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doi: 10.3934/dcdsb.2020038

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

Sun-Sig Byun 1,, , Yumi Cho 2, and  Shuang Liang 3,

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  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.

Keywords: Double obstacle problem, regularity of solution, nonstandard growth, Calderón-Zygmund estimate.
Mathematics Subject Classification: Primary: 35J87; Secondary: 35B65.
Citation: Sun-Sig Byun, Yumi Cho, Shuang Liang. Calderón-Zygmund estimates for quasilinear elliptic double obstacle problems with variable exponent and logarithmic growth. Discrete & Continuous Dynamical Systems - B, 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.  Google Scholar

[2]

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

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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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M. Bildhauer, M. 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.  Google Scholar

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

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

[7]

S. S. Byun, Y. 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.  Google Scholar

[8]

S. S. Byun, Y. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

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

M. Bildhauer, M. 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.  Google Scholar

[5]

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

[6]

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

[7]

S. S. Byun, Y. 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.  Google Scholar

[8]

S. S. Byun, Y. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

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