Optimal Poisson kernel regularity for elliptic operators with H\"older-continuous coefficients in vanishing chord-arc domains

Abstract

We show that if $\Omega$ is a vanishing chord-arc domain and $L$ is a divergence-form elliptic operator with H\"older-continuous coefficient matrix, then $\log k_L \in VMO$, where $k_L$ is the elliptic kernel for $L$ in the domain $\Omega$. This extends the previous work of Kenig and Toro in the case of the Laplacian.