The Lp Dirichlet problem and nondivergence harmonic measure

We consider the Dirichlet problem [GRAPHICS] for two second-order elliptic operators L(k)u = Sigma(i,j=1)(n) a(k)(i,j)(x) partial derivative(ij)u(x), k = 0,1, in a bounded Lipschitz domain D subset of R-n. The coefficients a(k)(i,j) belong to the space of bounded mea oscillation BMO with a suitable small BMO modulus. We assume that L-0 is regular in L-p (partial derivativeD, dsigma) for some p, 1 < infinity, that is, parallel to Nu parallel to L-p <= C parallel to g parallel to L-p for all continuous boundary data g. Here sigma is the surface measure partial derivative D and Nu is the nontangential maximal operator. The aim of this paper is to establish sufficient conditions on the difference of the coefficients epsilon(i,j)(x) = a(1)(i,j)(x) - a(o)(i,j)(x) that will assure the perturbed operator L-1 to be regular in L-q (partial derivativeD, dsigma) for some q, 1 < q < infinity.