May  2020, 19(5): 2777-2796. doi: 10.3934/cpaa.2020121

$ L^{p(\cdot)} $-regularity of Hessian for nondivergence parabolic and elliptic equations with measurable coefficients

1. 

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

2. 

College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, Hebei 050016, China

3. 

College of Mathematics, Inner Mongolia University for Nationalities, Tongliao 028043, China

* Corresponding author

Received  May 2019 Revised  October 2019 Published  March 2020

Fund Project: The first author is supported by Science Foundation of Hebei Normal University (No. L2019B02), Hebei Natural Science Foundation of China (No. A2019205218). The third author is supported by Foundation of the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region (No. 2018MS01008), and the Science Research Program of institution of higher education at Universities of Inner Mongolia Autonomous Region (No. NJZY18164)

We prove the interior $ L^{p(\cdot)} $-estimates for the Hessian of strong solutions to nondivergence parabolic equations $ u_{t}(x,t)-a_{ij}(x,t)D_{ij}u(x,t) = f(x,t) $ and elliptic equations $ a_{ij}(x)D_{ij}u(x) = f(x) $, respectively. Besides a natural assumption that $ p(\cdot) $ is $ \log $-Hölder continuous, we also assume that the coefficients $ a_{ij}(x,t) $ and $ a_{ij}(x) $ are merely measurable in one of spatial variables and have small BMO semi-norms with respect to other variables.

Citation: Junjie Zhang, Shenzhou Zheng, Haiyan Yu. $ L^{p(\cdot)} $-regularity of Hessian for nondivergence parabolic and elliptic equations with measurable coefficients. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2777-2796. doi: 10.3934/cpaa.2020121
References:
[1]

E. Acerbi and G. Mingione, Gradient estimates for the $p(x)$-Laplacian systems, J. Reine Angew. Math., 584 (2005), 117-148.  doi: 10.1515/crll.2005.2005.584.117.

[2]

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.

[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 and V. Bögelein, Calderón-Zygmund estimates for parabolic $p(x, t)$-Laplacian systems, Rev. Mat. Iberoam., 30 (2014), 1355-1386.  doi: 10.4171/RMI/817.

[5]

M. Bramanti and M. C. Cerutti, $W^{1, 2}_{p}$-solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO coefficients, Commun. Partial Differ. Equ., 18 (1993), 1735-1763.  doi: 10.1080/03605309308820991.

[6]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2011. doi: 10.1007/978-0-387-70914-7.

[7]

S. S. ByunJ. Ok and L. H. Wang, $W^{1, p(\cdot)}$-regularity for elliptic equations with measurable coefficients in nonsmooth domains, Commun. Math. Phys., 329 (2014), 937-958.  doi: 10.1007/s00220-014-1962-8.

[8]

S. S. ByunM. Lee and J. Ok, $W^{2, p(\cdot)}$-regulairty for elliptic equations in nondivergence form with BMO coefficients, Math. Ann., 363 (2015), 1023-1052.  doi: 10.1007/s00208-015-1194-z.

[9]

A. P. Calderón and A. Zygmund, On the existence of certain singular integrals, Acta Math., 88 (1952), 85-139.  doi: 10.1007/BF02392130.

[10]

F. ChiarenzaM. Frasca and P. Longo, Interior $W^{2, p}$-estimates for nondivergence elliptic equations with discontinuous coefficients, Ricerche Mat., 40 (1991), 149-168. 

[11]

F. ChiarenzaM. Frasca and P. Longo, $W^{2, p}$-solvability of the Dirichlet problem for nonlinear elliptic equations with VMO coefficients, Trans. Amer. Math. Soc., 336 (1993), 841-853.  doi: 10.2307/2154379.

[12]

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

[13]

Q. Han and F. H. Lin, Elliptic Partial Differential Equation, Courant Institute of Mathematical Sciences, New York University, New York, 1997.

[14]

D. Kim and N. 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.

[15]

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

[16]

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

[17]

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.

[18]

J. J. Zhang and S. Z. Zheng, Weighted Lorentz estimates for nondivergence linear elliptic equations with partially BMO coefficients, Commun. Pure Appl. Anal., 16 (2017), 899-914.  doi: 10.3934/cpaa.2017043.

[19]

C. Zhang and S. L. Zhou, Hölder regularity for the gradients of solutions of the strong $p(x)$-Laplacian, J. Math. Anal. Appl., 389 (2012), 1066-1077.  doi: 10.1016/j.jmaa.2011.12.047.

[20]

C. Zhang and S. L. Zhou, Global weighted estimates for quasilinear elliptic equations with non-standard 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 the $p(x)$-Laplacian systems, J. Reine Angew. Math., 584 (2005), 117-148.  doi: 10.1515/crll.2005.2005.584.117.

[2]

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.

[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 and V. Bögelein, Calderón-Zygmund estimates for parabolic $p(x, t)$-Laplacian systems, Rev. Mat. Iberoam., 30 (2014), 1355-1386.  doi: 10.4171/RMI/817.

[5]

M. Bramanti and M. C. Cerutti, $W^{1, 2}_{p}$-solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO coefficients, Commun. Partial Differ. Equ., 18 (1993), 1735-1763.  doi: 10.1080/03605309308820991.

[6]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2011. doi: 10.1007/978-0-387-70914-7.

[7]

S. S. ByunJ. Ok and L. H. Wang, $W^{1, p(\cdot)}$-regularity for elliptic equations with measurable coefficients in nonsmooth domains, Commun. Math. Phys., 329 (2014), 937-958.  doi: 10.1007/s00220-014-1962-8.

[8]

S. S. ByunM. Lee and J. Ok, $W^{2, p(\cdot)}$-regulairty for elliptic equations in nondivergence form with BMO coefficients, Math. Ann., 363 (2015), 1023-1052.  doi: 10.1007/s00208-015-1194-z.

[9]

A. P. Calderón and A. Zygmund, On the existence of certain singular integrals, Acta Math., 88 (1952), 85-139.  doi: 10.1007/BF02392130.

[10]

F. ChiarenzaM. Frasca and P. Longo, Interior $W^{2, p}$-estimates for nondivergence elliptic equations with discontinuous coefficients, Ricerche Mat., 40 (1991), 149-168. 

[11]

F. ChiarenzaM. Frasca and P. Longo, $W^{2, p}$-solvability of the Dirichlet problem for nonlinear elliptic equations with VMO coefficients, Trans. Amer. Math. Soc., 336 (1993), 841-853.  doi: 10.2307/2154379.

[12]

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

[13]

Q. Han and F. H. Lin, Elliptic Partial Differential Equation, Courant Institute of Mathematical Sciences, New York University, New York, 1997.

[14]

D. Kim and N. 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.

[15]

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

[16]

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

[17]

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.

[18]

J. J. Zhang and S. Z. Zheng, Weighted Lorentz estimates for nondivergence linear elliptic equations with partially BMO coefficients, Commun. Pure Appl. Anal., 16 (2017), 899-914.  doi: 10.3934/cpaa.2017043.

[19]

C. Zhang and S. L. Zhou, Hölder regularity for the gradients of solutions of the strong $p(x)$-Laplacian, J. Math. Anal. Appl., 389 (2012), 1066-1077.  doi: 10.1016/j.jmaa.2011.12.047.

[20]

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

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