Sharp Morrey regularity for an even order elliptic system

TL;DR

This paper develops a precise Morrey regularity theory for high-order elliptic systems of Rivi e type, extending previous L^p results and covering minimal regularity assumptions on coefficients and forcing terms.

Contribution

It establishes sharp Morrey regularity results for even order elliptic systems with minimal coefficient regularity, generalizing prior second and fourth order theories.

Findings

Proves Morrey regularity under minimal assumptions

Extends L^p-regularity to Morrey space setting

Generalizes previous results for second and fourth order systems

Abstract

In this short note, we establish a sharp Morrey regularity theory for an even order elliptic system of Rivi\`ere type: \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)+f\qquad \text{in} B^{2m} \end{equation*} under minimal regularity assumptions on the coefficients functions V^l, w^l and that f belongs to certain Morrey space. This can be regarded as a further extension of the recent L^p-regularity theory obtained by Guo-Xiang-Zheng [15], and generalizes [7, 27] for second and fourth order elliptic systems.