Carleson perturbations of elliptic operators on domains with low dimensional boundaries

We prove an analogue of a perturbation result for the Dirichlet problem divergence form elliptic operators by Fefferman, Kenig and Pipher, degenerate David, Feneuil Mayboroda, which were developed to study geometric analytic properties sets with boundaries whose co-dimension is higher than 1. These are ? div A ? , where weighted matrix crafted weigh distance high boundary in way that allows nourishment PDE theory. When this d -Alhfors-David regular set R n ? [ 1 ) ? 3 we membership harmonic measure ? preserved under Carleson perturbations coefficients, yielding turn L p -solvability also stable these (with possibly different ). If suitably small, establish solvability same space. One corollaries our results together previous Engelstein that, given any -ADR ? 2 there family described above absolutely continuous respect -dimensional Hausdorff on ?.