The L^p Dirichlet Problem and Nondivergence Harmonic Measure

Abstract

We consider the Dirichlet problem Lu = 0 in D u = g on E = boundary of D for two second order elliptic operators L_k(u) = \sum_{i,j=1}^n a_k^{ij}(x) \partial_{ij} u(x), k=0,1, in a bounded Lipschitz domain D in R^n. The coefficients a_k^{ij} belong to the space of bounded mean oscillation BMO with a suitable small BMO modulus. We assume that L_0 is regular in L^p(E,ds) for some p, 1<p<\infty, that is, |Nu|_{L^p}< C |g|_{L^p} for all continuous boundary data g. Here ds is the surface measure on E and Nu is the nontangential maximal operator. The aim of this paper is to establish sufficient conditions on the difference of the coefficients a_1^{ij}(x)-a_0^{ij}(x) that will assure the perturbed operator L_1 to be regular in L^q(E,ds) for some q, 1<q<\infty.