JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2020, Vol. 55 ›› Issue (6): 76-83.doi: 10.6040/j.issn.1671-9352.0.2019.871

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Local gradient estimates for weak solutions of obstacle problems to a class of A-harmonic equations

ZHANG Ya-nan, YANG Ya-qi, TONG Yu-xia*   

  1. College of Science, North China University of Science and Technology, Tangshan 063210, Hebei, China
  • Published:2020-06-01

Abstract: This paper deals with the local gradient estimates for weak solutions of obstacle problems to a class of A-harmonic equations. Lp-type estimates to weak solutions of obstacle problems are derived. The Lφ-type estimates in Orlicz space are also obtained by a new normalization method and a new iteration-covering approach.

Key words: A-harmonic equation, obstacle problem, weak solution, gradient estimate

CLC Number: 

  • O175.25
[1] CHOE H J. A regularity theory for a general class of quasilinear elliptic partial differential equations and obstacle problems[J]. Archive for Rational Mechanics and Analysis, 1991, 114(4):383-394.
[2] LIANG Shuang, ZHENG Shenzhou. Gradient estimate in Orlicz spaces for elliptic obstacle problems with partially BMO nonlinearities[J]. Electronic Journal of Differential Equations, 2018, 2018(58):1-15.
[3] YAO Fengping. Gradient estimates for weak solutions of A-harmonic equations[J]. Journal of Inequalities and Applications, 2010(1):1-19.
[4] BYUN S S, LEE M. Weighted estimates for nondivergence parabolic equations in Orlicz spaces[J]. Journal of Functional Analysis, 2015, 269(8):2530-2563.
[5] BYUN S S, YAO Fengping, ZHOU Shulin. Gradient estimates in Orlicz space for nonlinear elliptic equations[J]. Journal of Functional Analysis, 2008, 255(8):1851-1873.
[6] GIAQUINTA M. Multiple integrals in the calculus of variations and nonlinear elliptic systems[M]. Princeton: Princeton University Press, 1983.
[7] 高红亚, 贾苗苗. 障碍问题解的局部正则性和局部有界性[J]. 数学物理学报, 2017, 37(4):706-713. GAO Hongya, JIA Miaomiao. Local regularity and local boundedness for solutions to obstacle problems[J]. Acta Mathematica Scientia, 2017, 37(4):706-713.
[8] YAO Fengping, SUN Yu, ZHOU Shulin. Gradient estimates in Orlicz spaces for quasilinear elliptic equation[J]. Nonlinear Analysis, 2008, 69(8):2553-2565.
[9] HEINONEN J, KILPELANEN T, MARTIO O. Nonlinear potential theory of degenerate elliptic equations[M]. New York: Clarendon Press, 1993.
[10] LIEBRMAN G M. The natural generalization of the natural conditions of Ladyzhenskaya and Urall'tseva for elliptic equations[J]. Communications in Partial Differential Equations, 1991, 16(2/3):311-361.
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