Endpoint solvability results for divergence form, complex elliptic equations

We consider divergence form elliptic equations Lu := ∇ · (A∇u) = 0 in the half space R + := {(x, t) ∈ R × (0,∞)}, whose coefficient matrix A is complex elliptic, bounded and measurable. In addition, we suppose that A satisfies some additional regularity in the direction transverse to the boundary, namely that the discrepancy A(x, t) − A(x, 0) satisfies a Carleson measure condition of Fefferman-Kenig-Pipher type, with small Carleson norm. Under these conditions, we obtain solvability of the Dirichlet problem for L, with data in Λα(R) (which is defined to be BMO(R) when α = 0 and the space of Hölder continuous functions C(R) when α ∈ (0, 1)) for α < α0, where α0 is the De Giorgi-Nash exponent, and solvability of the Neumann and Regularity problems, with data in the spaces H(R) and H(R) respectively, for p ∈ ( n n+α0 , 1], assuming that we have bounded Layer Potentials in L(R) and invertible Layer Potentials in Λα(R) and H(R) for the t-independent operator L0 := −∇ · (A(·, 0)∇).