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
