Solvability of the Dirichlet problem using a weaker Carleson condition in the upper half plane

Abstract

We study an elliptic operator $L:=\mathrm{div}(A\nabla \cdot )$ on the upper half plane ${\mathbb{R}}_{+}^2$. There are several conditions on the behavior of the matrix $A$ in the transversal $t$-direction that yield $\omega \in A_{\infty }(\sigma )$. These include the $t$-independence condition, a mixed $L^1-L^{\infty }$ condition on ${\partial }_{t}A$, and Dini-type conditions. We introduce an $L^1$ Carleson condition on ${\partial }_{t}A(x,t)$ that extends the class of elliptic operators for which we have $\omega \in A_{\infty }(\sigma )$, i.e. solvability of the $L^p$ Dirichlet problem for some $1<p<\infty$.

arXiv admin note: text overlap with arXiv:2311.00614