Regularity of \(p\)-harmonic maps from the \(p\)-dimensional ball into a sphere. (English) Zbl 0797.58019

Suppose that \(u: B^ p \to S^{n-1}\), \(n,p \geq 2\), is a weak solution of \(-\text{div}(| \nabla u |^{p-2} \nabla u_ i) = u_ i| \nabla u|^ p\), \(i = 1,\dots,n\), in the \(p\)-dimensional unit ball, i.e. a weakly \(p\)-harmonic map from \(B^ p\) to \(S^{n - 1}\). Then \(u\) is smooth. This result extends earlier work of F. Hélein [Manuscr. Math. 70, No. 2, 203-218 (1991; Zbl 0718.58019)] and occurs as a special case of the reviewer [Manuscr. Math. 81, No. 1-2, 89-94 (1993)].


58E20 Harmonic maps, etc.
35D10 Regularity of generalized solutions of PDE (MSC2000)


Zbl 0718.58019
Full Text: DOI EuDML


[1] Bethuel, F.: On the singular set of stationary harmonic maps. Manuscripta Math.78, 417–443 (1992) · Zbl 0792.53039
[2] Coifman, R.; P. L. Lions, Y. Meyer, S. Semmes: Compacité par compensation et espaces de Hardy. C. R. Acad. Sci. Paris309, 945–949 (1989);and Compensated compactness and Hardy spaces, Cahiers Mathematiques de la Decision, preprint no. 9123, CEREMADE · Zbl 0684.46044
[3] Evans, L.C.: Partial Regularity for Stationary Harmonic Maps into Spheres. Arch. Rat. Mech. Anal.116, 101–113 (1991) · Zbl 0754.58007
[4] Fefferman, C.: Characterizations of bounded mean oscillation. Bull. AMS77, 585–587 (1971) · Zbl 0229.46051
[5] Fefferman, C.; E. Stein:H p spaces of several variables. Acta Math.129, 137–193 (1972) · Zbl 0257.46078
[6] Fuchs, M.:p-harmonic obstacle problems. Part II: Extensions of maps and applications. Manuscripta Math.63, 381–419 (1989) · Zbl 0715.49004
[7] Fuchs, M.:p-harmonic obstacle problems. Part I: Partial regularity theory; Part III: Boundary regularity. Annali Mat. Pura Appl.156, 127–180 (1990) · Zbl 0715.49003
[8] Fuchs, M.: The blow up ofp-harmonic maps. Preprint no. 1511, Technische Hochshule Darmstadt 1992; to appear in Manuscripta Math.
[9] Garcia-Cuerva, J.; J.L. Rubio de Francia: Weighted Norm Inequalities and Related Topics. Mathematics Studies 116, Elsevier Science Publishers B.V., 1985
[10] Giaquinta, M.: Multiple integrals in the Calculus of Variations and Nonlinear Elliptic systems Annals of Math. Study no. 105, Princeton University Press, Princeton 1983. · Zbl 0516.49003
[11] Hardt, R.; F. H. Lin: Mappings minimizing theL p norm of the gradient. Comm. Pure Appl. Math.40, 555–588 (1987) · Zbl 0646.49007
[12] Hélein, F.: Regularité des applications faiblement harmoniques entre une surface et une sphère. C. R. Acad. Sci. Paris311, 519–524 (1990) · Zbl 0728.35014
[13] Hélein, F.: Regularity of weakly harmonic maps from a surface into a manifold with symmetries. Manuscripta Math.70, 203–218 (1991) · Zbl 0718.58019
[14] Hélein, F.: Regularité des applications faiblement harmoniques entre une surface et une variètè riemannienne. C. R. Acad. Sci. Paris312, 591–596 (1991) · Zbl 0728.35015
[15] Iwaniec, T.; A. Lutoborski: Integral estimates for null Lagrangians. Preprint, Syracuse University 1992 (to appear in Arch. Rat. Mech. Anal.) · Zbl 0793.58002
[16] Luckhaus, S.: Partial Hölder continuity for minima of certain energies among maps into a Riemannian manifold. Indiana Univ. Math. J.37, no. 2, 346–367 (1988) · Zbl 0641.58012
[17] Morrey, C.B.: Multiple integrals in the calculus of variations. Springer Verlag 1968. · Zbl 0142.38701
[18] Müller, S.: Higher integrability of determinants and weak convergence inL 1. J. Reine Angew. Math.412, 20–34 (1990) · Zbl 0713.49004
[19] Schoen, R.; K. Uhlenbeck: A regularity theory for harmonic maps. J. Diff. Geom.17, 307–335 (1982) · Zbl 0521.58021
[20] Torchinsky, A.: Real-Variable Methods in Harmonic Analysis. Acad. Press 1986 · Zbl 0621.42001
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