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

TL;DR

This paper proves that in vanishing chord-arc domains, elliptic operators with H"older-continuous coefficients have their elliptic kernel logarithm in VMO, extending previous results for the Laplacian.

Contribution

It establishes the regularity of the elliptic kernel for a broader class of elliptic operators in complex domains, generalizing prior work on the Laplacian.

Findings

Logarithm of the elliptic kernel belongs to VMO in vanishing chord-arc domains.

Extension of Kenig and Toro's results from Laplacian to H"older-continuous coefficient elliptic operators.

Provides new regularity insights for elliptic kernels in complex geometric settings.

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.