## Elliptic equations in divergence form with drifts in $L^2$

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## Abstract:

We consider the Dirichlet problem for second-order linear elliptic equations in divergence form \begin{equation*} -div(A\nabla u)+\mathbf {b} \cdot \nabla u+\lambda u=f+div\mathbf {F}\quad \text {in } \Omega \quad \text {and}\quad u=0\quad \text {on } \partial \Omega , \end{equation*} in bounded Lipschitz domain $\Omega$ in $\mathbb {R}^2$, where $A:\mathbb {R}^2\rightarrow \mathbb {R}^{2^2}$, $\mathbf {b} : \Omega \rightarrow \mathbb {R}^2$, and $\lambda \geq 0$ are given. If $2<p<\infty$ and $A$ has a small mean oscillation in small balls, $\Omega$ has small Lipschitz constant, and $divA,\,\mathbf {b} \in L^{2}(\Omega ;\mathbb {R}^2)$, then we prove existence and uniqueness of weak solutions in $W^{1,p}_0(\Omega )$ of the problem. Similar result also holds for the dual problem.## References

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## Additional Information

**Hyunwoo Kwon**- Affiliation: Department of Mathematics, Republic of Korea Air Force Academy, Postbox 335-2, 635, Danjae-ro Sangdang-gu, Cheongju-si 28187, Chungcheongbuk-do, Republic of Korea
- MR Author ID: 1430021
- ORCID: 0000-0002-7199-3631
- Email: willkwon@sogang.ac.kr; and willkwon@afa.ac.kr
- Received by editor(s): April 6, 2021
- Received by editor(s) in revised form: August 19, 2021, August 31, 2021, and September 2, 2021
- Published electronically: April 29, 2022
- Communicated by: Ariel Barton
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**150**(2022), 3415-3429 - MSC (2020): Primary 35J15, 35J25
- DOI: https://doi.org/10.1090/proc/15828
- MathSciNet review: 4439464