# American Institute of Mathematical Sciences

• Previous Article
Generic absence of finite blocking for interior points of Birkhoff billiards
• DCDS Home
• This Issue
• Next Article
Regularity criteria for the Boussinesq system with temperature-dependent viscosity and thermal diffusivity in a bounded domain
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. [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. [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. [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. [5] H. Dong, Parabolic equations with variably partially VMO coefficients,, Algebra i Analiz, 23 (2011), 150. doi: 10.1090/S1061-0022-2012-01206-9. [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. [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. [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. [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. [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. [11] D. Kim, Parabolic equations with measurable coefficients. II,, J. Math. Anal. Appl., 334 (2007), 534. doi: 10.1016/j.jmaa.2006.12.077. [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. [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. [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. [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. [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. [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. [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. [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. [20] N. V. Krylov, Parabolic and elliptic equations with VMO coefficients,, Comm. Partial Differential Equations, 32 (2007), 453. doi: 10.1080/03605300600781626. [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. [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. [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. [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. [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. [26] A. Kufner, Weighted Sobolev Spaces,, A Wiley-Interscience Publication. John Wiley & Sons, (1985). [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. [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.

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. [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. [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. [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. [5] H. Dong, Parabolic equations with variably partially VMO coefficients,, Algebra i Analiz, 23 (2011), 150. doi: 10.1090/S1061-0022-2012-01206-9. [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. [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. [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. [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. [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. [11] D. Kim, Parabolic equations with measurable coefficients. II,, J. Math. Anal. Appl., 334 (2007), 534. doi: 10.1016/j.jmaa.2006.12.077. [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. [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. [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. [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. [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. [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. [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. [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. [20] N. V. Krylov, Parabolic and elliptic equations with VMO coefficients,, Comm. Partial Differential Equations, 32 (2007), 453. doi: 10.1080/03605300600781626. [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. [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. [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. [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. [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. [26] A. Kufner, Weighted Sobolev Spaces,, A Wiley-Interscience Publication. John Wiley & Sons, (1985). [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. [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.
 [1] Peter Weidemaier. Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L_p$-norm. Electronic Research Announcements, 2002, 8: 47-51. [2] Kyeong-Hun Kim, Kijung Lee. A weighted $L_p$-theory for second-order parabolic and elliptic partial differential systems on a half space. Communications on Pure & Applied Analysis, 2016, 15 (3) : 761-794. doi: 10.3934/cpaa.2016.15.761 [3] Ping Li, Pablo Raúl Stinga, José L. Torrea. On weighted mixed-norm Sobolev estimates for some basic parabolic equations. Communications on Pure & Applied Analysis, 2017, 16 (3) : 855-882. doi: 10.3934/cpaa.2017041 [4] Youngwoo Koh, Ihyeok Seo. Strichartz estimates for Schrödinger equations in weighted $L^2$ spaces and their applications. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4877-4906. doi: 10.3934/dcds.2017210 [5] Ildoo Kim. An $L_p$-Lipschitz theory for parabolic equations with time measurable pseudo-differential operators. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2751-2771. doi: 10.3934/cpaa.2018130 [6] Stefan Meyer, Mathias Wilke. Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces. Evolution Equations & Control Theory, 2013, 2 (2) : 365-378. doi: 10.3934/eect.2013.2.365 [7] Pierre-Étienne Druet. Higher $L^p$ regularity for vector fields that satisfy divergence and rotation constraints in dual Sobolev spaces, and application to some low-frequency Maxwell equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 475-496. doi: 10.3934/dcdss.2015.8.475 [8] N. V. Krylov. Some $L_{p}$-estimates for elliptic and parabolic operators with measurable coefficients. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2073-2090. doi: 10.3934/dcdsb.2012.17.2073 [9] Shouming Zhou. The Cauchy problem for a generalized $b$-equation with higher-order nonlinearities in critical Besov spaces and weighted $L^p$ spaces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4967-4986. doi: 10.3934/dcds.2014.34.4967 [10] Tahar Z. Boulmezaoud, Amel Kourta. Some identities on weighted Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 427-434. doi: 10.3934/dcdss.2012.5.427 [11] Der-Chen Chang, Jie Xiao. $L^q$-Extensions of $L^p$-spaces by fractional diffusion equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1905-1920. doi: 10.3934/dcds.2015.35.1905 [12] Mathias Wilke. $L_p$-theory for a Cahn-Hilliard-Gurtin system. Evolution Equations & Control Theory, 2012, 1 (2) : 393-429. doi: 10.3934/eect.2012.1.393 [13] Jian Lu, Huaiyu Jian. Topological degree method for the rotationally symmetric $L_p$-Minkowski problem. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 971-980. doi: 10.3934/dcds.2016.36.971 [14] Xiaojun Li, Xiliang Li, Kening Lu. Random attractors for stochastic parabolic equations with additive noise in weighted spaces. Communications on Pure & Applied Analysis, 2018, 17 (3) : 729-749. doi: 10.3934/cpaa.2018038 [15] Ming Wang. Sharp global well-posedness of the BBM equation in $L^p$ type Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5763-5788. doi: 10.3934/dcds.2016053 [16] Alberto Fiorenza, Anna Mercaldo, Jean Michel Rakotoson. Regularity and uniqueness results in grand Sobolev spaces for parabolic equations with measure data. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 893-906. doi: 10.3934/dcds.2002.8.893 [17] Lucas C. F. Ferreira, Elder J. Villamizar-Roa. On the stability problem for the Boussinesq equations in weak-$L^p$ spaces. Communications on Pure & Applied Analysis, 2010, 9 (3) : 667-684. doi: 10.3934/cpaa.2010.9.667 [18] Xinghong Pan, Jiang Xu. Global existence and optimal decay estimates of the compressible viscoelastic flows in $L^p$ critical spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2021-2057. doi: 10.3934/dcds.2019085 [19] Chérif Amrouche, Mohamed Meslameni, Šárka Nečasová. Linearized Navier-Stokes equations in $\mathbb{R}^3$: An approach in weighted Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 901-916. doi: 10.3934/dcdss.2014.7.901 [20] Angelo Favini, Alfredo Lorenzi, Hiroki Tanabe, Atsushi Yagi. An $L^p$-approach to singular linear parabolic equations with lower order terms. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 989-1008. doi: 10.3934/dcds.2008.22.989

2017 Impact Factor: 1.179