October  2020, 25(10): 3843-3855. doi: 10.3934/dcdsb.2020038

Calderón-Zygmund estimates for quasilinear elliptic double obstacle problems with variable exponent and logarithmic growth

1. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea

2. 

Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea

3. 

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

* Corresponding author: Sun-Sig Byun

Received  June 2019 Published  October 2020 Early access  February 2020

Fund Project: S. Byun was supported by NRF-2017R1A2B2003877. Y. Cho was supported by NRF-2019R1I1A1A01064053. S. Liang was partially supported by NRF-2015R1A4A1041675

Quasilinear elliptic double obstacle problems with variable exponent and logarithmic growth are studied. We obtain a global Calderón-Zygmund estimate for such an irregular obstacle problem by proving that the gradient of the solution is as integrable as both the nonhomogeneous term and the gradient of the associated double obstacles under minimal regularity requirements on the elliptic operator over the boundary of the nonsmooth domain.

Citation: Sun-Sig Byun, Yumi Cho, Shuang Liang. Calderón-Zygmund estimates for quasilinear elliptic double obstacle problems with variable exponent and logarithmic growth. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 3843-3855. doi: 10.3934/dcdsb.2020038
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]

H. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341.  doi: 10.1007/BF01176474.

[3]

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

[4]

M. BildhauerM. Fuchs and G. Mingione, A priori gradient bounds and local $C^{1, \alpha}$-estimates for (double) obstacle problems under non-standard growth conditions, Z. Anal. Anwendungen, 20 (2001), 959-985.  doi: 10.4171/ZAA/1054.

[5]

V. BögeleinF. Duzzar and G. Mingione, Degenerate problems with irregular obstacles, J. Reine Angew. Math., 650 (2011), 107-160.  doi: 10.1515/CRELLE.2011.006.

[6]

V. Bögelein and C. Scheven, Higher integrability in parabolic obstacle problems, Forum Math., 24 (2012), 931-972.  doi: 10.1515/form.2011.091.

[7]

S. S. ByunY. Cho and J. Ok, Global gradient estimates for nonlinear obstacle problems with nonstandard growth, Forum Math., 28 (2016), 729-747.  doi: 10.1515/forum-2014-0153.

[8]

S. S. ByunY. Cho and L. Wang, Calderón-Zygmund theory for nonlinear elliptic problems with irregular obstacles, J. Funct. Anal., 263 (2012), 3117-3143.  doi: 10.1016/j.jfa.2012.07.018.

[9]

S. S. Byun and S. Ryu, Gradient estimates for nonlinear elliptic double obstacle problems, Nonlinear Anal., in press. doi: 10.1016/j.na.2018.08.011.

[10]

S. S. Byun and L. Wang, Elliptic equations with BMO coefficients in Reifenberg domains, Comm. Pure Appl. Math., 57 (2004), 1283-1310.  doi: 10.1002/cpa.20037.

[11]

G. Dal MasoU. Mosco and M. A. Vivaldi, A pointwise regularity theory for the two-obstacle problem, Acta. Math., 163 (1989), 57-107.  doi: 10.1007/BF02392733.

[12]

A. Erhardt, Calderón-Zygmund theory for parabolic obstacle problems with nonstandard growth, Adv. Nonlinear Anal., 3 (2014), 15-44.  doi: 10.1515/anona-2013-0024.

[13]

A. Erhardt, Higher integrability for solutions to parabolic problems with irregular obstacles and nonstandard growth, J. Math. Anal. Appl., 435 (2016), 1772-1803.  doi: 10.1016/j.jmaa.2015.11.028.

[14] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies, 105, Princeton University Press, Princeton, NJ, 1983. 
[15]

P. Harjulehto and P. Hästö, Orlicz spaces and generalized Orlicz spaces, Lecture Notes in Mathematics, 2236, Springer, Cham, 2019. doi: 10.1007/978-3-030-15100-3.

[16]

T. Kilpeläinen and W. P. Ziemer, Pointwise regularity of solutions to nonlinear double obstacle problems, Ark. Mat., 29 (1991), 83-106.  doi: 10.1007/BF02384333.

[17]

T. Kuusi, G. Mingione and K. Nyström, Sharp regularity for evolutionary obstacle problems, interpolative geometries and removable sets, J. Math. Pures Appl. (9), 101 (2014), 119-151. doi: 10.1016/j.matpur.2013.03.004.

[18]

A. Lemenant and E. Milakis, On the extension property of Reifenberg-flat domains, Ann. Acad. Sci. Fenn. Math., 39 (2014), 51-71.  doi: 10.5186/aasfm.2014.3907.

[19]

H. Li and X. Chai, A two-obstacle problem with variable exponent and measure data, Turkish J. Math., 41 (2017), 717-724.  doi: 10.3906/mat-1409-9.

[20]

G. M. Lieberman, Regularity for solutions to some degenerate double obstacle problems, Indiana Univ. Math. J., 40 (1991), 1009-1028.  doi: 10.1512/iumj.1991.40.40045.

[21]

E. Milakis and T. Toro, Divergence form operators in Reifenberg flat domains, Math. Z., 264 (2010), 15-41.  doi: 10.1007/s00209-008-0450-2.

[22]

J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, 1034, Springer Verlag, Berlin, 1983. doi: 10.1007/BFb0072210.

[23]

J. Ok, Gradient estimates for elliptic equations with $L^{p(\cdot)}\log L$ growth, Calc. Var. Partial Differential Equations, 55 (2016), 30pp. doi: 10.1007/s00526-016-0965-z.

[24]

J. Ok, Calderón-Zygmund estimates for a class of obstacle problems with nonstandard growth, NoDEA Nonlinear Differential Equations Appl., 23 (2016) 21pp. doi: 10.1007/s00030-016-0404-z.

[25]

J. F. Rodrigues and R. Teymurazyan, On the two obstacles problem in Orlicz-Sobolev spaces and applications, Complex Var. Elliptic Equ., 56 (2011), 769-787.  doi: 10.1080/17476933.2010.505016.

[26]

C. Scheven, Elliptic obstacle problems with measure data: Potentials and low order regularity, Publ. Mat., 56 (2012), 327-374.  doi: 10.5565/PUBLMAT_56212_04.

[27]

C. Scheven, Existence of localizable solutions to nonlinear parabolic problems with irregular obstacles, Manuscripta Math., 146 (2015), 7-63.  doi: 10.1007/s00229-014-0684-8.

[28]

T. Toro, Doubling and flatness: Geometry of measures, Notices Amer. Math. Soc., 44 (1997), 1087-1094. 

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]

H. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341.  doi: 10.1007/BF01176474.

[3]

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

[4]

M. BildhauerM. Fuchs and G. Mingione, A priori gradient bounds and local $C^{1, \alpha}$-estimates for (double) obstacle problems under non-standard growth conditions, Z. Anal. Anwendungen, 20 (2001), 959-985.  doi: 10.4171/ZAA/1054.

[5]

V. BögeleinF. Duzzar and G. Mingione, Degenerate problems with irregular obstacles, J. Reine Angew. Math., 650 (2011), 107-160.  doi: 10.1515/CRELLE.2011.006.

[6]

V. Bögelein and C. Scheven, Higher integrability in parabolic obstacle problems, Forum Math., 24 (2012), 931-972.  doi: 10.1515/form.2011.091.

[7]

S. S. ByunY. Cho and J. Ok, Global gradient estimates for nonlinear obstacle problems with nonstandard growth, Forum Math., 28 (2016), 729-747.  doi: 10.1515/forum-2014-0153.

[8]

S. S. ByunY. Cho and L. Wang, Calderón-Zygmund theory for nonlinear elliptic problems with irregular obstacles, J. Funct. Anal., 263 (2012), 3117-3143.  doi: 10.1016/j.jfa.2012.07.018.

[9]

S. S. Byun and S. Ryu, Gradient estimates for nonlinear elliptic double obstacle problems, Nonlinear Anal., in press. doi: 10.1016/j.na.2018.08.011.

[10]

S. S. Byun and L. Wang, Elliptic equations with BMO coefficients in Reifenberg domains, Comm. Pure Appl. Math., 57 (2004), 1283-1310.  doi: 10.1002/cpa.20037.

[11]

G. Dal MasoU. Mosco and M. A. Vivaldi, A pointwise regularity theory for the two-obstacle problem, Acta. Math., 163 (1989), 57-107.  doi: 10.1007/BF02392733.

[12]

A. Erhardt, Calderón-Zygmund theory for parabolic obstacle problems with nonstandard growth, Adv. Nonlinear Anal., 3 (2014), 15-44.  doi: 10.1515/anona-2013-0024.

[13]

A. Erhardt, Higher integrability for solutions to parabolic problems with irregular obstacles and nonstandard growth, J. Math. Anal. Appl., 435 (2016), 1772-1803.  doi: 10.1016/j.jmaa.2015.11.028.

[14] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies, 105, Princeton University Press, Princeton, NJ, 1983. 
[15]

P. Harjulehto and P. Hästö, Orlicz spaces and generalized Orlicz spaces, Lecture Notes in Mathematics, 2236, Springer, Cham, 2019. doi: 10.1007/978-3-030-15100-3.

[16]

T. Kilpeläinen and W. P. Ziemer, Pointwise regularity of solutions to nonlinear double obstacle problems, Ark. Mat., 29 (1991), 83-106.  doi: 10.1007/BF02384333.

[17]

T. Kuusi, G. Mingione and K. Nyström, Sharp regularity for evolutionary obstacle problems, interpolative geometries and removable sets, J. Math. Pures Appl. (9), 101 (2014), 119-151. doi: 10.1016/j.matpur.2013.03.004.

[18]

A. Lemenant and E. Milakis, On the extension property of Reifenberg-flat domains, Ann. Acad. Sci. Fenn. Math., 39 (2014), 51-71.  doi: 10.5186/aasfm.2014.3907.

[19]

H. Li and X. Chai, A two-obstacle problem with variable exponent and measure data, Turkish J. Math., 41 (2017), 717-724.  doi: 10.3906/mat-1409-9.

[20]

G. M. Lieberman, Regularity for solutions to some degenerate double obstacle problems, Indiana Univ. Math. J., 40 (1991), 1009-1028.  doi: 10.1512/iumj.1991.40.40045.

[21]

E. Milakis and T. Toro, Divergence form operators in Reifenberg flat domains, Math. Z., 264 (2010), 15-41.  doi: 10.1007/s00209-008-0450-2.

[22]

J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, 1034, Springer Verlag, Berlin, 1983. doi: 10.1007/BFb0072210.

[23]

J. Ok, Gradient estimates for elliptic equations with $L^{p(\cdot)}\log L$ growth, Calc. Var. Partial Differential Equations, 55 (2016), 30pp. doi: 10.1007/s00526-016-0965-z.

[24]

J. Ok, Calderón-Zygmund estimates for a class of obstacle problems with nonstandard growth, NoDEA Nonlinear Differential Equations Appl., 23 (2016) 21pp. doi: 10.1007/s00030-016-0404-z.

[25]

J. F. Rodrigues and R. Teymurazyan, On the two obstacles problem in Orlicz-Sobolev spaces and applications, Complex Var. Elliptic Equ., 56 (2011), 769-787.  doi: 10.1080/17476933.2010.505016.

[26]

C. Scheven, Elliptic obstacle problems with measure data: Potentials and low order regularity, Publ. Mat., 56 (2012), 327-374.  doi: 10.5565/PUBLMAT_56212_04.

[27]

C. Scheven, Existence of localizable solutions to nonlinear parabolic problems with irregular obstacles, Manuscripta Math., 146 (2015), 7-63.  doi: 10.1007/s00229-014-0684-8.

[28]

T. Toro, Doubling and flatness: Geometry of measures, Notices Amer. Math. Soc., 44 (1997), 1087-1094. 

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