AI and the World's Hard Problems · Mathematics · arXiv · 2026-10-21
Elliptic theory and dimension drop for operators with divergence-free form-bounded drifts in one-sided chord-arc domains
The authors develop boundary value and potential theory for elliptic operators with form-bounded, divergence-free drift in one-sided chord-arc domains, establishing local regularity estimates, solvability of the Dirichlet problem, and the associated harmonic measure. In the divergence-free case they additionally prove Green's function and gradient estimates, CFMS estimates, doubling of harmonic measure, a comparison principle for nonnegative solutions with zero trace, and a change of pole formula. Using AI, they then construct, in the upper half-space, an operator of the form -Δ+B·∇ with B form-bounded and divergence-free whose harmonic measure exhibits dimension drop.