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

Abstract

In this paper, we develop an elementary and unified treatment, in the spirit of Rivière 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 Hölder continuity for solutions of the higher order elliptic system of de Longueville and Gastel in critical dimensions Δ k u = ∑ i = 0 k − 1 Δ i ⟨ V i , d u ⟩ + ∑ i = 0 k − 2 Δ i δ ( w i d u ) in  B 2 k , \begin{equation*} \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}, \end{equation*} under critical regularity assumptions on the coefficient functions. This verifies an expectation of Rivière, and provides an affirmative answer to an open question of Struwe in dimension four when k = 2 k=2 . The Hölder continuity is also an improvement of the continuity result of Lamm and Rivière and de Longueville and Gastel.