×

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)].

MSC:

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

Citations:

Zbl 0718.58019
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.