• Previous Article
    Regularity criteria for the Boussinesq system with temperature-dependent viscosity and thermal diffusivity in a bounded domain
  • DCDS Home
  • This Issue
  • Next Article
    Generic absence of finite blocking for interior points of Birkhoff billiards
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

[1]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

[2]

Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250

[3]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[4]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469

[5]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377

[6]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168

[7]

Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHum approach. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020055

[8]

Federico Rodriguez Hertz, Zhiren Wang. On $ \epsilon $-escaping trajectories in homogeneous spaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 329-357. doi: 10.3934/dcds.2020365

[9]

Aihua Fan, Jörg Schmeling, Weixiao Shen. $ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 297-327. doi: 10.3934/dcds.2020363

[10]

Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020274

[11]

Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252

[12]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[13]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[14]

Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103

[15]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445

[16]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

[17]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

[18]

Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020378

[19]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[20]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (79)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]