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September  2016, 36(9): 4895-4914. doi: 10.3934/dcds.2016011

Neumann problem for non-divergence elliptic and parabolic equations with BMO$_x$ coefficients in weighted Sobolev spaces

1. 

Department of Mathematics, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul, 02841, South Korea

2. 

Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, United States

Received  May 2015 Revised  February 2016 Published  May 2016

We prove the unique solvability in weighted Sobolev spaces of non-divergence form elliptic and parabolic equations on a half space with the homogeneous Neumann boundary condition. All the leading coefficients are assumed to be only measurable in the time variable and have small mean oscillations in the spatial variables. Our results can be applied to Neumann boundary value problems for stochastic partial differential equations with BMO$_x$ coefficients.
Citation: Doyoon Kim, Hongjie Dong, Hong Zhang. Neumann problem for non-divergence elliptic and parabolic equations with BMO$_x$ coefficients in weighted Sobolev spaces. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4895-4914. doi: 10.3934/dcds.2016011
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show all references

References:
[1]

Proc. Amer. Math. Soc., 91 (1984), 213-216.  Google Scholar

[2]

Ricerche Mat., 40 (1991), 149-168.  Google Scholar

[3]

Trans. Amer. Math. Soc., 336 (1993), 841-853. doi: 10.2307/2154379.  Google Scholar

[4]

Comment. Math. Univ. Carolin., 37 (1996), 537-556.  Google Scholar

[5]

Algebra i Analiz, 23 (2011), 150-174. doi: 10.1090/S1061-0022-2012-01206-9.  Google Scholar

[6]

J. Funct. Anal., 258 (2010), 2145-2172. doi: 10.1016/j.jfa.2010.01.003.  Google Scholar

[7]

Calc. Var. Partial Differential Equations, 40 (2011), 357-389. doi: 10.1007/s00526-010-0344-0.  Google Scholar

[8]

Arch. Ration. Mech. Anal., 199 (2011), 889-941. doi: 10.1007/s00205-010-0345-3.  Google Scholar

[9]

Adv. Math., 274 (2015), 681-735. doi: 10.1016/j.aim.2014.12.037.  Google Scholar

[10]

Trans. Amer. Math. Soc., 368 (2016), 7413-7460. doi: 10.1090/tran/6605.  Google Scholar

[11]

J. Math. Anal. Appl., 334 (2007), 534-548. doi: 10.1016/j.jmaa.2006.12.077.  Google Scholar

[12]

J. Math. Anal. Appl., 412 (2014), 589-612. doi: 10.1016/j.jmaa.2013.10.079.  Google Scholar

[13]

J. Theoret. Probab., 27 (2014), 107-136. doi: 10.1007/s10959-012-0459-7.  Google Scholar

[14]

SIAM J. Math. Anal., 36 (2004), 618-642. doi: 10.1137/S0036141003421145.  Google Scholar

[15]

J. Differential Equations, 254 (2013), 368-407. doi: 10.1016/j.jde.2012.08.002.  Google Scholar

[16]

In Proceedings of the St. Petersburg Mathematical Society. Vol. XV. Advances in mathematical analysis of partial differential equations, volume 232 of Amer. Math. Soc. Transl. Ser. 2, 177-191. Amer. Math. Soc., Providence, RI, 2014.  Google Scholar

[17]

Math. Nachr., 282 (2009), 1220-1241. doi: 10.1002/mana.200910796.  Google Scholar

[18]

Probab. Theory Related Fields, 98 (1994), 389-421. doi: 10.1007/BF01192260.  Google Scholar

[19]

Comm. Partial Differential Equations, 24 (1999), 1611-1653. doi: 10.1080/03605309908821478.  Google Scholar

[20]

Comm. Partial Differential Equations, 32 (2007), 453-475. doi: 10.1080/03605300600781626.  Google Scholar

[21]

American Mathematical Society, Providence, RI, 2008. doi: 10.1090/gsm/096.  Google Scholar

[22]

Stochastic Process. Appl., 119 (2009), 2095-2117. doi: 10.1016/j.spa.2008.11.003.  Google Scholar

[23]

SIAM J. Math. Anal., 31 (1999), 19-33. doi: 10.1137/S0036141098338843.  Google Scholar

[24]

SIAM J. Math. Anal., 30 (1999), 298-325. doi: 10.1137/S0036141097326908.  Google Scholar

[25]

J. Funct. Anal., 250 (2007), 521-558. doi: 10.1016/j.jfa.2007.04.003.  Google Scholar

[26]

A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1985. Translated from the Czech.  Google Scholar

[27]

Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24 (1997), 537-549.  Google Scholar

[28]

Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 5 (1967), 250-254.  Google Scholar

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