I have two questions, one specific and one general, on this result for compactly supported solutions to the divergence equation in star-like domains. The result which can be found in Galdi's "An introduction to the mathematical theory of the Navier-Stokes equations" volume 1, and may have originated with Bogovskii in 1979 or so, states:
If $\Omega$ is an open bounded domain of $\mathbb{R}^n$ ($n\geq 2$) that is star-like with respect to every point in the unit ball $B_1(0)=B,\ \overline{B}\subset \Omega$, and $g\in W_0^{k,p}(\Omega)$ for $k\geq 0$ and $p>1$ with $\int_\Omega g\ dV=0$ then letting $\omega$ be smooth, compactly supporting in $B$, and $\int_B\omega\ dV=1$, the vector field $$ \mathbf{v}(\mathbf{x})= \int_\Omega g(\mathbf{y})\left[ \frac{\mathbf{x}-\mathbf{y}}{|\mathbf{x}-\mathbf{y}|^n}\int_{|\mathbf{x}-\mathbf{y}|}^\infty \omega\left(\mathbf{y}+\xi\frac{\mathbf{x}-\mathbf{y}}{|\mathbf{x}-\mathbf{y}|}\right)\xi^{n-1}d\xi\right]dV_\mathbf{y} $$ satisfies $\nabla \cdot \mathbf{v}(\mathbf{x})=g(\mathbf{x})$ and $\mathbf{v}\in W_0^{k+1,p}(\Omega)^n$. Specifically, $\mathbf{v}$ has support in the convex combinations of the support of $g$ and $B$.
If $g\in L^p(\Omega)$ this provides a solution $\mathbf{v}\in W_0^{1,p}(\Omega)^n$, however if $g$ has more regularity (but is not compactly supported) it does not appear that $\mathbf{v}$ necessarily possesses the extra regularity up to the boundary (does it?). Other methods show there are solutions with higher regularity up to the boundary but may not retain the property of compact support.
My specific question is if there is a manner to adapt this result to a case like $\Omega_+=\{\mathbf{x}\in \Omega: \mathbf{e}_0\cdot \mathbf{x}>0\}$ and given data $g\in W^{k,p}_0(\Omega)$ get a solution $\mathbf{v}\in W^{k,p}(\Omega_+)^n$, which satisfies Dirichlet boundary conditions when $\mathbf{x}\cdot\mathbf{e}_0=0$ and has compact support in $\overline{\Omega_+}\cap \Omega$. I think this would be a useful result and wonder if there is a similar formula including a Green's function on the hyperplane. I am not very comfortable with the techniques and would appreciate the help of someones expertise.
More generally, what is the property of the divergence operator that allows the construction of such a compactly supported solution and are there other operators for which a similar result applies?
Thanks for any help and discussion!