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

HTML articles powered by AMS MathViewer

- by Hyunwoo Kwon
- Proc. Amer. Math. Soc.
**150**(2022), 3415-3429 - DOI: https://doi.org/10.1090/proc/15828
- Published electronically: April 29, 2022
- PDF | Request permission

## 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

- A. D. Aleksandrov,
*Uniqueness conditions and bounds for the solution of the Dirichlet problem*, Vestnik Leningrad. Univ. Ser. Mat. Meh. Astronom.**18**(1963), no. 3, 5–29 (Russian, with English summary). MR**0164135** - P. Auscher and M. Qafsaoui,
*Observations on $W^{1,p}$ estimates for divergence elliptic equations with VMO coefficients*, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8)**5**(2002), no. 2, 487–509 (English, with English and Italian summaries). MR**1911202** - Haim Brezis,
*Functional analysis, Sobolev spaces and partial differential equations*, Universitext, Springer, New York, 2011. MR**2759829**, DOI 10.1007/978-0-387-70914-7 - Sun-Sig Byun,
*Elliptic equations with BMO coefficients in Lipschitz domains*, Trans. Amer. Math. Soc.**357**(2005), no. 3, 1025–1046. MR**2110431**, DOI 10.1090/S0002-9947-04-03624-4 - Daniel Daners,
*Domain perturbation for linear and semi-linear boundary value problems*, Handbook of differential equations: stationary partial differential equations. Vol. VI, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008, pp. 1–81. MR**2569323**, DOI 10.1016/S1874-5733(08)80018-6 - G. Di Fazio,
*$L^p$ estimates for divergence form elliptic equations with discontinuous coefficients*, Boll. Un. Mat. Ital. A (7)**10**(1996), no. 2, 409–420 (English, with Italian summary). MR**1405255** - Hongjie Dong,
*Recent progress in the $L_p$ theory for elliptic and parabolic equations with discontinuous coefficients*, Anal. Theory Appl.**36**(2020), no. 2, 161–199. MR**4156495**, DOI 10.4208/ata.oa-0021 - Hongjie Dong and Doyoon Kim,
*Elliptic equations in divergence form with partially BMO coefficients*, Arch. Ration. Mech. Anal.**196**(2010), no. 1, 25–70. MR**2601069**, DOI 10.1007/s00205-009-0228-7 - Hongjie Dong and Doyoon Kim,
*Higher order elliptic and parabolic systems with variably partially BMO coefficients in regular and irregular domains*, J. Funct. Anal.**261**(2011), no. 11, 3279–3327. MR**2835999**, DOI 10.1016/j.jfa.2011.08.001 - Hongjie Dong and Doyoon Kim,
*On the $L_p$-solvability of higher order parabolic and elliptic systems with BMO coefficients*, Arch. Ration. Mech. Anal.**199**(2011), no. 3, 889–941. MR**2771670**, DOI 10.1007/s00205-010-0345-3 - Jérôme Droniou,
*Non-coercive linear elliptic problems*, Potential Anal.**17**(2002), no. 2, 181–203. MR**1908676**, DOI 10.1023/A:1015709329011 - Lawrence C. Evans and Ronald F. Gariepy,
*Measure theory and fine properties of functions*, Revised edition, Textbooks in Mathematics, CRC Press, Boca Raton, FL, 2015. MR**3409135**, DOI 10.1201/b18333 - N. Filonov,
*On the regularity of solutions to the equation $-\Delta u+b\cdot \nabla u=0$*, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)**410**(2013), no. Kraevye Zadachi Matematicheskoĭ Fiziki i Smezhnye Voprosy Teorii Funktsiĭ. 43, 168–186, 189 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.)**195**(2013), no. 1, 98–108. MR**3048265**, DOI 10.1007/s10958-013-1566-4 - Nikolay Filonov and Timofey Shilkin,
*On some properties of weak solutions to elliptic equations with divergence-free drifts*, Mathematical analysis in fluid mechanics—selected recent results, Contemp. Math., vol. 710, Amer. Math. Soc., [Providence], RI, [2018] ©2018, pp. 105–120. MR**3818670**, DOI 10.1090/conm/710/14366 - Claus Gerhardt,
*Stationary solutions to the Navier-Stokes equations in dimension four*, Math. Z.**165**(1979), no. 2, 193–197. MR**520820**, DOI 10.1007/BF01182469 - David Gilbarg and Neil S. Trudinger,
*Elliptic partial differential equations of second order*, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR**1814364**, DOI 10.1007/978-3-642-61798-0 - Byungsoo Kang and Hyunseok Kim,
*$W^{1,p}$-estimates for elliptic equations with lower order terms*, Commun. Pure Appl. Anal.**16**(2017), no. 3, 799–821. MR**3623550**, DOI 10.3934/cpaa.2017038 - Byungsoo Kang and Hyunseok Kim,
*On $L^p$-resolvent estimates for second-order elliptic equations in divergence form*, Potential Anal.**50**(2019), no. 1, 107–133. MR**3900848**, DOI 10.1007/s11118-017-9675-1 - Hyunseok Kim and Young-Heon Kim,
*On weak solutions of elliptic equations with singular drifts*, SIAM J. Math. Anal.**47**(2015), no. 2, 1271–1290. MR**3328143**, DOI 10.1137/14096270X - H. Kim and H. Kwon,
*Dirichlet and Neumann problems for elliptic equations with singular drifts on Lipschitz domains*, Trans. Amer. Math. Soc., to appear, arXiv:1811.12619, 2021. - Hyunseok Kim and Tai-Peng Tsai,
*Existence, uniqueness, and regularity results for elliptic equations with drift terms in critical weak spaces*, SIAM J. Math. Anal.**52**(2020), no. 2, 1146–1191. MR**4075335**, DOI 10.1137/19M1282969 - N. V. Krylov,
*Parabolic and elliptic equations with VMO coefficients*, Comm. Partial Differential Equations**32**(2007), no. 1-3, 453–475. MR**2304157**, DOI 10.1080/03605300600781626 - N. V. Krylov,
*On stochastic equations with drift in $L_d$*, Ann. Probab.**49**(2021), no. 5, 2371–2398. MR**4317707**, DOI 10.1214/21-aop1510 - N. V. Krylov,
*Elliptic equations with VMO $a, b\in L_d$, and $c\in L_{d/2}$*, Trans. Amer. Math. Soc.**374**(2021), no. 4, 2805–2822. MR**4223034**, DOI 10.1090/tran/8282 - Hyunwoo Kwon,
*Existence and uniqueness of weak solution in $W^{1,2+\varepsilon }$ for elliptic equations with drifts in weak-$L^n$ spaces*, J. Math. Anal. Appl.**500**(2021), no. 1, Paper No. 125165, 19. MR**4235253**, DOI 10.1016/j.jmaa.2021.125165 - Olga A. Ladyzhenskaya and Nina N. Ural’tseva,
*Linear and quasilinear elliptic equations*, Academic Press, New York-London, 1968. Translated from the Russian by Scripta Technica, Inc; Translation editor: Leon Ehrenpreis. MR**0244627** - Gioconda Moscariello,
*Existence and uniqueness for elliptic equations with lower-order terms*, Adv. Calc. Var.**4**(2011), no. 4, 421–444. MR**2844512**, DOI 10.1515/ACV.2011.007 - Guido Stampacchia,
*Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus*, Ann. Inst. Fourier (Grenoble)**15**(1965), no. fasc. 1, 189–258 (French). MR**192177**, DOI 10.5802/aif.204 - Neil S. Trudinger,
*Linear elliptic operators with measurable coefficients*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3)**27**(1973), 265–308. MR**369884**

## Bibliographic 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