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On rank-one convex and polyconvex conformal energy functions with slow growth

Published online by Cambridge University Press:  14 November 2011

Baisheng Yan
Baisheng Yan
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824, U.S.A., e-mail: [email protected]
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Abstract

We make some remarks about rank-one convex and polyconvex functions on the set of all real n × n matrices that vanish on the subset Kn consisting of all conformal matrices and grow like a power function at infinity. We prove that every non-negative rank-one convex function that vanishes on Kn and grows below a power of degree n/2 must vanish identically. In odd dimensions n ≧ 3, we prove that every non-negative polyconvex function that vanishes on Kn must vanish identically if it grows below a power of degree n; while in even dimensions, such polyconvex functions can exist that also grow like a power of half-dimension degree.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 3 , 1997 , pp. 651 - 663
Copyright
Copyright © Royal Society of Edinburgh 1997

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References

1Acerbi, E. and Fusco, N.. Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86(1984), 125–45.CrossRefGoogle Scholar
2Ball, J. M.. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63(1977), 337–403.CrossRefGoogle Scholar
3Ball, J. M.. Sets of gradients with no rank-one connections. J. Math. Pures Appl. 69 (1990), 241–59.Google Scholar
4Ball, J. M. and James, R. D.. Proposed experimental tests of a theory of fine microstructures and the two well problems. Philos. Trans. Roy. Soc. London. Ser. A 338 (1992), 389–450.CrossRefGoogle Scholar
5Ball, J. M. and Murat, F.. W 1,p-Quasiconvexity and variational problems for multiple integrals. J. Fund. Anal. 58 (1984), 225–53.CrossRefGoogle Scholar
6Bhattacharya, K., Firoozye, N., James, R. and Kohn, R.. Restrictions on microstructures. Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), 843–78.CrossRefGoogle Scholar
7Chipot, M. and Kinderlehrer, D.. Equilibrium configurations of crystals. Arch. Rational Mech. Anal. 103 (1988), 237–77.CrossRefGoogle Scholar
8Dacorogna, B.. Direct Methods in the Calculus of Variations (Berlin: Springer, 1989).CrossRefGoogle Scholar
9DiPerna, R. J.. Compensated compactness and general systems of conservation laws. Trans. Amer. Math. Soc. 292 (1985), 383–420.CrossRefGoogle Scholar
10Evans, L. C.. Quasiconvexity and partial regularity in the calculus of variations. Arch. Rational Mech. Anal. 95(1986), 227–52.CrossRefGoogle Scholar
11Fonseca, I.. Variational methods for elastic crystals. Arch. Rational Mech. Anal. 97 (1987), 189–220.CrossRefGoogle Scholar
12Giaquinta, M.. Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems (Princeton: Princeton University Press, 1983).Google Scholar
13Iwaniec, T., p-Harmonic tensors and quasiregular mappings. Ann. Math. 136 (1992), 589–624.CrossRefGoogle Scholar
14Iwaniec, T. and Lutoborski, A.. Integral estimates for null Lagrangians. Arch. Rational Mech. Anal. 125(1993), 25–79.CrossRefGoogle Scholar
15Iwaniec, T. and Lutoborski, A.. Polyconvex functionals for nearly conformal deformations. SIAM J. Math. Anal. 27 (1996), 609–19.CrossRefGoogle Scholar
16Iwaniec, T. and Martin, G.. Quasiregular mappings in even dimensions. Acta Math. 170 (1993), 29–81.CrossRefGoogle Scholar
17Kinderlehrer, D.. Remarks about equilibrium configurations of crystals. In Material Instabilities in Continuum Mechanics, ed. Ball, J. M. (Oxford: Oxford University Press, 1988).Google Scholar
18Kinderlehrer, D. and Pedregal, P.. Gradient Young measures generated by sequences in Sobolev spaces. J. Geom. Anal. 4 (1994), 59–90.CrossRefGoogle Scholar
19Kohn, R. V.. The relaxation of a double-well energy. Contin. Mech. Thermodyn. 3 (1991), 193–236.CrossRefGoogle Scholar
20Morrey, C. B.. Multiple Integrals in the Calculus of Variations (Berlin: Springer, 1966).CrossRefGoogle Scholar
21Müller, S.. On quasiconvex functions which are homogeneous of degree 1. Indiana Univ. Math. J. 41 (1992), 295–301.CrossRefGoogle Scholar
22Milller, S. and Šverák, V.. Attainment results for the two well problem by convex integration (preprint).Google Scholar
23Müler, S., Šverák, V. and Yan, B.. Sharp stability results for almost conformal maps in even dimensions. J. Geom. Anal, (to appear).Google Scholar
24Reshetnyak, Yu. G.. Stability Theorems in Geometry and Analysis (Dordrecht: Kluwer, 1994).CrossRefGoogle Scholar
25Šverák, V.. On regularity for the Monge–Ampère equation without convexity assumptions (preprint, 1992).Google Scholar
26Šverák, V.. Rank-one convexity does not imply quasiconvexity. Proc. Roy. Soc. Edinburgh Sect. A 120(1992), 185–9.CrossRefGoogle Scholar
27Šverák, V.. On the problem of two wells. In Microstructure and Phase Transition, eds. Kinderlehrer, D.et al., 183–90 (Berlin: Springer, 1993).CrossRefGoogle Scholar
28Šverák, V.. Lower semicontinuity for variational integral functionals and compensated compactness. In Proceedings of the International Congress of Mathematicians, Zürich, 1994, 1153–8 (Basel: Birkhäuser, 1995).CrossRefGoogle Scholar
29Tartar, L.. The compensated compactness method applied to systems of conservation laws. In Systems of Nonlinear Partial Differential Equations, ed. Ball, J. M., NATO ASI Series 103 (Dordrecht: D. Reidel, 1983).Google Scholar
30Yan, B.. On quasiconvexs hulls of sets of matrices and strong convergence of certain minimizing sequences (preprint, 1993).Google Scholar
31Yan, B.. Remarks about W 1.p-stability of the conformal set in higher dimensions. Ann. Inst. H. Poincaré, Anal. Non Linéaire 13 (1996), 691–705.CrossRefGoogle Scholar
32Zhang, K.. A construction of quasiconvex functions with linear growth at infinity. Ann. Scuola Norm. Sup. Pisa 19 (1992), 313–26.Google Scholar