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Non-Uniqueness for the p-Harmonic Flow

Published online by Cambridge University Press:  20 November 2018

Norbert Hungerbühler
Norbert Hungerbühler*
Affiliation:
ETH-Zentrum Departement Mathematik CH-8092 Zürich Switzerland, e-mail: [email protected]
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Abstract

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If f0: Ω ⊂ ℝm → Sn is a weakly p-harmonic map from a bounded smooth domain Ω in ℝm (with 2 < p < m) into a sphere and if f0 is not stationary p-harmonic, then there exist infinitely many weak solutions of the p-harmonic flow with initial and boundary data f0, i.e., there are infinitely many global weak solutions f :Ω × ℝ → ⊂ Sn of

We also show that there exist non-stationary weakly (m − 1)-harmonic maps f0: Bm → Sm−1.

Keywords

35K4035K5535K65
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 40 , Issue 2 , 01 June 1997 , pp. 174 - 182
Copyright
Copyright © Canadian Mathematical Society 1997

References

1. Bethuel, F., Coron, J.-M., Ghidaglia, J.-M. and Soyeur, A., Heat flows and relaxed energies for harmonic maps, Progr. Nonlinear Differential Equations Appl. 7 (1992), 99–110.Google Scholar
2. Chen, Y., The weak solutions to the evolution problem of harmonic maps, Math. Z. 201 (1989), 69–74.Google Scholar
3. Chen, Y., Hong, M.-C. and Hungerbühler, N., Heat flow of p-harmonic maps with values into spheres, Math. Z. 215 (1994), 25–35.Google Scholar
4. Chen, Y. and Struwe, M., Existence and partial regularity results for the heat flow for harmonic maps, Math. Z. 201 (1989), 83–103.Google Scholar
5. Coron, J.-M., Nonuniqueness for the heat flow of harmonic maps, Ann. Inst. H. Poincaré. Anal. Non Linéaire 7 (1990), 335–344.Google Scholar
6. Eells, J. and Sampson, H. J., Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–169.Google Scholar
7. Freire, A., Uniqueness for the harmonic map flow from surfaces to general targets, Comment. Math. Helv. 70 (1995), 310–338.Google Scholar
8. Hungerbühler, N., Compactness properties of the p-harmonic flow into homogeneous spaces, Nonlinear Anal., to appear.Google Scholar
9. Hungerbühler, N., Global weak solutions of the p-harmonic flow into homogeneous spaces, Indiana Univ. Math. J. (1)45 (1996), 275–288.Google Scholar
10. Struwe, M., On the evolution of harmonic maps of Riemannian surfaces, Comment.Math. Helv. 60 (1985), 558–581.Google Scholar
11. Struwe, M., On the evolution of harmonic maps in higher dimensions, J. Differential Geom. 28 (1988), 485–502.Google Scholar