Stationary biharmonic maps from \(\mathbb{R}^{m}\) into a Riemannian manifold. (English) Zbl 1055.58008

This paper continues the studies of [C. Wang, Math.Z.247, 65–87 (2004; Zbl 1064.58016)] about biharmonic maps into a Riemannian manifold. Now dimensions \(m\geq 5\) of the domain are studied. It is proved that every stationary biharmonic map in \(W^{2,2}\) is smooth outside a closed set of \((m{-}4)\)-dimensional Hausdorff measure zero. This is true for both extrinsic (critical for \(\int | \Delta u| ^2\)) and intrinsic (critical for \(\int | \tau(u)| ^2\), \(\tau(u)\) the tension field) biharmonic maps.
The proof uses a special frame in Coulomb gauge, based on a recent generalization of Uhlenbeck’s gauge theorem to Morrey spaces by Y. Meyer and T. Rivière [Rev. Mat. Iberoam. 19, No. 1, 195–219 (2003; Zbl 1127.35317)] and T. Tao and G. Tian [J. Am. Math. Soc. 17, No. 3, 557–593 (2004; Zbl 1086.53043)]. Another crucial ingredient is the monotonicity formula for biharmonic maps from [S.-Y. A. Chang, L. Wang and P. C. Yang, Commun.Pure Appl.Math.52, 1113–1137 (1999; Zbl 0953.58013)]. The proof is interesting for its consequent use of Morrey spaces.


58E20 Harmonic maps, etc.
35J35 Variational methods for higher-order elliptic equations
Full Text: DOI


[1] Adams, Duke Math J 42 pp 765– (1975)
[2] Adams, Proc Amer Math Soc 114 pp 155– (1992)
[3] Adams, Studia Math 74 pp 169– (1982)
[4] Bethuel, Manuscripta Math 78 pp 417– (1993)
[5] Chang, Comm Pure Appl Math 52 pp 1113– (1999)
[6] Chang, Comm Pure Appl Math 52 pp 1099– (1999)
[7] Eells, Bull London Math Soc 20 pp 385– (1988)
[8] Evans, Arch Rational Mech Anal 116 pp 101– (1991)
[9] Hélein, C R Acad Sci Paris Sér I Math 312 pp 591– (1991)
[10] Harmonic maps, conservation laws and moving frames. 2nd ed. Cambridge Tracts in Mathematics, 150. Cambridge University Press, Cambridge, 2002. · doi:10.1017/CBO9780511543036
[11] Iwaniec, Acta Math 170 pp 29– (1993)
[12] Interior and boundary regularity of intrinsic biharmonic maps to spheres. Preprint, 2000.
[13] Meyer, Rev Math Iberoamericana
[14] Nirenberg, Ann Scuola Norm Sup Pisa Cl Sci (4) 13 pp 115– (1959)
[15] Schoen, J Differential Geom 17 pp 307– (1982)
[16] ; A singularity removal theorem for Yang-Mills fields in higher dimensions. Preprint, 2002.
[17] Uhlenbeck, Comm Math Phys 83 pp 11– (1982)
[18] Uhlenbeck, Comm Math Phys 83 pp 31– (1982)
[19] Biharmonic maps from R4 into a Riemannian manifold. Preprint, 2003.
[20] Remarks on biharmonic maps into spheres. Preprint, 2003.
[21] Weakly differentiable functions. Sobolev spaces and functions of bounded variation. Graduate Texts in Mathematics, 120. Springer, New York, 1989. · Zbl 0692.46022 · doi:10.1007/978-1-4612-1015-3
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.