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.
Categories: math.AP math.DG