Elliptic theory and dimension drop for operators with divergence-free form-bounded drifts in one-sided chord-arc domains
Abstract
We develop a boundary value and potential theory for elliptic operators L=−Div(A∇)+B·∇ with form-bounded drift in one-sided chord-arc domains. Assuming that the positive part of DivB 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 DivB≤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 DivB=0 or B∈ Ln,1textuploc(Ω). 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 −Δ+B·∇ with B form-bounded and divergence-free, whose harmonic measure exhibits dimension drop.
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BibTeX
@misc{mayboroda2026elliptic,
title = {{Elliptic theory and dimension drop for operators with divergence-free form-bounded drifts in one-sided chord-arc domains}},
author = {Svitlana Mayboroda and Alberto Pacati},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19983},
eprint = {2610.19983},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19983},
}
RIS
TY - UNPB
TI - Elliptic theory and dimension drop for operators with divergence-free form-bounded drifts in one-sided chord-arc domains
AU - Svitlana Mayboroda
AU - Alberto Pacati
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.19983
UR - https://arxiv.org/abs/2610.19983
AB - We develop a boundary value and potential theory for elliptic operators L=−Div(A∇)+B·∇ with form-bounded drift in one-sided chord-arc domains. Assuming that the positive part of DivB 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 DivB≤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 DivB=0 or B∈ L^n,1_textuploc(Ω). 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 −Δ+B·∇ with B form-bounded and divergence-free, whose harmonic measure exhibits dimension drop.
ER -