$L^p$ regularity theory for even order elliptic systems with antisymmetric first order potentials

Abstract

Motivated by a challenging expectation of Rivi\`ere (2011), in the recent interesting work of deLongueville-Gastel (2019), de Longueville and Gastel proposed the following geometrical even order elliptic system \begin{equation*} \Delta^{m}u=\sum_{l=0}^{m-1}\Delta^{l}\left\langle V_{l},du\right\rangle +\sum_{l=0}^{m-2}\Delta^{l}\delta\left(w_{l}du\right)\qquad \text{ in } B^{2m}\label{eq: Longue-Gastel system} \end{equation*} which includes polyharmonic mappings as special cases. Under minimal regularity assumptions on the coefficient functions and an additional algebraic antisymmetry assumption on the first order potential, they successfully established a conservation law for this system, from which everywhere continuity of weak solutions follows. This beautiful result amounts to a significant advance in the expectation of Rivi\`ere. In this paper, we seek for the optimal interior regularity of the above system, aiming at a more complete solution to the aforementioned expectation of Rivi\`ere. Combining their conservation law and some new ideas together, we obtain optimal H\"older continuity and sharp $L^p$ regularity theory, similar to that of Sharp and Topping \cite{Sharp-Topping-2013-TAMS}, for weak solutions to a related inhomogeneous system. Our results can be applied to study heat flow and bubbling analysis for polyharmonic mappings.