自然增长条件下的非齐次A-调和方程弱解的梯度估计

数学物理学报 ›› 2020, Vol. 40 ›› Issue (2): 379-394.

自然增长条件下的非齐次A-调和方程弱解的梯度估计

张雅楠,闫硕,佟玉霞*()

^{华北理工大学理学院河北唐山 063210}

收稿日期:2018-12-28 出版日期:2020-04-26 发布日期:2020-05-21
通讯作者: 佟玉霞 E-mail:tongyuxia@126.com
基金资助:
河北省社会科学基金(HB17YJ094)

Gradient Estimates for Weak Solutions to Non-Homogeneous A-Harmonic Equations Under Natural Growth

Yanan Zhang,Shuo Yan,Yuxia Tong*()

^{College of Science, North China University of Science and Technology, Hebei Tangshan 063210}

Received:2018-12-28 Online:2020-04-26 Published:2020-05-21
Contact: Yuxia Tong E-mail:tongyuxia@126.com
Supported by:
the Social Science Fund of Hebei Province(HB17YJ094)

摘要/Abstract

摘要：

该文主要研究一类自然增长条件下的非齐次A-调和方程弱解的梯度估计，首先获得自然增长条件下的非齐次A-调和方程弱解的L^p估计，然后使用迭代覆盖逼近等方法，将其推广到Orlicz空间.

关键词: A-调和方程, 弱解, 梯度估计

Abstract:

The gradient estimates for weak solutions to non-homogeneous A-harmonic equations under natural growth is obtained. The L^p-type estimates for such equation is derived under natural growth, and then the L^φ-type estimate in Orlicz space is derived by a new iteration-covering approach.

Key words: A-Harmonic equation, Weak solution, Gradient estimate

中图分类号:

O175.25

张雅楠,闫硕,佟玉霞. 自然增长条件下的非齐次A-调和方程弱解的梯度估计[J]. 数学物理学报, 2020, 40(2): 379-394.

Yanan Zhang,Shuo Yan,Yuxia Tong. Gradient Estimates for Weak Solutions to Non-Homogeneous A-Harmonic Equations Under Natural Growth[J]. Acta mathematica scientia,Series A, 2020, 40(2): 379-394.

参考文献 25

1	DiBenedetto E , Manfredi J . On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems. American Journal of Mathematics, 1993, 115 (5): 1107- 1134
2	Iwaniec T . Projections onto gradient fields and L^p-estimates for degenerated elliptic operators. Studia Mathematica, 1983, 75 (3): 293- 312
3	Acerbi E , Mingione G . Gradient estimates for the p(x)-Laplacean system. Journal für die reine und angewandte Mathmatik, 2005, 584, 117- 148
4	Byun S S , Wang L H . Quasilinear elliptic equations with BMO coefficients in Lipschitz domains. Transactions of the American Mathematical Society, 2007, 359 (12): 5899- 5913
5	Kinnunen J , Zhou S L . A local estimate for nonlinear equations with discontinuous coefficients. Commuications in Partial Differential Equations, 1999, 24 (11/12): 2043- 2068
6	Byun S S , Wang L H . L^p-estimates for general nonlinear elliptic equations. Indiana University Mathematics Journal, 2007, 56 (6): 3193- 3221
7	Jia H , Li D S , Wang L H . Regularity in Orlicz spaces for the Poisson equation. Manuscripta Mathematica, 2007, 122 (3): 265- 275
8	Yao F P. Gradient estimates for weak solutions of $ \mathcal{A} $-harmonic equations. Journal of Inequalities and Applications, 2010, Article ID: 685046. DOI: 10.1155/2010/685046
9	Yao F P . Gradient estimates for parabolic A-harmonic equations. Nonlinear Analysis, 2011, 74, 1200- 1211
10	李惠珍,郑神州.非散度型线性椭圆方程强解的Hessian估计[D].北京:北京交通大学, 2017
	Li H Z, Zheng S Z. Hessian Estimates to Strong Solutions for Nondivergence Linear Elliptic Equations[D]. Beijing: Beijing Jiaotong University, 2017
11	Liang S , Zheng S Z . Gradient estimate in Orlicz spaces for elliptic obstracle problems with partially BMO nonlinearities. Electronic Journal of Differential Equations, 2018, 58, 1- 15
12	Zheng S Z , Zheng X L , Feng Z S . Regularity for a class of degenerate elliptic equations with discontinuous coefficients under natural growth. Journal of Mathematical Analysis and Applications, 2008, 346 (2): 359- 373
13	周树清, 胡振华, 彭冬云. 一类A -调和方程的障碍问题的很弱解的全局正则性. 数学物理学报, 2017, 37A (4): 706- 713
	Zhou S Q , Hu Z H , Peng D Y . Global regularity for very weak solutions to obstacle promlems corresponding to a class of A-harmonic equations. Acta Mathematica Scientia, 2017, 37A (4): 706- 713
14	高红亚,褚玉明.拟正则映射与A -调和方程.北京:科学出版社, 2013
	Gao H Y, Chu Y M. Quasiregular Mappings and A-Harmonic Equation. Beijing: Science Press, 2013
15	Zhang J J , Zheng S Z . Weighted Lorentz estimates for nondivergence linear elliptic equations with partially BMO coefficients. Communications on Pure and Applied Analysis, 2017, 16 (3): 899- 914
16	佟玉霞, 郑神州, 程林娜. 弱A -调和张量的奇点可去性. 数学物理学报, 2017, 37A (6): 1001- 1011
	Tong Y X , Zheng S Z , Cheng L N . Removable singularities of weakly A-harmonic tensors. Acta Mathematica Scientia, 2017, 37A (6): 1001- 1011
17	Byun S S , Lee M . Weighted estimates for nondivergence parabolic equations in Orlicz spaces. Journal of Functional Analysis, 2015, 269 (8): 2530- 2563
18	Byun S S , Yao F P , Zhou S L . Gradient estimates in Orlicz space for nonlinear elliptic equations. Journal of Functional Analysis, 2008, 255 (8): 1851- 1873
19	Wang L H , Yao F P , Zhou S L , Jia H L . Optimal regularity for the Poisson equation. Proceedings of the American Mathematical Society, 2009, 137 (6): 2037- 2047
20	Giaquinta M. Multiple integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton: Princeton University Press, 1983
21	高红亚, 贾苗苗. 障碍问题解的局部正则性和局部有界性. 数学物理学报, 2017, 37A (4): 706- 713
	Gao H Y , Jia M M . Local regularity and local boundedness for solutions to obstacle problems. Acta Mathematica Scientia, 2017, 37A (4): 706- 713
22	Lieberman G M . The natural generalization of the natural conditions of Ladyzhenskaya and Urall'tseva for elliptic equations. Communications in Partial Differential Equations, 1991, 16 (2/3): 311- 361
23	Yao F P , Sun Y , Zhou S L . Gradient estimates in Orlicz spaces for quasilinear elliptic equation. Nonlinear Analysis, 2008, 69 (8): 2553- 2565
24	Tolksdorf P . Regularity for more general class of quasilinear elliptic equations. Journal of Differential Equations, 1984, 51 (1): 126- 150
25	Byun S S , Wang L H , Zhou S L . Nonlinear elliptic equations with small BMO coefficients in Reifenberg domains. Journal of Functional Analysis, 2007, 250 (1): 167- 196

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