一类双调和映照型偏微分方程组正则性研究

一类双调和映照型偏微分方程组正则性研究

Research on the Regularity of a Class of Biharmonic Map-Type Partial Differential Equation Systems

摘要/Abstract

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数学物理学报 ›› 2025, Vol. 45 ›› Issue (2): 408-417.

刘安淇¹(),余婷^1,^*(),向长林²()

¹三峡大学数理学院湖北宜昌 443002
²三峡大学三峡数学研究中心湖北宜昌 443002

收稿日期:2024-07-22 修回日期:2024-12-16 出版日期:2025-04-26 发布日期:2025-04-09
通讯作者: 余婷 E-mail:anqi.liu@ctgu.edu.cn;yuting@ctgu.edu.cn;changlin.xiang@ctgu.edu.cn
作者简介:刘安淇，E-mail:anqi.liu@ctgu.edu.cn;|向长林，E-mail:changlin.xiang@ctgu.edu.cn
基金资助:
国家自然科学基金(12271296);湖北省自然科学基金杰出青年项目(2024AFA061)

Anqi Liu¹(),Ting Yu^1,^*(),Changlin Xiang²()

¹College of Mathematics and Physics, China Three Gorges University, Hubei Yichang 443002
²2Three Gorges Mathematical Research Center, China Three Gorges University, Hubei Yichang 443002

Received:2024-07-22 Revised:2024-12-16 Online:2025-04-26 Published:2025-04-09
Contact: Ting Yu E-mail:anqi.liu@ctgu.edu.cn;yuting@ctgu.edu.cn;changlin.xiang@ctgu.edu.cn
Supported by:
NSFC(12271296);SFC of Hubei province(2024AFA061)

摘要：

双调和映照是一类重要的几何映照, 但是满足的偏微分方程非常复杂, 导致其正则性研究很困难. 为了研究这一类问题, 该文考虑一类双调和映照型四阶椭圆偏微分方程组

$\Delta^{2}u=Q_{1}(x,u,\nabla u,\nabla^{2}u)+{\rm div}\,\boldsymbol{Q}_{2}(x,u,\nabla u,\nabla^{2}u),\qquad x\in B_{1},$

其中 $B_1=\{x\in\mathbb{R}^{n}:|x|<1\}$ , $n\ge4$ , $Q_{1},{\boldsymbol{Q}_{2}}$ 满足关于 $\nabla u$ 和 $\nabla^2 u$ 的临界增长条件. 则在适当的小性条件假设下, 该文证明该方程组的解均具有 Hölder 正则性, 从而推广了文献中的相关结果. 该结果有助于加深对双调和映照结构的理解与正则性理论的研究.

关键词: 双调和映照, 临界非线性椭圆方程组, H?lder 正则性, 反向 Poincaré 不等式, 衰减估计

Abstract:

Biharmonic mappings are an important class of geometric mappings, but the partial differential equations that are satisfied are very complex, making their regularity study difficult. In order to study this class of problems, in this note we consider a class of biharmonic map-type fourth order elliptic partial differential equation system

$\Delta^{2}u=Q_{1}(x,u,\nabla u,\nabla^{2}u)+{\rm div}\,\boldsymbol{Q}_{2}(x,u,\nabla u,\nabla^{2}u)\qquad\text{in }B_{1},$

where $B_1=\{x\in\mathbb{R}^{n}:|x|<1\}$ with $n\ge4$ , and $Q_{1},{\rm Q_{2}}$ satisfy critical growth conditions with respect to $\nabla u$ and $\nabla^2 u$ . Then, under suitable smallness assumption, this note proves that the solutions of this system of equations all have Hölder regularity, thus generalising related results in the literature. This result helps to deepen the understanding of the structure of biharmonic mappings and the research on the regularity theory.

Key words: biharmonic mappings, elliptic systems with critical nonlinearity, H?lder regularity, reverse Poincaré inequality, decay estimate

中图分类号:

O175.25

刘安淇,余婷,向长林. 一类双调和映照型偏微分方程组正则性研究[J]. 数学物理学报, 2025, 45(2): 408-417.

Anqi Liu,Ting Yu,Changlin Xiang. Research on the Regularity of a Class of Biharmonic Map-Type Partial Differential Equation Systems[J]. Acta mathematica scientia,Series A, 2025, 45(2): 408-417.

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