Regularity and Neumann problems for operators with real coefficients satisfying Carleson conditions

Abstract

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 Rn, 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 Lq Dirichlet problem is solvable for some 1<q<∞. Moreover, further studies completely resolved the range of Lq 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 preg>1 such that for all 1<p<preg the Lp Regularity problem for the operator L=div(A∇⋅) is solvable. Furthermore 1preg+1q⁎=1 where q⁎>1 is the number such that the Lq Dirichlet problem for the adjoint operator L⁎ is solvable for all q>q⁎.Additionally when n=2, there exists pneum>1 such that for all 1<p<pneum the Lp Neumann problem for the operator L=div(A∇⋅) is solvable. Furthermore 1preg+1q⁎=1 where q⁎>1 is the number such that the Lq Dirichlet problem for the operator L1=div(A1∇⋅) with matrix A1=A/det⁡A is solvable for all q>q⁎.