The Dirichlet Problem for elliptic equations with singular drift terms
Steve Hofmann
TL;DR
This paper proves finite-$p$ solvability for the Dirichlet problem in a 1-sided chord-arc domain for elliptic operators with a singular drift, Lu=$-\operatorname{div}(A\nabla u)+\mathbf{B}\cdot\nabla u$, under the assumption that the homogeneous operator $L_0=-\operatorname{div}(A\nabla u)$ is $L^{p_0}$-solvable and that the drift satisfies $|\mathbf{B}(X)|\le C\delta(X)^{-1}$ with $|\mathbf{B}|^2\delta dX$ a Carleson measure. The method blends perturbation theory for drift terms with an extrapolation-of-Carleson-measures technique, working within the flexible framework of 1-sided CADs and ADR boundaries, and relies on a robust foundation of a priori estimates, Green function, and elliptic measure theory. A truncation-approximation scheme plus a dyadic, sawtooth-based extrapolation yields the $A_\infty$ (hence $L^p$ solvability) relation between $L$ and the base operator $L_0$, and the approach also delivers CFMS-type Green function bounds and doubling of elliptic measure. Collectively, the results extend previous Lipschitz-domain perturbation theory to more general rough domains, with quantitative dependence on domain geometry and the Carleson size of the drift.
Abstract
We establish $L^p$ solvability of the Dirichlet problem, for some finite $p$, in a 1-sided chord-arc domain $Ω$ (i.e., a uniform domain with Ahlfors-David regular boundary), for elliptic equations of the form \[ Lu=-\text{div}(A\nabla u) + {\bf B}\cdot \nabla u=:L_0 u+ {\bf B}\cdot \nabla u=0, \] given that the analogous result holds (typically with a different value of $p$) for the homogeneous second order operator $L_0$. Essentially, we assume that $|{\bf B}(X)|\lesssim \text{dist}(X,\partial Ω)^{-1}$, and that $|{\bf B}(X)|^2\text{dist}(X,\partial Ω) dX$ is a Carleson measure in $Ω$.