Consider this "Neumann problem" with Laplace's Equation: Find a solution to $\nabla^2\phi=0$ inside an N-dimensional hypercube which has a given (on the $2N$ faces of the hypercube) normal gradient $\eta$ (assume that mixed partial derivatives of all orders of $\eta$ exist on the faces). By the way, if a solution $\phi$ exists, we know it's unique to within a constant.
There are some obvious boundary checks for $\eta$. At the "hypercube edges" (where two faces touch), assume the curl of $\eta$ is zero. At the "hypercube corners" (where N faces touch), assume the divergence of $\eta$ is zero. As a whole, assume the integral over all faces of $\eta$ is zero.
So, does there then always exist a solution $\phi$?
I am wondering, first, if I have made enough assumptions (especially "that mixed partial derivatives of all orders of $\eta$ exist on the faces") to solve the 3D case, and second, if the same holds in higher dimensions. If I need some more assumptions, please specify them.
(I am also curious about the "Dirichlet problem" for which the only difference above would be that "obvious boundary checks" instead assume only that the Laplacian is zero at "hypercube edges" - Please answer for that too if it is easy.)