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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Regularity of weak solutions to higher order elliptic systems in critical dimensions
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by Chang-Yu Guo and Chang-Lin Xiang PDF
Trans. Amer. Math. Soc. 374 (2021), 3579-3602 Request permission


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 \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 $k=2$. 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:
  • 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
  • MR Author ID: 1119398
  • ORCID: 0000-0002-8135-4438
  • Email:
  • 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).
  • © Copyright 2021 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 374 (2021), 3579-3602
  • MSC (2020): Primary 35J48, 35G50, 35B65
  • DOI:
  • MathSciNet review: 4237957