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

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SHARP MORREY REGULARITY THEORY FOR A FOURTH ORDER GEOMETRICAL EQUATION

Changlin XIANG1, Gaofeng ZHENG2,*   

  1. 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 ZHENG2,*   

  1. 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: 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