THE L p DIRICHLET PROBLEM AND NONDIVERGENCE HARMONIC MEASURE

We consider the Dirichlet problem { Lu = 0 in D u = g on ∂D for two second order elliptic operators Lku = ∑n i,j=1 a i,j k (x) ∂iju(x), k = 0, 1, in a bounded Lipschitz domain D ⊂ IR. The coefficients a k belong to the space of bounded mean oscillation BMO with a suitable small BMO modulus. We assume that L0 is regular in L(∂D, dσ) for some p, 1 < p < ∞, that is, ‖Nu‖Lp ≤ C ‖g‖Lp for all continuous boundary data g. Here σ is the surface measure on ∂D and Nu is the nontangential maximal operator. The aim of this paper is to establish sufficient conditions on the difference of the coefficients ε(x) = a 1 (x) − a i,j 0 (x) that will assure the perturbed operator L1 to be regular in L(∂D, dσ) for some q, 1 < q < ∞.