Suppose we have the Boundary Value Problem (BVP):
$\Delta{u}=f$ , in the domain (topos) $D$
$\partial_{n}{u}=h$, in $\partial{D}$
It is easy to prove using $1$st Green's identity that a necessary condition for the above mentioned BVP is $\int_{\partial{D}}h=\int_{{D}}f$ . But how about the sufficient case here ? Is also this condition enough in order the BVP to have a solution. Any hint or ref is really appreciated.