While doing my research, I encounter a problem as follows: Is there any regularity result on a elliptic PDE with piecewise smooth boundary and Neumann Data, for example, the boundary a union of two smooth hypersurface with orthogonal normal field on their intersection. Let $f$ be the solution of the following PDE: \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 $g$, $g_1$, $g_2$ for the regularitiy? (we can even assume all off them are smooth.) Any commons are welcomed.

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?
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.