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On connected subsets of M2×2 without rank-one connections

Published online by Cambridge University Press:  14 November 2011

Kewei Zhang
Kewei Zhang
Affiliation:
School of Mathematics, Physics, Computing and Electronics, Macquarie University, Sydney, NSW 2109, Australia* e-mail: kewei@macadam.mpce.mq.edu.au
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Abstract

We prove that connected subsets of M2×2 without rank-one connections are Lipschitz graphs of mappings from subsets of a fixed two-dimensional subspace to its orthogonal complement. Under a weaker condition that the set does not have rank-one connections locally, we are able to establish some global results on the set. We also establish some results on Lipschitz extensions of the functions thus obtained.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 1 , 1997 , pp. 207 - 216
Copyright
Copyright © Royal Society of Edinburgh 1997

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References

1Ball, J. M.. Sets of gradients with no rank-one connections. J. Math. Pures Appl. 69 (1990), 241–59.Google Scholar
2Bhattacharya, K., Firoozye, N. B., James, R. D. and Kohn, R. V.. Restrictions on microstructures. Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), 843–78.CrossRefGoogle Scholar
3Bredon, G. E.. Topology and Geometry, Graduate Texts in Mathematics 139 (Berlin: Springer, 1993).CrossRefGoogle Scholar
4Šverák, V.. On the problems of two-wells. In Microstructure and Phase Transition, eds Kinderlehrer, D. et al. (Berlin: Springer, 1992).Google Scholar
5Šverák, V.. On Tartar's conjecture. Ann. Inst. H. Poincaré Anal. Non Lin. 10 (1993), 405–12.CrossRefGoogle Scholar
6Tartar, L.. The compensated compactness methods applied to systems of conservation laws. In Systems of Nonlinear Partial Differential Equations, ed. Ball, J. M., NATO ASI Series C111 (Dordrecht: Reidel, 1982).Google Scholar
7Wells, J. H. and Williams, L. R.. Embeddings and Extensions in Analysis (Berlin: Springer, 1975).CrossRefGoogle Scholar