since $A_\varepsilon u_\varepsilon=f$ in $H^{-1}$, but other than that I am not sure.
 A: This statement is not true in general. Consider the following elliptic pde:
$$
-\Delta u + \epsilon \chi_{K_\epsilon} u = f
$$
plus homogeneous Dirichlet-boundary conditions. Here, $K_\epsilon =\Omega \cap B(x_0, \epsilon^{-3/2d})$, $x_0\in\Omega$.
Set $A_\epsilon:=-\Delta + \epsilon \chi_{K_\epsilon}I$. The inverses are uniformly bounded from $H^{-1}$ to $H_0^1(\Omega)$.
The operators $A_\epsilon$ are uniformly bounded from $H_0^1(\Omega)$ to $H^{-1}(\Omega)$:
If $B(x_0, \epsilon^{-3/2d})\subset \Omega$ then
$$
\left| \int_{K_\epsilon} uvdx\right|
\le c |B(x_0, \epsilon^{-3/2d})|^{2/3}\|u\|_{L^6(\Omega)}\|v\|_{L^6(\Omega)}
\le c \epsilon^{-1}\|u\|_{H^1(\Omega)}\|v\|_{H^1(\Omega)}
$$
using continuity of the embedding $H^1\hookrightarrow L^6$, $n\le 3$.
If $B(x_0, \epsilon^{-3/2d})\not\subset \Omega$ then $\epsilon^{-3/2d}\ge dist(x_0,\partial \Omega)$ and boundedness follows as well.
The operators $A_\epsilon$ are not uniformly bounded from $H^2(\Omega)$ to $L^2(\Omega)$.
Take any function $u\in H^2(\Omega)\cap H_0^1(\Omega)$ with $u\ge 1$ on $K_{\epsilon_0}$, $\epsilon_0$ large enough, then
$$
\|u\|_{L^2(K_\epsilon)} \ge meas(K_\epsilon)^{1/2} = \epsilon^{-3/4}
$$
for $\epsilon>\epsilon_0$.

However, if the operators $A_\epsilon$ are second-order elliptic operators then it suffices to assume that the coefficient functions are uniformly bounded with respect to $\epsilon$. See Gilbarg/Trudinger, 'Elliptic Partial Differential Equations of Second Order', Section 8.4.
