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
MathSciNet review:
4237957
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Abstract | References | Similar Articles | Additional Information
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 \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.
- Robert A. Adams and John J. F. Fournier, Sobolev spaces, 2nd ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. MR 2424078
- Gilles Angelsberg and David Pumberger, A regularity result for polyharmonic maps with higher integrability, Ann. Global Anal. Geom. 35 (2009), no. 1, 63–81. MR 2480664, DOI https://doi.org/10.1007/s10455-008-9122-z
- Fabrice Bethuel, Un résultat de régularité pour les solutions de l’équation de surfaces à courbure moyenne prescrite, C. R. Acad. Sci. Paris Sér. I Math. 314 (1992), no. 13, 1003–1007 (French, with English summary). MR 1168525
- Fabrice Bethuel, On the singular set of stationary harmonic maps, Manuscripta Math. 78 (1993), no. 4, 417–443. MR 1208652, DOI https://doi.org/10.1007/BF02599324
- Sun-Yung A. Chang, Lihe Wang, and Paul C. Yang, A regularity theory of biharmonic maps, Comm. Pure Appl. Math. 52 (1999), no. 9, 1113–1137. MR 1692148, DOI https://doi.org/10.1002/%28SICI%291097-0312%28199909%2952%3A9%3C1113%3A%3AAID-CPA4%3E3.0.CO%3B2-7
- Frédéric L. de Longueville, Regularität der Lösungen von Systemen $(2m)$-ter Ordnung vom polyharmonischen Typ in kritischer Dimension, Dissertation Universität Duisburg-Essen 2018 (see https://d-nb.info/1191692124/34).
- Frédéric L. de Longueville and Andreas Gastel, Conservation laws for even order systems of polyharmonic map type, Preprint 2019.
- R. DeVore and K. Scherer, Interpolation of linear operators on Sobolev spaces, Ann. of Math. (2) 109 (1979), no. 3, 583–599. MR 534764, DOI https://doi.org/10.2307/1971227
- Lawrence C. Evans, Partial regularity for stationary harmonic maps into spheres, Arch. Rational Mech. Anal. 116 (1991), no. 2, 101–113. MR 1143435, DOI https://doi.org/10.1007/BF00375587
- Andreas Gastel and Christoph Scheven, Regularity of polyharmonic maps in the critical dimension, Comm. Anal. Geom. 17 (2009), no. 2, 185–226. MR 2520907, DOI https://doi.org/10.4310/CAG.2009.v17.n2.a2
- Mariano Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton University Press, Princeton, NJ, 1983. MR 717034
- Paweł Goldstein, Paweł Strzelecki, and Anna Zatorska-Goldstein, On polyharmonic maps into spheres in the critical dimension, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 4, 1387–1405. MR 2542730, DOI https://doi.org/10.1016/j.anihpc.2008.10.008
- Chang-Yu Guo and Chang-Lin Xiang, Regularity of solutions for a fourth-order elliptic system via conservation law, J. Lond. Math. Soc. (2) 101 (2020), no. 3, 907–922. MR 4111928, DOI https://doi.org/10.1112/jlms.12289
- Chang-Yu Guo and Chang-Lin Xiang, Regularity of weak solutions to higher order elliptic systems in critical dimensions. arXiv:2010.09149
- Frédéric Hélein, Harmonic maps, conservation laws and moving frames, 2nd ed., Cambridge Tracts in Mathematics, vol. 150, Cambridge University Press, Cambridge, 2002. Translated from the 1996 French original; With a foreword by James Eells. MR 1913803
- Stefan Hildebrandt, Nonlinear elliptic systems and harmonic mappings, Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Vol. 1, 2, 3 (Beijing, 1980) Sci. Press Beijing, Beijing, 1982, pp. 481–615. MR 714341
- Kwok-Pun Ho, Sobolev-Jawerth embedding of Triebel-Lizorkin-Morrey-Lorentz spaces and fractional integral operator on Hardy type spaces, Math. Nachr. 287 (2014), no. 14-15, 1674–1686. MR 3266132, DOI https://doi.org/10.1002/mana.201300217
- Jasmin Hörter and Tobias Lamm, Conservation laws for even order elliptic systems in the critical dimensions - a new approach, Preprint 2020.
- Yin Bon Ku, Interior and boundary regularity of intrinsic biharmonic maps to spheres, Pacific J. Math. 234 (2008), no. 1, 43–67. MR 2375314, DOI https://doi.org/10.2140/pjm.2008.234.43
- Tobias Lamm and Tristan Rivière, Conservation laws for fourth order systems in four dimensions, Comm. Partial Differential Equations 33 (2008), no. 1-3, 245–262. MR 2398228, DOI https://doi.org/10.1080/03605300701382381
- Tobias Lamm and Changyou Wang, Boundary regularity for polyharmonic maps in the critical dimension, Adv. Calc. Var. 2 (2009), no. 1, 1–16. MR 2494504, DOI https://doi.org/10.1515/ACV.2009.001
- Yves Meyer and Tristan Rivière, A partial regularity result for a class of stationary Yang-Mills fields in high dimension, Rev. Mat. Iberoamericana 19 (2003), no. 1, 195–219. MR 1993420, DOI https://doi.org/10.4171/RMI/343
- Charles B. Morrey Jr., The problem of Plateau on a Riemannian manifold, Ann. of Math. (2) 49 (1948), 807–851. MR 27137, DOI https://doi.org/10.2307/1969401
- Richard O’Neil, Convolution operators and $L(p,\,q)$ spaces, Duke Math. J. 30 (1963), 129–142. MR 146673
- Tristan Rivière, Conservation laws for conformally invariant variational problems, Invent. Math. 168 (2007), no. 1, 1–22. MR 2285745, DOI https://doi.org/10.1007/s00222-006-0023-0
- Tristan Rivière, The role of integrability by compensation in conformal geometric analysis, Analytic aspects of problems in Riemannian geometry: elliptic PDEs, solitons and computer imaging, Sémin. Congr., vol. 22, Soc. Math. France, Paris, 2011, pp. 93–127 (English, with English and French summaries). MR 3060451
- Tristan Rivière, Conformally invariant variational problems, Lecture notes at ETH Zurich, available at https://people.math.ethz.ch/~riviere/lecture-notes, 2012.
- Tristan Rivière and Michael Struwe, Partial regularity for harmonic maps and related problems, Comm. Pure Appl. Math. 61 (2008), no. 4, 451–463. MR 2383929, DOI https://doi.org/10.1002/cpa.20205
- Michael Struwe, Partial regularity for biharmonic maps, revisited, Calc. Var. Partial Differential Equations 33 (2008), no. 2, 249–262. MR 2413109, DOI https://doi.org/10.1007/s00526-008-0175-4
- Paweł Strzelecki, On biharmonic maps and their generalizations, Calc. Var. Partial Differential Equations 18 (2003), no. 4, 401–432. MR 2020368, DOI https://doi.org/10.1007/s00526-003-0210-4
- Terence Tao and Gang Tian, A singularity removal theorem for Yang-Mills fields in higher dimensions, J. Amer. Math. Soc. 17 (2004), no. 3, 557–593. MR 2053951, DOI https://doi.org/10.1090/S0894-0347-04-00457-6
- Karen K. Uhlenbeck, Connections with $L^{p}$ bounds on curvature, Comm. Math. Phys. 83 (1982), no. 1, 31–42. MR 648356
- Changyou Wang, Remarks on biharmonic maps into spheres, Calc. Var. Partial Differential Equations 21 (2004), no. 3, 221–242. MR 2094320, DOI https://doi.org/10.1007/s00526-003-0252-7
- Changyou Wang, Biharmonic maps from $\mathbf R^4$ into a Riemannian manifold, Math. Z. 247 (2004), no. 1, 65–87. MR 2054520, DOI https://doi.org/10.1007/s00209-003-0620-1
- Changyou Wang, Stationary biharmonic maps from $\Bbb R^m$ into a Riemannian manifold, Comm. Pure Appl. Math. 57 (2004), no. 4, 419–444. MR 2026177, DOI https://doi.org/10.1002/cpa.3045
- William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. MR 1014685
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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
MR Author ID:
1119398
ORCID:
0000-0002-8135-4438
Email:
changlin.xiang@yangtzeu.edu.cn
Keywords:
Higher order elliptic systems,
Hölder 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:
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American Mathematical Society