Hello all,


could someone point me to a reference that ties the smoothness of the solution $u$ to the classical elliptic problem


$\nabla \cdot ( q \nabla u ) = f \;,\; x \in \Omega$


$u = g \;,\; x \in \Gamma = \partial \Omega$ 


to the smoothness of $f$, $q$ and $g$? 


$\Omega$ is a convex polygonal domain in $\Re^d$ with $d \in \{2,3\}$. The boundary $\Gamma$ is piecewise linear (can have corners, e.g., if $\Omega$ is the unit square).


I am particularly interested in the (minimal) smoothness requirements for the forcing and boundary data $f$ and $g$, such that $u \in {\cal H}^2(\Omega)$ (not just locally). 


I went through Evans' book on PDEs but he assumes homogeneous boundaries and proves only local smoothness $u \in {\cal H}\_{\rm loc}^s(\Omega)$ based on assumptions on the forcing $f$. My $g$ is generally nonzero.


Also, would the smoothness theory for the BVP above extend to a Helmholtz problem with a pure Neumann BC?


Thanks for any good pointers!


Kind regards, 
 -- Mihai



smoothness of solution for second order elliptic problem
Grisvard's book is a standard reference for elliptic problems in domains with corners.


