×

zbMATH — the first resource for mathematics

Partial regularity for stationary harmonic maps into spheres. (English) Zbl 0754.58007
Let \(\Omega\subset \mathbb{R}^ n\) denote an open set and consider a weak solution \(u\in H^{1,2}(\Omega,S^{m-1})\) of \(-\Delta u=u\cdot |\nabla u|^ 2\), i.e. a weak solution of the Euler-Lagrange equation of the energy functional \(\int_ \Omega|\nabla u|^ 2dx\) among all mappings from \(\Omega\) into the sphere \(S^{m-1}\). If in addition the monotonicity inequality holds for \(u\) then \(u\in C^ \infty(\Omega-\Sigma,\mathbb{R}^ m)\) for some relatively closed subset \(\Sigma\) of \(\Omega\) s.t. \(H^{n-2}(\Sigma)=0\).

MSC:
58E20 Harmonic maps, etc.
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] F. Bethuel & H. Brezis, Regularity of minimizers of relaxed problems for harmonic maps, preprint 1990.
[2] S. Chanillo, Sobolev inequalities involving divergence free maps, Comm. Part. Diff. Eqs., to appear. · Zbl 0778.42011
[3] Y. M. Chen, Weak solutions to the evolution problem for harmonic maps, preprint.
[4] R. Coifman, P.-L. Lions, Y. Meyer & S. Semmes, Compacité par compensation et espaces de Hardy, Comptes Rendus Acad. Sci. Serie I, t. 309 (1981), 945-949. · Zbl 0684.46044
[5] C. Fefferman, Characterizations of bounded mean oscillation, Bulletin AMS 77 (1971), 585-587. · Zbl 0229.46051
[6] C. Fefferman & E. Stein, HP spaces of several variables, Acta Math 129 (1972), 137-193. · Zbl 0257.46078
[7] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton Univ. Press, 1989.
[8] M. Giaquinta, G. Modica & J. Sou?ek, The Dirichlet energy of mappings with values in a sphere, Manuscripta Math. 65 (1989), 489-507. · Zbl 0678.49006
[9] M. Grüter, Regularity of weak H-surfaces, J. Reine Angew. Math. 329 (1981), 1-15. · Zbl 0461.53029
[10] R. Hardt, D. Kinderlehrer & F-H. Lin, Stable defects of minimizers of constrained variational principles, Ann. IHP, Analyse Nonlinéaire 5 (1988), 297-322. · Zbl 0657.49018
[11] R. Hardt & F-H. Lin, Mappings minimizing the L P norm of the gradient, Comm. Pure Appl. Math. 40 (1987), 556-588. · Zbl 0646.49007
[12] F. Hélefn, Regularité des applications faiblement harmoniques entre une surface et une sphere, Comptes Rendus Acad. Sci., to appear.
[13] F. Hélein, Regularity of weakly harmonic maps from a surface in a manifold with symmetries, preprint, 1990.
[14] S. Luckhaus, Partial Hölder continuity for minima of certain energies among maps into a Riemannian manifold, Indiana Univ. Math. Jour. 37 (1988), 349-367. · Zbl 0641.58012
[15] C. B. Morrey, Jr., Multiple Integrals in the Calcules of Variations, Springer-Verlag, 1966.
[16] S. Müller, A surprising higher integrability property of mappings with positive determinant, Bull. Amer. Math. Soc. 21 (1989), 245-248. · Zbl 0689.49006
[17] P. Price, A monotonicity formula for Yang-Mills fields, Manuscripta Math. 43 (1983), 131-166. · Zbl 0521.58024
[18] R. Schoen, Analytic aspects of the harmonic map problem, in Seminar on Nonlinear Partial Differential Equations (edited by S. S. Chern), MSRI Publications 2, Springer-Verlag, 1984. · Zbl 0551.58011
[19] R. Schoen & K. Uhlenbeck, A regularity theory for harmonic maps, Jour. Diff. Geom. 17 (1982), 307-335. · Zbl 0521.58021
[20] J. Shatah, Weak solutions and development of singularities of the SU(2) ?-model, Comm. Pure Appl Math. 41 (1988), 459-469. · Zbl 0686.35081
[21] A. Torchinsky, Real Variable Methods in Harmonic Analysis, Academic Press, 1986. · Zbl 0621.42001
[22] W. C. Wente, The Dirichlet problem with a volume constraint, Manuscripta Math. 11 (1974), 141-157. · Zbl 0268.35031
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.