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

Guo, Chang-Yu; Xiang, Chang-Lin

doi:10.1090/tran/8326

Regularity of weak solutions to higher order elliptic systems in critical dimensions
HTML articles powered by AMS MathViewer

by Chang-Yu Guo and Chang-Lin Xiang PDF

Trans. Amer. Math. Soc. 374 (2021), 3579-3602 Request permission

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.

References

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 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 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 10.1002/(SICI)1097-0312(199909)52:9<1113::AID-CPA4>3.0.CO;2-7
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 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 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 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
PawełGoldstein, PawełStrzelecki, and Anna Zatorska-Goldstein, On polyharmonic maps into spheres in the critical dimension, Ann. Inst. H. Poincaré C Anal. Non Linéaire 26 (2009), no. 4, 1387–1405. MR 2542730, DOI 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 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, DOI 10.1017/CBO9780511543036
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 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 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 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 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 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 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 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 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 10.1007/s00526-008-0175-4
PawełStrzelecki, On biharmonic maps and their generalizations, Calc. Var. Partial Differential Equations 18 (2003), no. 4, 401–432. MR 2020368, DOI 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 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, DOI 10.1007/BF01947069
Changyou Wang, Remarks on biharmonic maps into spheres, Calc. Var. Partial Differential Equations 21 (2004), no. 3, 221–242. MR 2094320, DOI 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 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 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, DOI 10.1007/978-1-4612-1015-3