# American Institute of Mathematical Sciences

August  2022, 21(8): 2587-2613. doi: 10.3934/cpaa.2022062

## Parabolic Systems with measurable coefficients in weighted Sobolev spaces

 a. Department of Mathematics, Korea University, Anam-ro 145, Sungbuk-gu, Seoul, 02841, Republic of Korea b. Department of Mathematics, Ajou University, World cup-ro 206, Yeongtong-gu, Suwon, 16499, Republic of Korea

*Corresponding author

Received  September 2021 Revised  February 2022 Published  August 2022 Early access  March 2022

Fund Project: The first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (2019R1A2C1084683).The second author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (2020R1A2C1A01003354). The third author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (2019R1F1A1058988)

We present a weighted $L_p$-theory of parabolic systems on a half space ${\mathbb{R}}^d_+$. The leading coefficients are assumed to be only measurable in time $t$ and have small bounded mean oscillations (BMO) with respect to the spatial variables $x$, and the lower order coefficients are allowed to blow up near the boundary.

Citation: Doyoon Kim, Kyeong-Hun Kim, Kijung Lee. Parabolic Systems with measurable coefficients in weighted Sobolev spaces. Communications on Pure and Applied Analysis, 2022, 21 (8) : 2587-2613. doi: 10.3934/cpaa.2022062
##### References:
 [1] Sun-Sig Byun and Lihe Wang, Elliptic equations with measurable coefficients in Reifenberg domains, Adv. Math., 225 (2010), 2648-2673.  doi: 10.1016/j.aim.2010.05.014. [2] Hongjie Dong, Recent progress in the $L_p$ theory for elliptic and parabolic equations with discontinuous coefficients, Anal. Theory Appl., 36 (2020), 161-199.  doi: 10.4208/ata.oa-0021. [3] Hongjie Dong and Doyoon Kim, Elliptic equations in divergence form with partially BMO coefficients, Arch. Ration. Mech. Anal., 196 (2010), 25-70.  doi: 10.1007/s00205-009-0228-7. [4] Hongjie Dong and Doyoon Kim, $L_p$ solvability of divergence type parabolic and elliptic systems with partially BMO coefficients, Calc. Var. Partial Differ. Equ., 40 (2011), 357-389.  doi: 10.1007/s00526-010-0344-0. [5] 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), 889-941.  doi: 10.1007/s00205-010-0345-3. [6] Hongjie Dong and Doyoon Kim, Elliptic and parabolic equations with measurable coefficients in weighted Sobolev spaces, Adv. Math., 274 (2015), 681-735.  doi: 10.1016/j.aim.2014.12.037. [7] Hongjie Dong and Doyoon Kim, On $L_p$-estimates for elliptic and parabolic equations with $A_p$ weights, Trans. Amer. Math. Soc., 370 (2018), 5081-5130.  doi: 10.1090/tran/7161. [8] P. Grisvard, Elliptic problems in nonsmooth domains, volume 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985. [9] Doyoon Kim and N. V. Krylov, Elliptic differential equations with coefficients measurable with respect to one variable and VMO with respect to the others, SIAM J. Math. Anal., 39 (2007), 489-506.  doi: 10.1137/050646913. [10] Doyoon Kim and N. V. Krylov, Parabolic equations with measurable coefficients, Potential Anal., 26 (2007), 345-361.  doi: 10.1007/s11118-007-9042-8. [11] Ildoo Kim, Kyeong-Hun Kim and Kijung Lee, A weighted $L_p$-theory for divergence type parabolic PDEs with BMO coefficients on $C^1$-domains, J. Math. Anal. Appl., 412 (2014), 589-612.  doi: 10.1016/j.jmaa.2013.10.079. [12] Kyeong-Hun Kim and N. V. Krylov, On the Sobolev space theory of parabolic and elliptic equations in $C^1$ domains, SIAM J. Math. Anal., 36 (2004), 618-642.  doi: 10.1137/S0036141003421145. [13] Kyeong-Hun Kim and Kijung Lee, A weighted $L_p$-theory for parabolic PDEs with BMO coefficients on $C^1$-domains, J. Differ. Equ., 254 (2013), 368-407.  doi: 10.1016/j.jde.2012.08.002. [14] Kyeong-Hun Kim and Kijung Lee, A weighted $L_p$-theory for second-order parabolic and elliptic partial differential systems on a half space, Commun. Pure Appl. Anal., 15 (2016), 761-794.  doi: 10.3934/cpaa.2016.15.761. [15] Vladimir Kozlov and Alexander Nazarov, The Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients, Math. Nachr., 282 (2009), 1220-1241.  doi: 10.1002/mana.200910796. [16] N. V. Krylov, A $W^n_2$-theory of the Dirichlet problem for SPDEs in general smooth domains, Probab. Theory Related Fields, 98 (1991), 389-421.  doi: 10.1007/BF01192260. [17] N. V. Krylov, Weighted Sobolev spaces and Laplace's equation and the heat equations in a half space, Commun. Partial Differ. Equ., 24 (1999), 1611-1653.  doi: 10.1080/03605309908821478. [18] N. V. Krylov, Weighted Sobolev spaces and the heat equation in the whole space, Appl. Anal., 71 (1999), 111-126.  doi: 10.1080/00036819908840707. [19] N. V. Krylov, Some properties of traces for stochastic and deterministic parabolic weighted Sobolev spaces, J. Funct. Anal., 183 (2001), 1-41.  doi: 10.1006/jfan.2000.3728. [20] N. V. Krylov, Parabolic and elliptic equations with VMO coefficients, Commun. Partial Differ. Equ., 32 (2007), 453-475.  doi: 10.1080/03605300600781626. [21] N. V. Krylov and S. V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients in a half space, SIAM J. Math. Anal., 31 (1999), 19-33.  doi: 10.1137/S0036141098338843. [22] Alois Kufner, Weighted Sobolev spaces, A Wiley-Interscience Publication, John Wiley & Sons Inc., New York, 1985 [23] Lindemulder Nick, Maximal regularity with weights for parabolic problems with inhomogeneous boundary conditions, J. Evol. Equ., 20 (2020), 59-108.  doi: 10.1007/s00028-019-00515-7. [24] José L. Rubio de Francia, Factorization theory and $A_{p}$ weights, Amer. J. Math., 106 (1984), 533-547.  doi: 10.2307/2374284.

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##### References:
 [1] Sun-Sig Byun and Lihe Wang, Elliptic equations with measurable coefficients in Reifenberg domains, Adv. Math., 225 (2010), 2648-2673.  doi: 10.1016/j.aim.2010.05.014. [2] Hongjie Dong, Recent progress in the $L_p$ theory for elliptic and parabolic equations with discontinuous coefficients, Anal. Theory Appl., 36 (2020), 161-199.  doi: 10.4208/ata.oa-0021. [3] Hongjie Dong and Doyoon Kim, Elliptic equations in divergence form with partially BMO coefficients, Arch. Ration. Mech. Anal., 196 (2010), 25-70.  doi: 10.1007/s00205-009-0228-7. [4] Hongjie Dong and Doyoon Kim, $L_p$ solvability of divergence type parabolic and elliptic systems with partially BMO coefficients, Calc. Var. Partial Differ. Equ., 40 (2011), 357-389.  doi: 10.1007/s00526-010-0344-0. [5] 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), 889-941.  doi: 10.1007/s00205-010-0345-3. [6] Hongjie Dong and Doyoon Kim, Elliptic and parabolic equations with measurable coefficients in weighted Sobolev spaces, Adv. Math., 274 (2015), 681-735.  doi: 10.1016/j.aim.2014.12.037. [7] Hongjie Dong and Doyoon Kim, On $L_p$-estimates for elliptic and parabolic equations with $A_p$ weights, Trans. Amer. Math. Soc., 370 (2018), 5081-5130.  doi: 10.1090/tran/7161. [8] P. Grisvard, Elliptic problems in nonsmooth domains, volume 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985. [9] Doyoon Kim and N. V. Krylov, Elliptic differential equations with coefficients measurable with respect to one variable and VMO with respect to the others, SIAM J. Math. Anal., 39 (2007), 489-506.  doi: 10.1137/050646913. [10] Doyoon Kim and N. V. Krylov, Parabolic equations with measurable coefficients, Potential Anal., 26 (2007), 345-361.  doi: 10.1007/s11118-007-9042-8. [11] Ildoo Kim, Kyeong-Hun Kim and Kijung Lee, A weighted $L_p$-theory for divergence type parabolic PDEs with BMO coefficients on $C^1$-domains, J. Math. Anal. Appl., 412 (2014), 589-612.  doi: 10.1016/j.jmaa.2013.10.079. [12] Kyeong-Hun Kim and N. V. Krylov, On the Sobolev space theory of parabolic and elliptic equations in $C^1$ domains, SIAM J. Math. Anal., 36 (2004), 618-642.  doi: 10.1137/S0036141003421145. [13] Kyeong-Hun Kim and Kijung Lee, A weighted $L_p$-theory for parabolic PDEs with BMO coefficients on $C^1$-domains, J. Differ. Equ., 254 (2013), 368-407.  doi: 10.1016/j.jde.2012.08.002. [14] Kyeong-Hun Kim and Kijung Lee, A weighted $L_p$-theory for second-order parabolic and elliptic partial differential systems on a half space, Commun. Pure Appl. Anal., 15 (2016), 761-794.  doi: 10.3934/cpaa.2016.15.761. [15] Vladimir Kozlov and Alexander Nazarov, The Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients, Math. Nachr., 282 (2009), 1220-1241.  doi: 10.1002/mana.200910796. [16] N. V. Krylov, A $W^n_2$-theory of the Dirichlet problem for SPDEs in general smooth domains, Probab. Theory Related Fields, 98 (1991), 389-421.  doi: 10.1007/BF01192260. [17] N. V. Krylov, Weighted Sobolev spaces and Laplace's equation and the heat equations in a half space, Commun. Partial Differ. Equ., 24 (1999), 1611-1653.  doi: 10.1080/03605309908821478. [18] N. V. Krylov, Weighted Sobolev spaces and the heat equation in the whole space, Appl. Anal., 71 (1999), 111-126.  doi: 10.1080/00036819908840707. [19] N. V. Krylov, Some properties of traces for stochastic and deterministic parabolic weighted Sobolev spaces, J. Funct. Anal., 183 (2001), 1-41.  doi: 10.1006/jfan.2000.3728. [20] N. V. Krylov, Parabolic and elliptic equations with VMO coefficients, Commun. Partial Differ. Equ., 32 (2007), 453-475.  doi: 10.1080/03605300600781626. [21] N. V. Krylov and S. V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients in a half space, SIAM J. Math. Anal., 31 (1999), 19-33.  doi: 10.1137/S0036141098338843. [22] Alois Kufner, Weighted Sobolev spaces, A Wiley-Interscience Publication, John Wiley & Sons Inc., New York, 1985 [23] Lindemulder Nick, Maximal regularity with weights for parabolic problems with inhomogeneous boundary conditions, J. Evol. Equ., 20 (2020), 59-108.  doi: 10.1007/s00028-019-00515-7. [24] José L. Rubio de Francia, Factorization theory and $A_{p}$ weights, Amer. J. Math., 106 (1984), 533-547.  doi: 10.2307/2374284.
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