Notes on the regularity of harmonic map systems

Huang, Tao; Wang, Changyou

doi:10.1090/S0002-9939-10-10344-X

Notes on the regularity of harmonic map systems
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by Tao Huang and Changyou Wang PDF

Proc. Amer. Math. Soc. 138 (2010), 2015-2023 Request permission

Abstract:

In this paper, we provide an alternative proof of $C^{1,\alpha }$-regularity of continuous weak solutions to the system of harmonic maps or heat flow of harmonic maps by Riesz potential estimates between Morrey spaces.

References

David R. Adams, A note on Riesz potentials, Duke Math. J. 42 (1975), no. 4, 765–778. MR 458158
Fabrice Bethuel, On the singular set of stationary harmonic maps, Manuscripta Math. 78 (1993), no. 4, 417–443. MR 1208652, DOI 10.1007/BF02599324
Yun Mei Chen, Jiayu Li, and Fang-Hua Lin, Partial regularity for weak heat flows into spheres, Comm. Pure Appl. Math. 48 (1995), no. 4, 429–448. MR 1324408, DOI 10.1002/cpa.3160480403
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
James Eells Jr. and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160. MR 164306, DOI 10.2307/2373037
Mikhail Feldman, Partial regularity for harmonic maps of evolution into spheres, Comm. Partial Differential Equations 19 (1994), no. 5-6, 761–790. MR 1274539, DOI 10.1080/03605309408821034
Frédéric Hélein, Régularité des applications faiblement harmoniques entre une surface et une variété riemannienne, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 8, 591–596 (French, with English summary). MR 1101039
Mariano Giaquinta and Enrico Giusti, The singular set of the minima of certain quadratic functionals, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984), no. 1, 45–55. MR 752579
Mariano Giaquinta and Stefan Hildebrandt, A priori estimates for harmonic mappings, J. Reine Angew. Math. 336 (1982), 124–164. MR 671325, DOI 10.1515/crll.1982.336.124
Stéfan Hildebrandt, Helmut Kaul, and Kjell-Ove Widman, An existence theorem for harmonic mappings of Riemannian manifolds, Acta Math. 138 (1977), no. 1-2, 1–16. MR 433502, DOI 10.1007/BF02392311
Jürgen Jost, Harmonic mappings between Riemannian manifolds, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 4, Australian National University, Centre for Mathematical Analysis, Canberra, 1984. MR 756629
Richard M. Schoen, Analytic aspects of the harmonic map problem, Seminar on nonlinear partial differential equations (Berkeley, Calif., 1983) Math. Sci. Res. Inst. Publ., vol. 2, Springer, New York, 1984, pp. 321–358. MR 765241, DOI 10.1007/978-1-4612-1110-5_{1}7
Richard Schoen and Karen Uhlenbeck, A regularity theory for harmonic maps, J. Differential Geometry 17 (1982), no. 2, 307–335. MR 664498
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095