Endpoint solvability results for divergence form, complex elliptic equations

Abstract

We consider divergence form elliptic equations Lu := ∇ • (A∇u) = 0 in the half space ℝn+1+ := {(x,t)∈ ℝn x (0,∞)}, whose coeffi cient matrix A is complex elliptic, bounded and measurable. In addition, we suppose that A satisfi es some additional regularity in the direction transverse to the boundary, namely that the discrepancy A(x,t) - A(x,0) satis fies a Carleson measure condition of Feff erman-Kenig-Pipher type, with small Carleson norm. Under these conditions, we obtain solvability of the Dirichlet problem for L, with data in Λα (ℝn) (which is defi ned to be BMO(ℝn) when α = 0 and the space of Holder continuous functions C α(ℝn) when α ∈(0, 1)) for α < α0, where 0 is the De Giorgi-Nash exponent, and solvability of the Neumann and Regularity problems, with data in the spaces Hp(Rn) and H1,p(ℝn) respectively, for p ∈ ( n/n+α 0, 1], assuming that we have bounded Layer Potentials in L2(ℝn) and invertible Layer Potentials in Λα (ℝn) and Hp(ℝn) for the t-independent operator L0 := -∇ • (A( • ,0)∇).