一类四阶几何方程的精确Morrey正则性理论
Sharp Morrey regularity theory for a fourth order geometrical equation
AI总结：
该研究延续前人工作，推导了最低正则性假设下四阶非齐次Lamm-Rivière方程弱解的精确Morrey正则性理论，深化了Lᵖ型正则性理论并将二阶问题结果推广至四阶情形。
摘要：
本文是Guo-Xiang-Zheng近期研究工作（文献[Guo-Xiang-Zheng-2021-CV]）的延续。受双调和映射流等诸多几何问题的启发，我们推导了四阶非齐次Lamm-Rivière方程弱解的精确Morrey正则性理论，方程形式为在B⁴中Δ²u=Δ(V∇u)+div(w∇u)+(∇ω+F)·∇u+f，其中V、w、ω、F满足最低正则性假设，f属于某些Morrey空间。我们的结果深化了文献[Guo-Xiang-Zheng-2021-CV]的Lᵖ型正则性理论，并将Du、Kang和Wang（文献[Du-Kang-Wang-2022]）关于二阶问题的工作推广到了本文的四阶问题。
英文摘要：
This paper is a continuation of the recent work of Guo-Xiang-Zheng \cite{Guo-Xiang-Zheng-2021-CV}. We deduce sharp Morrey regularity theory for weak solutions to the fourth order nonhomogeneous Lamm-Rivière equation \begin{equation*} Δ^{2}u=Δ(V\nabla u)+div(w\nabla u)+(\nablaω+F)\cdot\nabla u+f\qquad\text{in }B^{4},\end{equation*} under smallest regularity assumptions of $V,w,ω, F$ and that $f$ belongs to some Morrey spaces, which was motivated by many geometrical problems such as the flow of biharmonic mappings. Our results deepens the $L^p$ type regularity theory of \cite{Guo-Xiang-Zheng-2021-CV}, and generalizes the work of Du, Kang and Wang \cite{Du-Kang-Wang-2022} on a second order problem to our fourth order problems.
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