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

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

Junjie Zhang and  Shenzhou Zheng ,

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.

Keywords: Nondivergence elliptic equations, Muckenhoupt weights, weighted Lorentz estimates, partially BMO, Hardy-Littlewood maximal operator.
Mathematics Subject Classification: Primary: 35J15, 35B65; Secondary: 45E30.
Citation: Junjie Zhang, Shenzhou Zheng. Weighted lorentz estimates for nondivergence linear elliptic equations with partially BMO coefficients. Communications on Pure & 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.  Google Scholar

[2]

S. Agmon, A. 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.  Google Scholar

[3]

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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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

[12]

F. Chiarenza, M. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

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

[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.   Google Scholar

[25]

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

[26]

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

[27]

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

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

[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.  Google Scholar

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.  Google Scholar

[2]

S. Agmon, A. 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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

F. Chiarenza, M. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

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

[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.   Google Scholar

[25]

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

[26]

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

[27]

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

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

[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.  Google Scholar

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