偶数阶椭圆方程组的临界正则性与紧性理论——献给李工宝教授 70 寿辰

数学物理学报 ›› 2025, Vol. 45 ›› Issue (6): 1806-1813.

偶数阶椭圆方程组的临界正则性与紧性理论——献给李工宝教授 70 寿辰

向长林¹(), 王杰²(), 张彬航²(), 周艳平²()

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

收稿日期:2025-02-13 修回日期:2025-04-27 出版日期:2025-12-26 发布日期:2025-11-18
作者简介:向长林,E-mail：changlin.xiang@ctgu.edu.cn;
王杰,E-mail：jie.wang@ctgu.edu.cn;
张彬航,E-mail：evanustc@mail.ustc.edu.cn;
周艳平,E-mail：zhyp5208@163.com
基金资助:
国家自然科学基金(12271296)

Borderline Regularity and Compactness Theory for An Even Order Elliptic Systems

Changlin Xiang¹(), Jie Wang²(), Binhang Zhang²(), Yanping Zhou²()

¹There Gorges Mathematical Research, China Three Gorges University, Hubei Yichang 443002
²College of Mathematics and Physics, China Three Gorges University, Hubei Yichang 443002

Received:2025-02-13 Revised:2025-04-27 Online:2025-12-26 Published:2025-11-18
Supported by:
NSFC(12271296)

摘要/Abstract

摘要：

该文研究如下偶数阶几何型椭圆偏微分方程组

$\begin{equation*} \Delta^{k}u=\sum_{l=0}^{k-1}\Delta^{l}\left\langle V_{l},{\rm d}u\right\rangle +\sum_{l=0}^{k-2}\Delta^{l}\delta\left(w_{l}{\rm d}u\right)+f \qquad \text{在}~ B_1 \subset\mathbb{R}^m~\text{上}, \end{equation*}$

其中, 假设所有系数 $ \{V_l, w_l\}_{l} $ 具有最小正则性, 并且 $ f $ 属于临界函数空间 $ L\log L(B_1) $. 该文获得上述方程弱解的最优高阶正则性结果. 作为应用, 本文进一步证明相关的紧性结果.

关键词: 偶数阶椭圆方程组, 正则性, 紧性, 衰减估计

Abstract:

We deduce optimal higher order regularity result for the even order geometrical elliptic system

$\begin{equation*} \Delta^{k}u=\sum_{l=0}^{k-1}\Delta^{l}\left\langle V_{l},{\rm d}u\right\rangle +\sum_{l=0}^{k-2}\Delta^{l}\delta\left(w_{l}{\rm d}u\right)+f \quad \text{in } B_1 \subset\mathbb{R}^m,\end{equation*}$

where all the coefficients $ \{V_l, w_l\}_{l} $ are assumed to have the smallnest regularity and $ f $ lies in the borderline function space $ L\log L(B_1) $. As an application, we also obtain a compactness result.

Key words: even order elliptic system, regularity, compactness, decay estimate

中图分类号:

O175.25

向长林, 王杰, 张彬航, 周艳平. 偶数阶椭圆方程组的临界正则性与紧性理论——献给李工宝教授 70 寿辰[J]. 数学物理学报, 2025, 45(6): 1806-1813.

Changlin Xiang, Jie Wang, Binhang Zhang, Yanping Zhou. Borderline Regularity and Compactness Theory for An Even Order Elliptic Systems[J]. Acta mathematica scientia,Series A, 2025, 45(6): 1806-1813.

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