SHARP MORREY REGULARITY THEORY FOR A FOURTH ORDER GEOMETRICAL EQUATION

doi:10.1007/s10473-024-0202-3

数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (2): 420-430.doi: 10.1007/s10473-024-0202-3

SHARP MORREY REGULARITY THEORY FOR A FOURTH ORDER GEOMETRICAL EQUATION

Changlin XIANG¹, Gaofeng ZHENG^2,*

1. Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, China;
2. School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

收稿日期:2022-11-20 修回日期:2023-01-08 出版日期:2024-04-25 发布日期:2023-12-06
通讯作者: *Gaofeng ZHENG, E-mail: gfzheng@ccnu.edu.cn
作者简介:Changlin XIANG, E-mail: changlin.xiang@ctgu.edu.cn
基金资助:
National Natural Science Foundation of China (12271296, 12271195).

SHARP MORREY REGULARITY THEORY FOR A FOURTH ORDER GEOMETRICAL EQUATION

Changlin XIANG¹, Gaofeng ZHENG^2,*

1. Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, China;
2. School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

Received:2022-11-20 Revised:2023-01-08 Online:2024-04-25 Published:2023-12-06
Contact: *Gaofeng ZHENG, E-mail: gfzheng@ccnu.edu.cn
About author:Changlin XIANG, E-mail: changlin.xiang@ctgu.edu.cn
Supported by:
National Natural Science Foundation of China (12271296, 12271195).

摘要/Abstract

摘要： This paper is a continuation of recent work by Guo-Xiang-Zheng[10]. We deduce the sharp Morrey regularity theory for weak solutions to the fourth order nonhomogeneous Lamm-Rivière equation $\begin{equation*} \Delta^{2}u=\Delta(V\nabla u)+{\rm div}(w\nabla u)+(\nabla\omega+F)\cdot\nabla u+f\qquad\text{in }B^{4},\end{equation*}$ under the smallest regularity assumptions of $V,w,\omega, F$, where $f$ belongs to some Morrey spaces. This work was motivated by many geometrical problems such as the flow of biharmonic mappings. Our results deepens the $L^p$ type regularity theory of [10], and generalizes the work of Du, Kang and Wang [4] on a second order problem to our fourth order problems.

关键词: fourth order elliptic equation, regularity theory, Morrey space, decay estimates, Riesz potential

Abstract: This paper is a continuation of recent work by Guo-Xiang-Zheng[10]. We deduce the sharp Morrey regularity theory for weak solutions to the fourth order nonhomogeneous Lamm-Rivière equation $\begin{equation*} \Delta^{2}u=\Delta(V\nabla u)+{\rm div}(w\nabla u)+(\nabla\omega+F)\cdot\nabla u+f\qquad\text{in }B^{4},\end{equation*}$ under the smallest regularity assumptions of $V,w,\omega, F$, where $f$ belongs to some Morrey spaces. This work was motivated by many geometrical problems such as the flow of biharmonic mappings. Our results deepens the $L^p$ type regularity theory of [10], and generalizes the work of Du, Kang and Wang [4] on a second order problem to our fourth order problems.

Key words: fourth order elliptic equation, regularity theory, Morrey space, decay estimates, Riesz potential

中图分类号:

35J47

Changlin XIANG, Gaofeng ZHENG. SHARP MORREY REGULARITY THEORY FOR A FOURTH ORDER GEOMETRICAL EQUATION[J]. 数学物理学报(英文版), 2024, 44(2): 420-430.

Changlin XIANG, Gaofeng ZHENG. SHARP MORREY REGULARITY THEORY FOR A FOURTH ORDER GEOMETRICAL EQUATION[J]. Acta mathematica scientia,Series B, 2024, 44(2): 420-430.

参考文献

[1] Adams D R. A note on Riesz potentials. Duke Math J, 1975, 42: 765-778
[2] Adams R C, Fournier J R. Sobolev Spaces. Amsterdam: Elsevier, 2003
[3] Chang S Y A, Wang L, Yang P C. A regularity theory of biharmonic maps. Commun Pure Appl Math, 1999, 52(9): 1113-1137
[4] Du H, Kang Y, Wang J.Morrey regularity theory of Rivière's equation. Proc Amer Math Soc, 2023. DOI: https://doi.org/10.1090/proc/16143
[5] Gastel A. The extrinsic polyharmonic map heat flow in the critical dimension. Adv Geom, 2006, 6: 501-521
[6] Giaquinta M.Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton, NJ: Princeton University Press, 1983
[7] Guo C Y, Wang C, Xiang C L. $L^p$-regularity for fourth order elliptic systems with antisymmetric potentials in higher dimensions. Calc Var Partial Differential Equations, 2023, 62: Art 31
[8] Guo C Y, Xiang C L. Regularity of solutions for a fourth order linear system via conservation law. J Lond Math Soc, 2020, 101: 907-922
[9] Guo C Y, Xiang C L. Regularity of weak solutions to higher order elliptic systems in critical dimensions. Tran Amer Math Soc, 2021, 374: 3579-3602
[10] Guo C Y, Xiang C L, Zheng G F. The Lamm-Riviere system I: $L^p$ regularity theory. Calc Var Partial Differential Equations, 2021, 60: Art 213
[11] Guo C Y, Xiang C L, Zheng G F. $L^p$ regularity theory for even order elliptic systems with antisymmetric first order potentials. J Math Pures Appl, 2022, 165: 286-324
[12] Lamm T. Heat flow for extrinsic biharmonic maps with small initial energy. Ann Global Anal Geom, 2004, 26: 369-384
[13] Lamm T, Rivière T. Conservation laws for fourth order systems in four dimensions. Comm Partial Differential Equations, 2008, 33: 245-262
[14] Laurain P, Rivière T. Angular energy quantization for linear elliptic systems with antisymmetric potentials and applications. Anal PDE, 2014, 7: 1-41
[15] Rivière T. Conservation laws for conformally invariant variational problems. Invent Math, 2007,168: 1-22
[16] Rivière T, Struwe M. Partial regularity for harmonic maps and related problems. Comm Pure Appl Math, 2008, 61: 451-463
[17] Sharp B, Topping P. Decay estimates for Rivière's equation, with applications to regularity and compactness. Trans Amer Math Soc, 2013, 365: 2317-2339
[18] Struwe M. Partial regularity for biharmonic maps, revisited. Calc Var Partial Differential Equations, 2008, 33: 249-262
[19] Wang C. Heat flow of biharmonic maps in dimensions four and its application. Pure Appl Math Q, 2007, 3: 595-613
[20] Wang C. Remarks on approximate harmonic maps in dimension two. Calc Var Partial Differential Equations, 2017, 56: Art 23
[21] Wang C Y. Biharmonic maps from $R^{4}$ into a Riemannian manifold. Math Z, 2004, 247: 65-87
[22] Wang C Y. Stationary biharmonic maps from $R^m$ into a Riemannian manifold. Comm Pure Appl Math, 2004, 57: 419-444

SHARP MORREY REGULARITY THEORY FOR A FOURTH ORDER GEOMETRICAL EQUATION

SHARP MORREY REGULARITY THEORY FOR A FOURTH ORDER GEOMETRICAL EQUATION

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