Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T01:58:23.093Z Has data issue: false hasContentIssue false
  • English
  • Français

Some new properties of Sobolev mappings: intersection theoretical approach

Published online by Cambridge University Press:  14 November 2011

Takeshi Isobe
Takeshi Isobe
Affiliation:
Department of Mathematics, Faculty of Science, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152, Japan
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we give some new examples of the energy gap phenomenon for functionals defined in Sobolev spaces. Our result is independent of that of Giaquinta, Modica and Soucek. We also give some new characterisations of Sobolev maps which can be approximated by smooth maps.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 2 , 1997 , pp. 337 - 358
Copyright
Copyright © Royal Society of Edinburgh 1997

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Bethuel, F.. A characterization of maps in H1(B3; S2) which can be approximated by smooth maps. Ann. Inst. H. Poincaré, Anal. Non Linéaire 7 (1990), 269–86.CrossRefGoogle Scholar
2Bethuel, F.. The approximation problem for Sobolev maps between manifolds. Ada Math. 167 (1991), 153–206.Google Scholar
3Bethuel, F. and Zheng, X.. Density of smooth functions between two manifolds in Sobolev spaces. J. Fund. Anal. 80 (1988), 60–75.CrossRefGoogle Scholar
4Bethuel, F., Brezis, H. and Coron, J. M.. Relaxed energies for harmonic maps. Progr. Nonlinear Differential Equations Appl. 4 (1990), 11–24.Google Scholar
5Bethuel, F., Coron, J. M., Demengel, F. and Helein, F.. A cohomological criterion for density of smooth maps in Sobolev spaces between two manifolds. In Nematics, Nato Adv. Sci. Inst. Ser. C Math. Phys. Sci. 332 (1991), 15–23.Google Scholar
6Bott, R. and Tu, W.. Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82 (Berlin: Springer, 1984).Google Scholar
7Bredon, G.. Topology and Geometry, Graduate Texts in Mathematics 139 (Berlin: Springer, 1993).CrossRefGoogle Scholar
8Brezis, H.. Sk-valued maps with singularities. In Topics in the Calculus of Variations, ed. Giaquinta, M., Lecture Notes in Mathematics 1365 (Berlin: Springer, 1989).Google Scholar
9Demengel, F.. Une caractérisation des applications de W1.p(BN; S1) qui peuvent être approchées par des fonctions C∞. C.R. Acad. Sci. Paris 310 (1990), 553–7.Google Scholar
10Gilkey, P.. Invariance Theory, The Heat Equation, and the Atiyah–Singer Index Theorem (Wilmington, Delaware: Publish or Perish, 1984).Google Scholar
11Giaquinta, M., Modica, G. and Soucek, J.. Cartesian currents and variational problems for mappings into spheres. Ann. Scuola Norm. Sup. Pisa 17 (1989), 393–485.Google Scholar
12Giaquinta, M., Modica, G. and Soucek, J.. The gap phenomenon for variational integrals in Sobolev spaces. Proc. Roy. Soc. Edinburgh Sect. A 120 (1992), 325–86.CrossRefGoogle Scholar
13Giaquinta, M., Modica, G. and Soucek, J.. The Dirichlet integral for mappings between manifolds: Cartesian currents and homology. Math. Ann. (1992), 325–86.CrossRefGoogle Scholar
14Hardt, R. and Lin, F. H.. A remark on H1-mappings. Manuscripta Math. 56 (1986), 1–10.CrossRefGoogle Scholar
15Hirsch, M.. Differential Topology, Graduate Texts in Mathematics 33 (Berlin: Springer, 1976).CrossRefGoogle Scholar
16Husemoller, D.. Fibre Bundles, 3rd edn, Graduate Texts in Mathematics 20 (Berlin: Springer, 1993).Google Scholar
17Isobe, T.. Characterization of the strong closure of C∞ (B4; S2) in W1.p(B4;S2) . J. Math. Anal. Appl. 190 (1995), 361–72.CrossRefGoogle Scholar
18Isobe, T.. Energy gap phenomenon and the existence of infinitely many weakly harmonic maps for the Dirichlet problem. J. Fund. Anal. 129 (1995), 243–67.CrossRefGoogle Scholar
19Morrey, C. B.. Multiple Integrals in the Calculus of Variations (Berlin: Springer, 1966).CrossRefGoogle Scholar
20Schoen, R. and Uhlembeck, K.. Boundary regularity and the Dirichlet problem for harmonic maps. J. Differential Geom. 18 (1983), 253–66.CrossRefGoogle Scholar
21Warner, F.. Foundations of Differentiable Manifolds and Lie Groups (Glenview, Illinois: Scott, Foresman, 1971).Google Scholar
22Wells, R.. Differential Analysis on Complex Manifolds (New York: Prentice-Hall, 1973).Google Scholar
23Zhou, Y.. On the density of smooth maps in Sobolev spaces between two manifolds (Ph.D. Thesis, Columbia University, 1993).Google Scholar