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

TL;DR

This paper establishes optimal regularity results, including H"older continuity and sharp $L^p$ estimates, for solutions to a class of even order elliptic systems with antisymmetric potentials, advancing the understanding of polyharmonic mappings.

Contribution

It extends the regularity theory for even order elliptic systems with antisymmetric potentials, providing sharp $L^p$ and H"older estimates under minimal assumptions.

Findings

Proves optimal H"older continuity of weak solutions.

Derives sharp $L^p$ regularity estimates.

Applicable to heat flow and bubbling analysis in polyharmonic mappings.

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.