May  2017, 16(3): 899-914. doi: 10.3934/cpaa.2017043

Weighted lorentz estimates for nondivergence linear elliptic equations with partially BMO coefficients

Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China

* Corresponding author

Received  August 2016 Revised  December 2016 Published  February 2017

Fund Project: The first author is supported by the Fundamental Research Funds for the Central Universities of China grant 2016YJS154, and the second author is supported by NSF of China grant 11371050.

We prove weighted Lorentz estimates of the Hessian of strong solution for nondivergence linear elliptic equations $a_{ij}(x)D_{ij}u(x)=f(x)$. The leading coefficients are assumed to be measurable with respect to one variable and have small BMO semi-norms with respect to the other variables. Here, an approximation method, Lorentz boundedness of the Hardy-Littlewood maximal operators and an equivalent representation of Lorentz norm are employed.

Citation: Junjie Zhang, Shenzhou Zheng. Weighted lorentz estimates for nondivergence linear elliptic equations with partially BMO coefficients. Communications on Pure and Applied Analysis, 2017, 16 (3) : 899-914. doi: 10.3934/cpaa.2017043
References:
[1]

E. Acerbi and G. Mingione, Gradient estimates for a class of parabolic systems, Duke Math. J., 136 (2007), 285-320.  doi: 10.1215/S0012-7094-07-13623-8.

[2]

S. AgmonA. Douglisa and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Comm. Pure Appl. Math., 17 (1964), 35-92.  doi: 10.1002/cpa.3160170104.

[3]

P. Baroni, Lorentz estimates for obstacle parabolic problems, Nonlinear Anal., 96 (2014), 167-188.  doi: 10.1016/j.na.2013.11.004.

[4]

P. Baroni, Lorentz estimates for degenerate and singular evolutionary systems, J. Differ. Equ., 255 (2013), 2927-2951.  doi: 10.1016/j.jde.2013.07.024.

[5]

M. Bramanti and M. Cerutti, Wp1,2 solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO coefficients, Comm. Partial Differ. Equ., 18 (1993), 1735-1763.  doi: 10.1080/03605309308820991.

[6]

S. S. Byun and Y. Kim, Elliptic equations with measurable nonlinearities in nonsmooth domains, Adv. Math., 288 (2016), 152-200.  doi: 10.1016/j.aim.2015.10.015.

[7]

S. S. Byun and M. Lee, On weighted W2,p estimates for elliptic equations with BMO coefficients in nondivergence form, International Journal of Mathematics, 26 (2015), 1550001.  doi: 10.1142/S0129167X15500019.

[8]

S. S. Byun and M. Lee, Weighted estimates for nondivergence parabolic equations in Orlicz spaces, J. Funct. Anal., 269 (2015), 2530-2563.  doi: 10.1016/j.jfa.2015.07.009.

[9]

S. S. Byun and D. K. Palagachev, Weighted Lp-estimates for elliptic equations with measurable coefficients in nonsmooth domains, Potential Anal., 41 (2014), 51-79.  doi: 10.1007/s11118-013-9363-8.

[10]

L. A. Caffarelli and I. Peral, On W1,p estimates for elliptic equations in divergence form, Comm. Pure Appl. Math., 51 (1998), 1-21.  doi: 10.1002/(SICI)1097-0312(199801)51:1<1::AID-CPA1>3.3.CO;2-N.

[11]

F. ChiarenzaM. Frasca and P. Longo, Interior W2,p estimates for nondivergence elliptic equations with discontinuous coeffcients, Ricerche Mat., 40 (1991), 149-168. 

[12]

F. ChiarenzaM. Frasca and P. Longo, W2,p solvability of the Dirichlet problem for nonlinear elliptic equations with VMO coeffcients, Trans. Amer. Math. Soc., 336 (1993), 841-853.  doi: 10.2307/2154379.

[13]

H. Dong, Solvability of second-order equations with hierarchically patially BMO coefficients, Trans. Amer. Math. Soc., 364 (2012), 493-517.  doi: 10.1090/S0002-9947-2011-05453-X.

[14]

H. Dong, Parabolic equations with variably partially VMO coefficients, St. Petersburg Math. J., 23 (2012), 521-539.  doi: 10.1090/S1061-0022-2012-01206-9.

[15]

H. Dong and D. Kim, On Lp-estimates for elliptic and parabolic equations with Ap weights, preprint, arXiv: 1603.07844.

[16]

Q. Han and F. Lin, Elliptic Partial Differential Equation, American Mathematical Soc., 2011.

[17]

D. Kim and A. V. Krylov, Elliptic differenrial equations with coefficients measurable with respact to one variable and VMO with respect to the others, SIAM J. Math. Anal., 32 (2007), 489-506.  doi: 10.1137/050646913.

[18]

D. Kim and N. V. Krylov, Parabolic equations with measurable coefficients, Potential Anal., 26 (2007), 345-361.  doi: 10.1007/s11118-007-9042-8.

[19]

N. V. Krylov, Parabolic and elliptic equations with VMO coefficients, Comm. Partial Differ. Equ., 32 (2007), 453-475.  doi: 10.1080/03605300600781626.

[20]

N. V. Krylov, Second order parabolic equations with variably partially VMO coeffcients, J. Funct. Anal., 257 (2009), 1695-1712.  doi: 10.1016/j.jfa.2009.06.014.

[21]

G. M. Lieberman, A mostly elementary proof of Morrey space estimates for elliptic and parabolic equations with VMO coefficients, J. Funct. Anal., 201 (2003), 457-479.  doi: 10.1016/S0022-1236(03)00125-3.

[22] A. MaugeriD. K. Palagachev and L. G. Softova, Elliptic and Parabolic Equations with Discontinuous Coefficients, 1 edition, Wiley-vch Verlag, Berlin, 2000.  doi: 10.1002/3527600868.
[23]

T. Mengesha and N. C. Phuc, Global estimates for quasilinear elliptic equations on Reifenberg flat domains, Arch. Rational Mech. Anal., 203 (2012), 189-216.  doi: 10.1007/s00205-011-0446-7.

[24]

N. G. Meyers, An Lp-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 17 (1963), 189-206. 

[25]

G. Mingione, Gradient estimates below the duality exponent, Ann. Math., 346 (2010), 571-627.  doi: 10.1007/s00208-009-0411-z.

[26]

M. V. Safonov, Harnack inequality for elliptic equations and the Hölder property of their solutions, J. Soviet Math., 21 (1983), 851-863. 

[27]

G. Talenti, Elliptic Equations and Rearrangements, Ann Scuola Norm. Sup. Pisa Cl. Sci., 3 (1976), 697-718. 

[28] B. O. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces, 1 edition, Springer-Verlag Berlin Heidelberg, New York, 2000.  doi: 10.1007/BFb0103908.
[29]

L. Wang, A geometric approach to the Calderon-Zygmmund estimates, Acta Math. Sin., 19 (2003), 381-396.  doi: 10.1007/s10114-003-0264-4.

[30]

C. Zhang and S. Zhou, Global weighted estimates for quasilinear elliptic equations with nonstandard growth, J. Funct. Anal., 267 (2014), 605-642.  doi: 10.1016/j.jfa.2014.03.022.

show all references

References:
[1]

E. Acerbi and G. Mingione, Gradient estimates for a class of parabolic systems, Duke Math. J., 136 (2007), 285-320.  doi: 10.1215/S0012-7094-07-13623-8.

[2]

S. AgmonA. Douglisa and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Comm. Pure Appl. Math., 17 (1964), 35-92.  doi: 10.1002/cpa.3160170104.

[3]

P. Baroni, Lorentz estimates for obstacle parabolic problems, Nonlinear Anal., 96 (2014), 167-188.  doi: 10.1016/j.na.2013.11.004.

[4]

P. Baroni, Lorentz estimates for degenerate and singular evolutionary systems, J. Differ. Equ., 255 (2013), 2927-2951.  doi: 10.1016/j.jde.2013.07.024.

[5]

M. Bramanti and M. Cerutti, Wp1,2 solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO coefficients, Comm. Partial Differ. Equ., 18 (1993), 1735-1763.  doi: 10.1080/03605309308820991.

[6]

S. S. Byun and Y. Kim, Elliptic equations with measurable nonlinearities in nonsmooth domains, Adv. Math., 288 (2016), 152-200.  doi: 10.1016/j.aim.2015.10.015.

[7]

S. S. Byun and M. Lee, On weighted W2,p estimates for elliptic equations with BMO coefficients in nondivergence form, International Journal of Mathematics, 26 (2015), 1550001.  doi: 10.1142/S0129167X15500019.

[8]

S. S. Byun and M. Lee, Weighted estimates for nondivergence parabolic equations in Orlicz spaces, J. Funct. Anal., 269 (2015), 2530-2563.  doi: 10.1016/j.jfa.2015.07.009.

[9]

S. S. Byun and D. K. Palagachev, Weighted Lp-estimates for elliptic equations with measurable coefficients in nonsmooth domains, Potential Anal., 41 (2014), 51-79.  doi: 10.1007/s11118-013-9363-8.

[10]

L. A. Caffarelli and I. Peral, On W1,p estimates for elliptic equations in divergence form, Comm. Pure Appl. Math., 51 (1998), 1-21.  doi: 10.1002/(SICI)1097-0312(199801)51:1<1::AID-CPA1>3.3.CO;2-N.

[11]

F. ChiarenzaM. Frasca and P. Longo, Interior W2,p estimates for nondivergence elliptic equations with discontinuous coeffcients, Ricerche Mat., 40 (1991), 149-168. 

[12]

F. ChiarenzaM. Frasca and P. Longo, W2,p solvability of the Dirichlet problem for nonlinear elliptic equations with VMO coeffcients, Trans. Amer. Math. Soc., 336 (1993), 841-853.  doi: 10.2307/2154379.

[13]

H. Dong, Solvability of second-order equations with hierarchically patially BMO coefficients, Trans. Amer. Math. Soc., 364 (2012), 493-517.  doi: 10.1090/S0002-9947-2011-05453-X.

[14]

H. Dong, Parabolic equations with variably partially VMO coefficients, St. Petersburg Math. J., 23 (2012), 521-539.  doi: 10.1090/S1061-0022-2012-01206-9.

[15]

H. Dong and D. Kim, On Lp-estimates for elliptic and parabolic equations with Ap weights, preprint, arXiv: 1603.07844.

[16]

Q. Han and F. Lin, Elliptic Partial Differential Equation, American Mathematical Soc., 2011.

[17]

D. Kim and A. V. Krylov, Elliptic differenrial equations with coefficients measurable with respact to one variable and VMO with respect to the others, SIAM J. Math. Anal., 32 (2007), 489-506.  doi: 10.1137/050646913.

[18]

D. Kim and N. V. Krylov, Parabolic equations with measurable coefficients, Potential Anal., 26 (2007), 345-361.  doi: 10.1007/s11118-007-9042-8.

[19]

N. V. Krylov, Parabolic and elliptic equations with VMO coefficients, Comm. Partial Differ. Equ., 32 (2007), 453-475.  doi: 10.1080/03605300600781626.

[20]

N. V. Krylov, Second order parabolic equations with variably partially VMO coeffcients, J. Funct. Anal., 257 (2009), 1695-1712.  doi: 10.1016/j.jfa.2009.06.014.

[21]

G. M. Lieberman, A mostly elementary proof of Morrey space estimates for elliptic and parabolic equations with VMO coefficients, J. Funct. Anal., 201 (2003), 457-479.  doi: 10.1016/S0022-1236(03)00125-3.

[22] A. MaugeriD. K. Palagachev and L. G. Softova, Elliptic and Parabolic Equations with Discontinuous Coefficients, 1 edition, Wiley-vch Verlag, Berlin, 2000.  doi: 10.1002/3527600868.
[23]

T. Mengesha and N. C. Phuc, Global estimates for quasilinear elliptic equations on Reifenberg flat domains, Arch. Rational Mech. Anal., 203 (2012), 189-216.  doi: 10.1007/s00205-011-0446-7.

[24]

N. G. Meyers, An Lp-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 17 (1963), 189-206. 

[25]

G. Mingione, Gradient estimates below the duality exponent, Ann. Math., 346 (2010), 571-627.  doi: 10.1007/s00208-009-0411-z.

[26]

M. V. Safonov, Harnack inequality for elliptic equations and the Hölder property of their solutions, J. Soviet Math., 21 (1983), 851-863. 

[27]

G. Talenti, Elliptic Equations and Rearrangements, Ann Scuola Norm. Sup. Pisa Cl. Sci., 3 (1976), 697-718. 

[28] B. O. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces, 1 edition, Springer-Verlag Berlin Heidelberg, New York, 2000.  doi: 10.1007/BFb0103908.
[29]

L. Wang, A geometric approach to the Calderon-Zygmmund estimates, Acta Math. Sin., 19 (2003), 381-396.  doi: 10.1007/s10114-003-0264-4.

[30]

C. Zhang and S. Zhou, Global weighted estimates for quasilinear elliptic equations with nonstandard growth, J. Funct. Anal., 267 (2014), 605-642.  doi: 10.1016/j.jfa.2014.03.022.

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