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

Abstract

In this paper, we continue the study of a class of second order elliptic operators of the form $\mathcal L=\mbox{div}(A\nabla\cdot)$ in a domain above a Lipschitz graph in ${\mathbb{R}}^n,$ 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 $1<q<\infty$. 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 $\mathcal L=\mbox{div}(A\nabla\cdot)$ is solvable. Furthermore $\frac{1}{p_{reg}}+\frac{1}{q_{\ast }}=1$ where $q_{\ast }>1$ is the number such that the $L^q$ Dirichlet problem for the adjoint operator ${\mathcal{L}}^{\ast }$ is solvable for all $q>q_{\ast }$. 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 $\mathcal L=\mbox{div}(A\nabla\cdot)$ is solvable. Furthermore $\frac{1}{p_{reg}}+\frac{1}{q^{\ast }}=1$ where $q^{\ast }>1$ is the number such that the $L^q$ Dirichlet problem for the operator $\mathcal L_1=\mbox{div}(A_1\nabla\cdot)$ with matrix $A_1=A/\det A$ is solvable for all $q>q^{\ast }$.

27 pages. V2 has an updated and shortened argument