The $L^p$ Dirichlet problem and nondivergence harmonic measure

We consider the Dirichlet problem

for two second-order elliptic operators $\mathcal{L}_k u=\sum_{i,j=1}^na_k^{i,j}(x)\,\partial_{ij} u(x)$, $k=0,1$, in a bounded Lipschitz domain $D\subset\mathbb{R} ^n$. The coefficients $a_k^{i,j}$ belong to the space of bounded mean oscillation ${{BMO}}$ with a suitable small ${{BMO}}$ modulus. We assume that ${\mathcal{L}}_0$ is regular in $L^p(\partial D, d\sigma)$ for some $p$, $1<p<\infty$, that is, $\Vert Nu\Vert _{L^p}\le C\,\Vert g\Vert _{L^p}$ for all continuous boundary data $g$. Here $\sigma$ is the surface measure on $\partial D$ and $Nu$ is the nontangential maximal operator. The aim of this paper is to establish sufficient conditions on the difference of the coefficients $\varepsilon^{i,j}(x)=a^{i,j}_1(x)-a^{i,j}_0(x)$ that will assure the perturbed operator $\mathcal{L}_1$ to be regular in $L^q(\partial D,d\sigma)$ for some $q$, $1<q<\infty$.