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Transactions of the American Mathematical Society

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


Authors: Chang-Yu Guo and Chang-Lin Xiang
Journal: Trans. Amer. Math. Soc. 374 (2021), 3579-3602
MSC (2020): Primary 35J48, 35G50, 35B65
DOI: https://doi.org/10.1090/tran/8326
Published electronically: January 21, 2021
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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

$\displaystyle \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 )$$\displaystyle \quad \text {in } B^{2k},$    

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$. The Hölder continuity is also an improvement of the continuity result of Lamm and Rivière and de Longueville and Gastel.

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Additional Information

Chang-Yu Guo
Affiliation: Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, Qingdao, People’s Republic of China
Email: changyu.guo@sdu.edu.cn

Chang-Lin Xiang
Affiliation: Three Gorges Math Research Center, China Three Gorges University, 443002, Yichang, People’s Republic of China; and School of Information and Mathematics, Yangtze University, 434023, Jingzh ou, People’s Republic of China
Email: changlin.xiang@yangtzeu.edu.cn

DOI: https://doi.org/10.1090/tran/8326
Keywords: Higher order elliptic systems, H\"older regularity, Lorentz-Sobolev spaces, Riesz potential theory, Gauge transform
Received by editor(s): July 26, 2019
Received by editor(s) in revised form: September 15, 2020
Published electronically: January 21, 2021
Additional Notes: Chang-Lin Xiang is the corresponding author
The first author was supported by Swiss National Science Foundation Grant 175985 and the Qilu funding of Shandong University (No. 62550089963197). The second author was financially supported by the National Natural Science Foundation of China (No. 11701045) and the Yangtze Youth Fund (No. 2016cqn56).
Article copyright: © Copyright 2021 American Mathematical Society