Regularity of weak solutions to higher order elliptic systems in critical dimensions

TL;DR

This paper establishes interior Hölder continuity for weak solutions of higher order elliptic systems in critical dimensions, using a unified approach inspired by Rivi e and Struwe, without relying on conservation laws.

Contribution

It provides a new, elementary method to prove regularity of solutions in critical dimensions, confirming a conjecture of Rivi e and answering an open question of Struwe.

Findings

Proves interior Hölder continuity for higher order elliptic systems in critical dimensions.

Extends regularity results to systems with critical regularity assumptions on coefficients.

Improves upon previous continuity results by Lamm, Rivi e, de Longueville, and Gastel.

Abstract

In this paper, we develop an elementary and unified treatment, in the spirit of Rivi\`ere and Struwe (Comm. Pure. Appl. Math. 2008), to explore regularity of weak solutions of higher order geometric elliptic systems in critical dimensions without using conservation law. As a result, we obtain an interior H\"older continuity for solutions of the higher order elliptic system of de Longueville and Gastel \cite{deLongueville-Gastel-2019} in critical dimensions $$\Delta^{k}u=\sum_{i=0}^{k-1}\Delta^{i}\left\langle V_{i},du\right\rangle +\sum_{i=0}^{k-2}\Delta^{i}\delta\left(w_{i}du\right) \quad \text{in } B^{2k},$$ under critical regularity assumptions on the coefficient functions. This verifies an expectation of Rivi\`ere, and provides an affirmative answer to an open question of Struwe in dimension four when $k=2$. The H\"older continuity is also an improvement of the continuity result of Lamm and Rivi\`ere and de Longueville and Gastel.