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. |
References:
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| [2] | Lemair, L.: On the existence of harmonic maps. Thesis, Warwick University 1977 |
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| [4] | Schoen, R.M., Uhlenbeck, K.: Boundary regularity and miscellaneous results on harmonic maps. Differ. Geom.18, 253-268 (1983) · Zbl 0547.58020 |
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| [6] | Struwe, M.: On the evolution of harmonic maps in higher dimensions. To appear in J. Differ. Geom. · Zbl 0631.58004 |
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