While doing my research, I encountered the following problem as:
is there any regularity result for solutions to the Neumann problem for an elliptic PDE on a domain with piecewise smooth boundary?<br> For example, consider a domain whose boundary is the union of two smooth hypersurfaces with orthogonal normal field on their intersection. Let $f$ be the solution of the following Neumann problem:
\begin{equation}
\left\{\begin{array}{ll}
\mbox{div}(\nabla f)=g,& \mbox{ in }\Omega\\
f_{\nu}=g_1,&\mbox{ on }\Sigma_1\\
f_{\nu}=g_2,& \mbox{ on }\Sigma_2
\end{array}\right.
\end{equation}
where $\partial\Omega=\Sigma_1\cup\Sigma_2$, and $\Sigma_1$ and $\Sigma_2$ are smooth and perpendicular hypersurfaces. Is there any restriction on the data $g$, $g_1$, $g_2$ for the regularitiy? (we can even assume all of them are smooth.)
Any comments are welcomed.