Sharp Morrey regularity theory for a fourth order geometrical equation

Abstract

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 $V,w,\omega ,F$ 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.