Elliptic theory and dimension drop for operators with divergence-free form-bounded drifts in one-sided chord-arc domains

We develop a boundary value and potential theory for elliptic operators $$L=-\text{Div}(A\mathrm{\nabla })+\text{B}\cdot \mathrm{\nabla }$$ with form-bounded drift in one-sided chord-arc domains. Assuming that the positive part of $\text{Div}\text{B}$ has sufficiently small form bound, we establish local regularity estimates and solvability of the Dirichlet problem, and construct the associated harmonic measure. Under the additional assumption $\text{Div}\text{B}\le 0$, we construct a Green’s function and prove pointwise and gradient estimates. For the characteristic distributional identity with a Dirac mass at the pole to hold, we will additionally assume either $\text{Div}\text{B}=0$ or $\text{B}\in L_{\text{loc}}^{n,1}(\mathrm{\Omega })$. Moreover, in the divergence-free case, we obtain a local lower bound for the Green’s function, CFMS estimates, doubling of harmonic measure, a comparison principle for nonnegative solutions with zero trace, and a change of pole formula. Finally, using AI, we construct, in the upper half-space, an operator of the form $-\mathrm{\Delta }+\text{B}\cdot \mathrm{\nabla }$ with $\text{B}$ form-bounded and divergence-free, whose harmonic measure exhibits dimension drop.