Large Perturbations of Elliptic Operators and the Solvability of the Lp Dirichlet Problem

In this article, we investigate the solvability of the Lp Dirichlet problem for an elliptic operator, in divergence form, with bounded measurable coefficients. To be specific, assume that we have two such operators in the unit ball of Rn. Suppose that we know that the Lp Dirichlet problem is solvable for the first operator. We derive simple, easy to check criteria on the difference between the coefficients of the two operators which imply that the second operator has its Lp Dirichlet problem solvable as well. We emphasize that the value of p is the same for both operators and the criteria on the difference between the two coefficients allow for "large" perturbations (the definition of the notion of "large" perturbation is given quite precisely).