-
Previous Article
Quantitative jacobian determinant bounds for the conductivity equation in high contrast composite media
- DCDS-B Home
- This Issue
-
Next Article
Generalized solutions to models of inviscid fluids
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 |
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.
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. 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. |
[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. |
[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. 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. |
[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. |
[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 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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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 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. |
[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.
|
[1] |
Kai Zhang, Xiaoqi Yang, Song Wang. Solution method for discrete double obstacle problems based on a power penalty approach. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2021018 |
[2] |
Claudia Lederman, Noemi Wolanski. An optimization problem with volume constraint for an inhomogeneous operator with nonstandard growth. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020391 |
[3] |
João Vitor da Silva, Hernán Vivas. Sharp regularity for degenerate obstacle type problems: A geometric approach. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1359-1385. doi: 10.3934/dcds.2020321 |
[4] |
João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138 |
[5] |
Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213 |
[6] |
Shengbing Deng, Tingxi Hu, Chun-Lei Tang. $ N- $Laplacian problems with critical double exponential nonlinearities. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 987-1003. doi: 10.3934/dcds.2020306 |
[7] |
Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261 |
[8] |
Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367 |
[9] |
Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020340 |
[10] |
Li Cai, Fubao Zhang. The Brezis-Nirenberg type double critical problem for a class of Schrödinger-Poisson equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020125 |
[11] |
Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298 |
[12] |
Yunfeng Jia, Yi Li, Jianhua Wu, Hong-Kun Xu. Cauchy problem of semilinear inhomogeneous elliptic equations of Matukuma-type with multiple growth terms. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3485-3507. doi: 10.3934/dcds.2019227 |
[13] |
Huijuan Song, Bei Hu, Zejia Wang. Stationary solutions of a free boundary problem modeling the growth of vascular tumors with a necrotic core. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 667-691. doi: 10.3934/dcdsb.2020084 |
[14] |
Ali Wehbe, Rayan Nasser, Nahla Noun. Stability of N-D transmission problem in viscoelasticity with localized Kelvin-Voigt damping under different types of geometric conditions. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020050 |
[15] |
Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020348 |
[16] |
Ali Mahmoodirad, Harish Garg, Sadegh Niroomand. Solving fuzzy linear fractional set covering problem by a goal programming based solution approach. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020162 |
[17] |
Elena Nozdrinova, Olga Pochinka. Solution of the 33rd Palis-Pugh problem for gradient-like diffeomorphisms of a two-dimensional sphere. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1101-1131. doi: 10.3934/dcds.2020311 |
[18] |
Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292 |
[19] |
Agnaldo José Ferrari, Tatiana Miguel Rodrigues de Souza. Rotated $ A_n $-lattice codes of full diversity. Advances in Mathematics of Communications, 2020 doi: 10.3934/amc.2020118 |
[20] |
Michael Winkler, Christian Stinner. Refined regularity and stabilization properties in a degenerate haptotaxis system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 4039-4058. doi: 10.3934/dcds.2020030 |
2019 Impact Factor: 1.27
Tools
Article outline
[Back to Top]