×

The weak solutions to the evolution problems of harmonic maps. (English) Zbl 0685.58015

The author proves the existence of global weak solution to the evolution problem of harmonic maps of a compact Riemannian manifold M into the Euclidean n-sphere \(S^ n\), i.e. the existence of a distribution solution of \(\partial_ tu-\Delta_ Mu+| u_*|^ 2u=0\), \(t>0\), \(| u|^ 2=1\), with \(u(0,.)=u_ 0\in H^{1,2}(M)\) that is \(L^{\infty}\)-bounded and weakly continuous in \(t>0\) with values in \(H^{1,2}(M)\).
Reviewer: G.Tóth

MSC:

58E20 Harmonic maps, etc.
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] Eells, J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Ann. J. Math.86, 109-160 (1964) · Zbl 0122.40102
[2] Lemair, L.: On the existence of harmonic maps. Thesis, Warwick University 1977
[3] Sacks, J., Uhlenbeck, K.: The existence of minimal immersions of two-spheres. Bull. Am. Math. Soc.83, 1033-1036 (1977) · Zbl 0375.49016
[4] Schoen, R.M., Uhlenbeck, K.: Boundary regularity and miscellaneous results on harmonic maps. Differ. Geom.18, 253-268 (1983) · Zbl 0547.58020
[5] Struwe, M.: On the evolution of harmonic maps of Riemannian surfaces. Commun. Math. Helv.60, 558-581 (1985) · Zbl 0595.58013
[6] Struwe, M.: On the evolution of harmonic maps in higher dimensions. To appear in J. Differ. Geom. · Zbl 0631.58004
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.