Regularity and Neumann Problems for Operators with Real Coefficients Satisfying Carleson Conditions

In this paper, we continue the study of a class of second order elliptic operators of the form L=div(A∇⋅) in a domain above a Lipschitz graph in R ⁿ, where the coefficients of the matrix A satisfy a Carleson measure condition, expressed as a condition on the oscillation on Whitney balls. For this class of operators, it is known (since 2001) that the L ^q Dirichlet problem is solvable for some 1<q<∞. Moreover, further studies completely resolved the range of L ^q solvability of the Dirichlet, Regularity, Neumann problems in Lipschitz domains, when the Carleson measure norm of the oscillation is sufficiently small. We show that there exists p _reg>1 such that for all 1<p<p _reg the L ^p Regularity problem for the operator L=div(A∇⋅) is solvable. Furthermore [Formula presented] where q _⁎>1 is the number such that the L ^q Dirichlet problem for the adjoint operator L ^⁎ is solvable for all q>q _⁎. Additionally when n=2, there exists p _neum>1 such that for all 1<p<p _neum the L ^p Neumann problem for the operator L=div(A∇⋅) is solvable. Furthermore [Formula presented] where q ^⁎>1 is the number such that the L ^q Dirichlet problem for the operator L ₁=div(A ₁∇⋅) with matrix A ₁=A/det⁡A is solvable for all q>q ^⁎.