Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T12:04:41.358Z Has data issue: false hasContentIssue false
  • English
  • Français

On non-negative quasiconvex functions with unbounded zero sets

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We construct nontrivial, non-negative quasiconvex functions denned on M2×2 with p-th order growth such that the zero sets of the functions are Lipschitz graphs of mappings from subsets of a fixed two-dimensional subspace to its orthogonal complement. We assume that the graphs do not have rank-one connections with the Lipschitz constants sufficiently small. In particular, we are able to construct quasiconvex functions which are homogeneous of degree p (p > 1) and ‘conjugating’ invariant.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 2 , 1997 , pp. 411 - 422
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

1Ball, J. M.. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63(1977), 337–403.CrossRefGoogle Scholar
2Dacorogna, B.. Direct Methods in the Calculus of Variations (Berlin: Springer, 1989).CrossRefGoogle Scholar
3Dacorogna, B.. On rank one convex functions which are homogeneous of degree one (Preprint, 1994).Google Scholar
4Iwaniec, T.. Quasiconformal mapping problem for general nonlinear systems of partial differential equations. Sympos. Math. 18 (1976), 501–18.Google Scholar
5Kinderlehrer, D. and Pedregal, P.. Characterizations of Young measures generated by gradients. Arch. Rational Mech. Anal 115 (1991), 329–65.CrossRefGoogle Scholar
6Müller, S.. On quasiconvex functions which are homogeneous of degree one. Indiana Univ. Math. J. 41 (1992), 295–300.CrossRefGoogle Scholar
7Šverák, V.. On the problems of two-wells. In Microstructure and Phase Transition, eds Kinderlehrer, D. et al. (Berlin: Springer, 1992).Google Scholar
8Šverák, V.. On Tartar's conjecture. Ann. Inst. H. Poincaré, Anal non linèaire 10 (1993), 405–12.CrossRefGoogle Scholar
9Wells, J. H. and Williams, L. R.. Embeddings and extensions in analysis (Berlin: Springer, 1975).CrossRefGoogle Scholar
10Yan, B.-S.. Remarks on the set of quasi-confonnal matrices in higher dimensions (Preprint, 1994).Google Scholar
11Zhang, K.-W.. A construction of quasiconvex functions with linear growth at infinity. Ann. Scuola Norm. Sup. Pisa Ser. IV 19 (1992), 313–26.Google Scholar
12Zhang, K.-W.. On connected subsets of M2×2 without rank-one connections (Preprint).Google Scholar