Elliptic integral-differential operators resembling the classical elliptic partial differential equations are defined over a compact d-dimensional p-adic domain together with associated Sobolev spaces relying on coordinate Vladimirov-type Laplacians dating back to an idea of Wilson Zúñiga-Galindo in his previous work. The associated Poisson equations under boundary conditions are solved and their $L_2$-spectra are determined. Under certain finiteness conditions, a Markov semigroup acting on the Sobolev spaces which are also Hilbert spaces can be associated with such an operator and the boundary condition. It is shown that this also has an explicitly given heat kernel as an $L_2$-function, which allows a Green function to be derived from it.