Solvability of differential equations on open subsets of the Sierpinski gasket

We give necessary and sufficient conditions on the polynomial p for the differential equation p(Delta)u = f, based on the Laplacian, to be solvable on any open subset of the Sierpinski gasket for any f continuous on that subset. For general p, we find the open subsets on which p(Delta)u = f is solvable for any continuous f.