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 & 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. 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. 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. ChiarenzaM. 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. 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. 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. 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. 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.
[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. 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. 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. ChiarenzaM. 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. 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. 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. 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. 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.
[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

[1]

Aleksandra Čižmešija, Iva Franjić, Josip Pečarić, Dora Pokaz. On a family of means generated by the Hardy-Littlewood maximal inequality. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 223-231. doi: 10.3934/naco.2012.2.223

[2]

Guoqing Zhang, Jia-yu Shao, Sanyang Liu. Linking solutions for N-laplace elliptic equations with Hardy-Sobolev operator and indefinite weights. Communications on Pure & Applied Analysis, 2011, 10 (2) : 571-581. doi: 10.3934/cpaa.2011.10.571

[3]

Wenxiong Chen, Chao Jin, Congming Li, Jisun Lim. Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations. Conference Publications, 2005, 2005 (Special) : 164-172. doi: 10.3934/proc.2005.2005.164

[4]

Shuang Liang, Shenzhou Zheng. Variable lorentz estimate for stationary stokes system with partially BMO coefficients. Communications on Pure & Applied Analysis, 2019, 18 (6) : 2879-2903. doi: 10.3934/cpaa.2019129

[5]

Luigi Greco, Gioconda Moscariello, Teresa Radice. Nondivergence elliptic equations with unbounded coefficients. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 131-143. doi: 10.3934/dcdsb.2009.11.131

[6]

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

[7]

Claudia Anedda, Giovanni Porru. Boundary estimates for solutions of weighted semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3801-3817. doi: 10.3934/dcds.2012.32.3801

[8]

Huyuan Chen, Feng Zhou. Isolated singularities for elliptic equations with hardy operator and source nonlinearity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2945-2964. doi: 10.3934/dcds.2018126

[9]

MEI MING. WEIGHTED ELLIPTIC ESTIMATES FOR A MIXED BOUNDARY SYSTEM RELATED TO THE DIRICHLET-NEUMANN OPERATOR ON A CORNER DOMAIN. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6039-6067. doi: 10.3934/dcds.2019264

[10]

Yingshu Lü, Zhongxue Lü. Some properties of solutions to the weighted Hardy-Littlewood-Sobolev type integral system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3791-3810. doi: 10.3934/dcds.2016.36.3791

[11]

Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987

[12]

Giuseppe Da Prato, Alessandra Lunardi. Maximal dissipativity of a class of elliptic degenerate operators in weighted $L^2$ spaces. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 751-760. doi: 10.3934/dcdsb.2006.6.751

[13]

Hua Jin, Wenbin Liu, Huixing Zhang, Jianjun Zhang. Ground States of Nonlinear Fractional Choquard Equations with Hardy-Littlewood-Sobolev Critical Growth. Communications on Pure & Applied Analysis, 2020, 19 (1) : 123-144. doi: 10.3934/cpaa.2020008

[14]

Ze Cheng, Congming Li. An extended discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1951-1959. doi: 10.3934/dcds.2014.34.1951

[15]

Ze Cheng, Genggeng Huang, Congming Li. On the Hardy-Littlewood-Sobolev type systems. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2059-2074. doi: 10.3934/cpaa.2016027

[16]

Dachun Yang, Sibei Yang. Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated to non-negative self-adjoint operators satisfying Gaussian estimates. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2135-2160. doi: 10.3934/cpaa.2016031

[17]

Elisa Calzolari, Roberta Filippucci, Patrizia Pucci. Existence of radial solutions for the $p$-Laplacian elliptic equations with weights. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 447-479. doi: 10.3934/dcds.2006.15.447

[18]

Elisa Calzolari, Roberta Filippucci, Patrizia Pucci. Dead cores and bursts for p-Laplacian elliptic equations with weights. Conference Publications, 2007, 2007 (Special) : 191-200. doi: 10.3934/proc.2007.2007.191

[19]

Jun Yang, Yaotian Shen. Weighted Sobolev-Hardy spaces and sign-changing solutions of degenerate elliptic equation. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2565-2575. doi: 10.3934/cpaa.2013.12.2565

[20]

Mark F. Demers, Hong-Kun Zhang. Spectral analysis of the transfer operator for the Lorentz gas. Journal of Modern Dynamics, 2011, 5 (4) : 665-709. doi: 10.3934/jmd.2011.5.665

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (8)
  • HTML views (62)
  • Cited by (3)

Other articles
by authors

[Back to Top]