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

TL;DR

This paper extends the understanding of boundary value problems for elliptic operators with coefficients satisfying Carleson measure conditions, establishing new solvability ranges for Regularity and Neumann problems in Lipschitz domains.

Contribution

It proves the existence of specific p-parameters for which the L^p Regularity and Neumann problems are solvable for such operators, linking these to Dirichlet problem solvability of the adjoint operator.

Findings

Existence of p_reg > 1 for L^p Regularity problem solvability.

Existence of p_neum > 1 for L^p Neumann problem solvability in 2D.

Relationship between p_reg, p_neum, and Dirichlet problem solvability of the adjoint.

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 $\frac1{p_{reg}}+\frac1{q_*}=1$ where $q_*>1$ is the number such that the $L^q$ Dirichlet problem for the adjoint operator $\mathcal L^*$ 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 $\mathcal L=\mbox{div}(A\nabla\cdot)$ is solvable. Furthermore $\frac1{p_{reg}}+\frac1{q^*}=1$ where $q^*>1$ 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 $q>q^*$.