Abstract
We examine the harmonic map heat flow problem for maps between the three-dimensional ball and the two-sphere. We give blow-up results for certain initial data. We establish convergence results for suitable axially symmetric initial data, and discuss generalizations to higher dimensions.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Coron, J.-M., and Ghidaglia, J.-M.,Explosion en temps fini pour le flot des applications harmoniques, C. R. Acad. Sci. Paris, 308, Série I, 1989, pp. 339–344.
Eells, J., and Lemaire, L.,Another report on harmonic maps, Bull. London Math. Soc. 20, 1988, pp. 385–524.
Eells, J., and Sampson, J.H.,Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86, 1964, pp. 109–160.
Eells, J., and Wood, J.L.,Restrictions on harmonic maps of surfaces, Topology 15, 1976, pp. 263–266.
Grotowski, J.F.,Harmonic Maps with Symmetry, Ph.D. Thesis, New York University, 1990.
Grotowski, J.F.,Heat Flow for Harmonic Maps, inNematics: Mathematical and Physical Aspects, Proceedings of the NATO Advanced Research Workshop on Liquid Crystals, Orsay, 1990, J-M. Coron, J-M. Ghidaglia and F. Hélein, eds., Kluwer, Dordrecht, 1991, pp. 129–140.
Grayson, M., and Hamilton, R.S.,The formation of singularities in the harmonic map heat flow, preprint.
Hamilton, R.S.,Harmonic maps of manifolds with boundary, Springer Lecture Notes 471, 1975.
Hardt, R., and Kinderlehrer, D.,Mathematical questions of liquid crystals, inTheory and Applications of Liquid Crystals, J.L. Ericksen and D. Kinderlehrer, eds., Springer-Verlag, New York, 1987, pp. 151–184.
Hardt, R., Kinderlehrer, D., and Lin, F.H.,The variety of configerations of static liquid crystals, inProceedings of variational problems workshop, Ecol. Norm. Sup., Birkhauser, 1990.
Jost, J.,Harmonic mappings between Riemannian manifolds, Proc. Centre Math. Analysis, Australian National University Press, Canberra, 1983.
Jost, J.,Ein Existenzbeweis für harmonische Abbildungen, die ein Dirichletproblem lösen, mittels der Methode des Wärmeflusses, Man. Math. 34, 1981, pp.17–25.
Karcher, H., and Wood, J.W.,Non-existence results and growth properties for harmonic maps and forms, J. reine angew. Math. 353, 1984, pp. 165–180.
Ladyzhenskaja, O.A., Solonnikov, V.A., and Ural'ceva, N.N.,Linear and quasilinear equations of parabolic type, Trans. Math. Mono. 23, AMS, 1968.
Ladyzhenskaja, O.A., and Ural'ceva, N.N.,Linear and quasilinear elliptic equations, Academic Press, New York, 1968.
Schoen, R.,Analytic aspects of the harmonic map problem, Math. Sci. Res. Inst. Publ. 2, Springer Verlag, Berlin, 1984, pp. 321–358.
Schoen, R., and Uhlenbeck, K.,Boundary regularity and the Dirichlet problem for harmonic maps, J. Diff. Geom. 18, 1983, pp. 253–268.
von Wahl, W.,Klassische Lösbarkeit im Grossen für nichtlineare parabolische Systeme und das Verhalten der Lösungen für t→∞, Nach. Akad. Wiss. Gott. 5, 1981, pp. 131–177.
Zhang, D.,The Existence of Nonminimal Regular Harmonic Maps from B 3 to S 2, Ann. Sc. Norm. Sup. Pisa 16, No. 3, Serie IV, 1989, pp. 355–365.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Grotowski, J.F. Harmonic map heat flow for axially symmetric data. Manuscripta Math 73, 207–228 (1991). https://doi.org/10.1007/BF02567639
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02567639