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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 - A, 2016, 36 (9) : 4895-4914. doi: 10.3934/dcds.2016011
##### References:
 [1] H. Aimar and R. A. Macías, Weighted norm inequalities for the Hardy-Littlewood maximal operator on spaces of homogeneous type,, Proc. Amer. Math. Soc., 91 (1984), 213.   Google Scholar [2] F. Chiarenza, M. Frasca and P. Longo, Interior $W^{2,p}$ estimates for nondivergence elliptic equations with discontinuous coefficients,, Ricerche Mat., 40 (1991), 149.   Google Scholar [3] F. Chiarenza, M. Frasca and P. Longo, $W^{2,p}$-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients,, Trans. Amer. Math. Soc., 336 (1993), 841.  doi: 10.2307/2154379.  Google Scholar [4] G. Di Fazio and D. K. Palagachev, Oblique derivative problem for elliptic equations in non-divergence form with VMO coefficients,, Comment. Math. Univ. Carolin., 37 (1996), 537.   Google Scholar [5] H. Dong, Parabolic equations with variably partially VMO coefficients,, Algebra i Analiz, 23 (2011), 150.  doi: 10.1090/S1061-0022-2012-01206-9.  Google Scholar [6] H. Dong, Solvability of parabolic equations in divergence form with partially BMO coefficients,, J. Funct. Anal., 258 (2010), 2145.  doi: 10.1016/j.jfa.2010.01.003.  Google Scholar [7] H. Dong and D. Kim, $L_p$ solvability of divergence type parabolic and elliptic systems with partially BMO coefficients,, Calc. Var. Partial Differential Equations, 40 (2011), 357.  doi: 10.1007/s00526-010-0344-0.  Google Scholar [8] H. Dong and D. Kim, On the $L_p$-solvability of higher order parabolic and elliptic systems with BMO coefficients,, Arch. Ration. Mech. Anal., 199 (2011), 889.  doi: 10.1007/s00205-010-0345-3.  Google Scholar [9] H. Dong and D. Kim, Elliptic and parabolic equations with measurable coefficients in weighted Sobolev spaces,, Adv. Math., 274 (2015), 681.  doi: 10.1016/j.aim.2014.12.037.  Google Scholar [10] H. Dong and H. Zhang, Conormal problem of higher-order parabolic systems,, Trans. Amer. Math. Soc., 368 (2016), 7413.  doi: 10.1090/tran/6605.  Google Scholar [11] D. Kim, Parabolic equations with measurable coefficients. II,, J. Math. Anal. Appl., 334 (2007), 534.  doi: 10.1016/j.jmaa.2006.12.077.  Google Scholar [12] I. Kim, K.-H. Kim and K. 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.  doi: 10.1016/j.jmaa.2013.10.079.  Google Scholar [13] K.-H. Kim, A weighted Sobolev space theory of parabolic stochastic PDEs on non-smooth domains,, J. Theoret. Probab., 27 (2014), 107.  doi: 10.1007/s10959-012-0459-7.  Google Scholar [14] K.-H. 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.  doi: 10.1137/S0036141003421145.  Google Scholar [15] K.-H. Kim and K. Lee, A weighted $L_p$-theory for parabolic PDEs with BMO coefficients on $C^1$-domains,, J. Differential Equations, 254 (2013), 368.  doi: 10.1016/j.jde.2012.08.002.  Google Scholar [16] V. Kozlov and A. Nazarov, Oblique derivative problem for non-divergence parabolic equations with time-discontinuous coefficients,, In Proceedings of the St. Petersburg Mathematical Society. Vol. XV. Advances in mathematical analysis of partial differential equations, (2014), 177.   Google Scholar [17] V. Kozlov and A. Nazarov, The Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients,, Math. Nachr., 282 (2009), 1220.  doi: 10.1002/mana.200910796.  Google Scholar [18] N. V. Krylov, A $W^n_2$-theory of the Dirichlet problem for SPDEs in general smooth domains,, Probab. Theory Related Fields, 98 (1994), 389.  doi: 10.1007/BF01192260.  Google Scholar [19] N. V. Krylov, Weighted Sobolev spaces and Laplace's equation and the heat equations in a half space,, Comm. Partial Differential Equations, 24 (1999), 1611.  doi: 10.1080/03605309908821478.  Google Scholar [20] N. V. Krylov, Parabolic and elliptic equations with VMO coefficients,, Comm. Partial Differential Equations, 32 (2007), 453.  doi: 10.1080/03605300600781626.  Google Scholar [21] N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, volume 96 of Graduate Studies in Mathematics,, American Mathematical Society, (2008).  doi: 10.1090/gsm/096.  Google Scholar [22] N. V. Krylov, On divergence form {SPDE}s with VMO coefficients in a half space,, Stochastic Process. Appl., 119 (2009), 2095.  doi: 10.1016/j.spa.2008.11.003.  Google Scholar [23] 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.  doi: 10.1137/S0036141098338843.  Google Scholar [24] N. V. Krylov and S. V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients on a half line,, SIAM J. Math. Anal., 30 (1999), 298.  doi: 10.1137/S0036141097326908.  Google Scholar [25] N. V. Krylov, Parabolic equations with VMO coefficients in sobolev spaces with mixed norms,, J. Funct. Anal., 250 (2007), 521.  doi: 10.1016/j.jfa.2007.04.003.  Google Scholar [26] A. Kufner, Weighted Sobolev Spaces,, A Wiley-Interscience Publication. John Wiley & Sons, (1985).   Google Scholar [27] N. Nadirashvili, Nonuniqueness in the martingale problem and the Dirichlet problem for uniformly elliptic operators,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24 (1997), 537.   Google Scholar [28] N. N. Ural'ceva, The impossibility of $W_q{}^{2}$ estimates for multidimensional elliptic equations with discontinuous coefficients,, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 5 (1967), 250.   Google Scholar

show all references

##### References:
 [1] H. Aimar and R. A. Macías, Weighted norm inequalities for the Hardy-Littlewood maximal operator on spaces of homogeneous type,, Proc. Amer. Math. Soc., 91 (1984), 213.   Google Scholar [2] F. Chiarenza, M. Frasca and P. Longo, Interior $W^{2,p}$ estimates for nondivergence elliptic equations with discontinuous coefficients,, Ricerche Mat., 40 (1991), 149.   Google Scholar [3] F. Chiarenza, M. Frasca and P. Longo, $W^{2,p}$-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients,, Trans. Amer. Math. Soc., 336 (1993), 841.  doi: 10.2307/2154379.  Google Scholar [4] G. Di Fazio and D. K. Palagachev, Oblique derivative problem for elliptic equations in non-divergence form with VMO coefficients,, Comment. Math. Univ. Carolin., 37 (1996), 537.   Google Scholar [5] H. Dong, Parabolic equations with variably partially VMO coefficients,, Algebra i Analiz, 23 (2011), 150.  doi: 10.1090/S1061-0022-2012-01206-9.  Google Scholar [6] H. Dong, Solvability of parabolic equations in divergence form with partially BMO coefficients,, J. Funct. Anal., 258 (2010), 2145.  doi: 10.1016/j.jfa.2010.01.003.  Google Scholar [7] H. Dong and D. Kim, $L_p$ solvability of divergence type parabolic and elliptic systems with partially BMO coefficients,, Calc. Var. Partial Differential Equations, 40 (2011), 357.  doi: 10.1007/s00526-010-0344-0.  Google Scholar [8] H. Dong and D. Kim, On the $L_p$-solvability of higher order parabolic and elliptic systems with BMO coefficients,, Arch. Ration. Mech. Anal., 199 (2011), 889.  doi: 10.1007/s00205-010-0345-3.  Google Scholar [9] H. Dong and D. Kim, Elliptic and parabolic equations with measurable coefficients in weighted Sobolev spaces,, Adv. Math., 274 (2015), 681.  doi: 10.1016/j.aim.2014.12.037.  Google Scholar [10] H. Dong and H. Zhang, Conormal problem of higher-order parabolic systems,, Trans. Amer. Math. Soc., 368 (2016), 7413.  doi: 10.1090/tran/6605.  Google Scholar [11] D. Kim, Parabolic equations with measurable coefficients. II,, J. Math. Anal. Appl., 334 (2007), 534.  doi: 10.1016/j.jmaa.2006.12.077.  Google Scholar [12] I. Kim, K.-H. Kim and K. 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.  doi: 10.1016/j.jmaa.2013.10.079.  Google Scholar [13] K.-H. Kim, A weighted Sobolev space theory of parabolic stochastic PDEs on non-smooth domains,, J. Theoret. Probab., 27 (2014), 107.  doi: 10.1007/s10959-012-0459-7.  Google Scholar [14] K.-H. 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.  doi: 10.1137/S0036141003421145.  Google Scholar [15] K.-H. Kim and K. Lee, A weighted $L_p$-theory for parabolic PDEs with BMO coefficients on $C^1$-domains,, J. Differential Equations, 254 (2013), 368.  doi: 10.1016/j.jde.2012.08.002.  Google Scholar [16] V. Kozlov and A. Nazarov, Oblique derivative problem for non-divergence parabolic equations with time-discontinuous coefficients,, In Proceedings of the St. Petersburg Mathematical Society. Vol. XV. Advances in mathematical analysis of partial differential equations, (2014), 177.   Google Scholar [17] V. Kozlov and A. Nazarov, The Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients,, Math. Nachr., 282 (2009), 1220.  doi: 10.1002/mana.200910796.  Google Scholar [18] N. V. Krylov, A $W^n_2$-theory of the Dirichlet problem for SPDEs in general smooth domains,, Probab. Theory Related Fields, 98 (1994), 389.  doi: 10.1007/BF01192260.  Google Scholar [19] N. V. Krylov, Weighted Sobolev spaces and Laplace's equation and the heat equations in a half space,, Comm. Partial Differential Equations, 24 (1999), 1611.  doi: 10.1080/03605309908821478.  Google Scholar [20] N. V. Krylov, Parabolic and elliptic equations with VMO coefficients,, Comm. Partial Differential Equations, 32 (2007), 453.  doi: 10.1080/03605300600781626.  Google Scholar [21] N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, volume 96 of Graduate Studies in Mathematics,, American Mathematical Society, (2008).  doi: 10.1090/gsm/096.  Google Scholar [22] N. V. Krylov, On divergence form {SPDE}s with VMO coefficients in a half space,, Stochastic Process. Appl., 119 (2009), 2095.  doi: 10.1016/j.spa.2008.11.003.  Google Scholar [23] 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.  doi: 10.1137/S0036141098338843.  Google Scholar [24] N. V. Krylov and S. V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients on a half line,, SIAM J. Math. Anal., 30 (1999), 298.  doi: 10.1137/S0036141097326908.  Google Scholar [25] N. V. Krylov, Parabolic equations with VMO coefficients in sobolev spaces with mixed norms,, J. Funct. Anal., 250 (2007), 521.  doi: 10.1016/j.jfa.2007.04.003.  Google Scholar [26] A. Kufner, Weighted Sobolev Spaces,, A Wiley-Interscience Publication. John Wiley & Sons, (1985).   Google Scholar [27] N. Nadirashvili, Nonuniqueness in the martingale problem and the Dirichlet problem for uniformly elliptic operators,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24 (1997), 537.   Google Scholar [28] N. N. Ural'ceva, The impossibility of $W_q{}^{2}$ estimates for multidimensional elliptic equations with discontinuous coefficients,, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 5 (1967), 250.   Google Scholar
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