The L^p Dirichlet problem for divergence form elliptic operators with non-smooth coefficients

We present conditions on the disagreement between two elliptic operators which ensure the preservation of L^p solvability for the Dirichlet problem on the unit ball in ℝⁿ. The conditions depend on p and differ from previous results on this question. Furthermore, the conditions are sharp in the sense that they are equivalent to L^p solvability for a special class of operators which arise in the n = 2 case.