Is this bootstrap argument correct?

Is this bootstrap argument correct?

The setup. Assume $\Omega \subset \mathbb{R}^3$ bounded. Let $v \in
W^{1,2}$ be a weak solution of $$ \begin{cases} -\Delta v = g & \text{in }
\Omega \\ v=0 & \text{on } \Omega\end{cases} \tag{1}$$
Now assume we can show that $g \in L^{12/11}(\Omega)$. From standard
elliptic theory we may infer that $v \in W^{2,12/11}(\Omega)$.
The problem. Now, I even know that $D g (:=\nabla g) \in
L^{12/11}(\Omega)$ and want to infer that $v \in W^{3,12/11}$ and I want
to use a bootstrap argument to do so.
My attempt. Since $v$ is a weak solution to (1) we know that $$
\int_\Omega Dv Dw = \int_\Omega g w $$ for all $w \in H^1(0,1)$.
We choose such test function $w \in C_c^\infty(\Omega)$ and set $$ u:=-Dw.$$
Inserting $u$ for $w$ in the equation above, we get $$ -\int_\Omega DvD^2w
= -\int_\Omega gDw. $$ Integration by parts yields $$ \int_\Omega D^2 v Dw
= \int_\Omega Dg w. $$
This means that $Dv$ is also weak solution to $$ \begin{cases} -\Delta v =
Dg & \text{in } \Omega \\ v=0 & \text{on } \Omega\end{cases} \tag{1}$$ and
the claim follows by standard elliptic arguments as above.
The question. Is this argument correct? If not: what can I do to make it
precise?