@Barry interesting could you elaborate how to find such representations?

May be I should change "the vector Laplacian" to "a vector Laplacian" because in discrete objects one can present many different Laplacians easily which could possibly generalize many different versions of Laplacians in continuous case.

@Willie Wong Actually I obtained this pde as a generalization of a discrete optimization problem. For the discrete case the operator was a circulant matrix with eigen values $\frac{n-1}{2}$ pairs of complex conjugate eigen values plus an extra eigen value of $1$. For special reasons my worry is only when $n$ is odd. With this Laplacian eigen values, is the operator still elliptic?

@David Thank you!

What is the class of equations formally called in the literature?

@David Roberts: Do you have a reference for the cube? @Willie Wong: WHere can I get more information on this? Is there a formal write up of explicit equations/methodology somewhere?

Added an useful restriction to $g$.