@article{Lee2025, 
author = {Haesung Lee},
title = {Well-posedness of linear elliptic equations with        L    d  -drifts under divergence-type conditions},
year = {2025},
journal = {Electronic Research Archive},
volume = {33},
number = {12},
pages = {7974-7998},
keywords = {linear elliptic equations, well-posedness, weak solutions, divergence-free transformation, Harnack inequality, weak maximum principle},
url = {https://www.sciopen.com/article/10.3934/era.2025351},
doi = {10.3934/era.2025351},
abstract = {We establish the well-posedness of linear elliptic equations with critical-order drifts in        L    d   and positive zero-order coefficients in        L    1   or        L                            2          d                          d          +          2                    , where classical methods are often too restrictive. Our approach relies on a divergence-free transformation and a structural condition on the drift vector field, which admits a decomposition into a regular component and another whose weak divergence belongs to        L                            q          ~                     for some              q      ~        >      d    2  . This condition is essential for constructing a suitable weight function    ρ via the weak maximum principle and the Harnack inequality. Within this framework, we prove the existence and uniqueness of weak solutions, significantly relaxing the regularity assumptions on the zero-order coefficients in        L                  d        2            .}
}