非齐次椭圆障碍问题弱解梯度的局部 Besov 估计

Abstract

Abstract:

This paper studies the obstacle problem corresponding to non-homogeneous elliptic equations. By establishing appropriate admissible functions and using techniques such as variational inequalities, Young's inequality for $N$-functions, and the finite difference method, a local Besov estimate for the gradient of the weak solution of the elliptic obstacle problem in a bounded open domain is established.

Key words: obstacle problem, weak solution, Besov estimate, finite difference

CLC Number:

O175.25

Jiahui Yao, Yuxia Tong. Local Besov Estimates for the Gradient of Weak Solutions to Nonhomogeneous Elliptic Obstacle Problems[J].Acta mathematica scientia,Series A, 2026, 46(6): 2225-2238.

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References 21

[1]	Cianchi A, Maz'ya V. Global Lipschitz regularity for a class of quasilinear elliptic equations. Communications in Partial Differential Equations, 2011, 36(1): 100-133 doi: 10.1080/03605301003657843
[2]	Cianchi A, Maz'ya V. Gradient regularity via rearrangements for $p$-Laplacian type elliptic boundary value problems. Journal of the European Mathematical Society, 2014, 16(3): 571-595 doi: 10.4171/jems
[3]	Yao F P, Zhou S L. Calderón-Zygmund estimates for a class of quasilinear elliptic equations. Journal of Functional Analysis, 2017, 272(4): 1524-1552 doi: 10.1016/j.jfa.2016.11.008
[4]	Yao F P, Zhang C, Zhou S L. Global regularity estimates for a class of quasilinear elliptic equations in the whole space. Nonlinear Analysis: Theory, Methods and Applications, 2020, 194: Art 111307
[5]	Ma R M, Yao F P. Besov estimates for the weak solutions of a class of quasilinear elliptic equations. Complex Variables and Elliptic Equations, 2021, 66(12): 1975-1990 doi: 10.1080/17476933.2020.1797702
[6]	佟玉霞, 郑神州, 于海燕. 变指数 $A$-调和方程弱解的梯度的局部 Hölder 连续性. 数学物理学报, 2015, 35A(4): 656-667
	Tong Y X, Zheng S Z, Yu H Y. Hölder continuity of the gradients of weak solutions to $A$-harmonic equation with variable exponents. Acta Mathematica Scientia, 2015, 35A(4): 656-667
[7]	Malý J, Ziemer W P. Fine Regularity of Solutions of Elliptic Partial Differential. Providence, Rhode Island: American Mathematical Society, 1997
[8]	Evans L C. Second Edition.Partial Differential Equations, Providence, Rhode Island: American Mathematical Society, 2010
[9]	Byun S S, Cho Y, Liang S. Calderón-Zygmund estimates for quasilinear elliptic double obstacle problems with variable exponent and logarithmic growth. Discrete and Continuous Dynamical Systems-Series B, 2020, 25(10): 3843-3855 doi: 10.3934/dcdsb.2020038
[10]	Byun S S, Song K, Youn Y. Fractional differentiability for elliptic double obstacle problems with measure data. Zeitschrift für Analysis und ihre Anwendungen, 2023, 42(1/2): 37-64 doi: 10.4171/zaa
[11]	Zhang X L, Zheng S Z. $\textit{W}^{1,\gamma(\cdot)}$-estimate to non-uniformly elliptic obstacle problems with borderline growth. Complex Variables and Elliptic Equations, 2023, 68(10): 1833-1856 doi: 10.1080/17476933.2022.2080199
[12]	佟玉霞, 郭艳敏, 谷建涛. 一类椭圆障碍问题弱解梯度的全局 BMO 估计. 数学物理学报, 2023, 43A(1): 159-168
	Tong Y X, Guo Y M, Gu J T. Global BMO estimation for the gradient of weak solutions to a class of elliptic obstacle problems. Acta Mathematica Scientia, 2023, 43A(1): 159-168
[13]	Li G B, Olli M. Local and global integrability of gradients in obstacle problems. Annales Academiæ Scientiarum Fennicæ Series A.I. Mathematica, 1994, 19: 25-34
[14]	Byun S S, Namkyeong C. Higher differentiability for solutions of a general class of nonlinear elliptic obstacle problems with Orlicz growth. Nonlinear Differential Equations and Applications, 2022, 29: Art 73
[15]	Liang S, Zheng S Z. Lorentz estimates to nonlinear elliptic obstacle problems of $p(x)$-growth in Reifenberg domains. Journal of Mathematical Analysis and Applications, 2021, 501(1): Art 123924
[16]	Cho Y, Scheven C. Hölder regularity for singular parabolic obstacle problems of porous medium type. International Mathematics Research Notices, 2020, 10: 1671-1711
[17]	Tian H, Zheng S Z. Lorentz estimate with a variable power for parabolic obstacle problems with non-standard growths. Journal of Differential Equations, 2019, 266(1): 352-405 doi: 10.1016/j.jde.2018.07.049
[18]	Byun S S, Ryu S. Existence and regularity for nonlinear parabolic double obstacle problems. London Mathematical Society, 2023, 36: 4785-4809
[19]	Lindfors C. Obstacle problem for a class of parabolic equations of generalized $p$-Laplacian type. Journal of Differential Equations, 2016, 261(10): 5499-5540 doi: 10.1016/j.jde.2016.08.015
[20]	Adams R A, Fournier J J F. Sobolev Spaces, Second Edition. New York: Academic Press, 2003
[21]	Zhao L J, Zheng S Z. Besov regularity for a class of elliptic obstacle problems with double-phase Orlicz growth. Journal of Mathematical Analysis and Applications, 2024, 535(1): Art 128119

Local Besov Estimates for the Gradient of Weak Solutions to Nonhomogeneous Elliptic Obstacle Problems

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