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On various semiconvex relaxations of the squared-distance function

Published online by Cambridge University Press:  14 November 2011

K. Zhang
K. Zhang
Affiliation:
Department of Mathematics, Macquarie University, Sydney 2109, Australia (kewei@maths. ics.mq.edu.au)
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Extract

For the Euclidean squared-distance function f(·) = dist2(·, K), with K ⊂ MN×n, we show that K is convex if and only if f(·) equals either its rank-one convex, quasiconvex or polyconvex relaxations. We also establish that if (i) K is compact and contractible or (ii) dim C(K) = k < Nn, K is convex if and only if f equals one of the semiconvex relaxations when dist2(P, K) is sufficiently large, and for case (i), P ∈MNxn; for case (ii), P ∈ Ek—a k-dimensional plane containing C(K). We also give some estimates of the difference between dist2(P, K) and its semiconvex relaxations. Some possible extensions to more general p-distance functions are also considered.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 6 , 1999 , pp. 1309 - 1323
Copyright
Copyright © Royal Society of Edinburgh 1999

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References

1Acerbi, E. and Fusco, N.. Semicontinuity problems in the calculus of variations. Arch. Ration. Mech. Analysis 86 (1984), 125–145.CrossRefGoogle Scholar
2Ball, J. M.. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Analysis 63 (1977), 337–403.CrossRefGoogle Scholar
3Ball, J. M. and James, R. D.. Fine phase mixtures as minimizers of energy. Arch. Ration. Mech. Analysis 100 (1987), 13–52.Google Scholar
4Ball, J. M. and James, R. D.Proposed experimental tests of a theory of fine microstructures and the two-well problem. Phil. Trans. R. Soc. Lond. A 338 (1992), 389–450.CrossRefGoogle Scholar
5Ball, J. M. and Zhang, K.-W.. Lower semicontinuity of multiple integrals and the biting lemma. Proc. R. Soc. Edmb. A 114 (1990), 367–379.CrossRefGoogle Scholar
6Bhattacharya, K., Firoozy, N. B., James, R. D. and fCohn, R. V.Restrictions on microstructures. Proc. R. Soc. Edinb. A 124 (1994), 843–878.CrossRefGoogle Scholar
7Chipot, M. and Kinderlehrer, D.. Equilibrium configurations of crystals. Arch. Ration. Mech. Analysis 103 (1988), 237–277.CrossRefGoogle Scholar
8Dacorogna, B.. Weak continuity and weak lower semicontinuity of nonlinear functionals. Lectures Notes in Mathematics, vol. 922 (Springer, 1980).Google Scholar
9Dacorogna, B.. Direct methods in the calculus of variations (Springer, 1989).CrossRefGoogle Scholar
10Fonseca, I.. The lower quasiconvex envelope of the stored energy function for an elastic crystal. J. Math. Pures Appl. 67 (1988), 175–195.Google Scholar
11Kinderlehrer, D. and Pedregal, P.. Characterizations of Young measures generated by gradients. Arch. Ration. Mech. Analysis 115 (1991), 329–365.CrossRefGoogle Scholar
12Kohn, R. V.. The relaxation of a double-well energy. Cont. Mech. Therm. 3 (1991), 981–1000.CrossRefGoogle Scholar
13Kohn, R. V. and Strang, D.. Optimal design and relaxation of variational problems. I, II, III. Commun. Pure Appl. Math. 39 (1986), 113–137, 139–182, 353–377.CrossRefGoogle Scholar
14Kristensen, J.. On the non-locality of quasiconvexity. Ann. Inst. H. Poincaré Analyse Non Linéare 16 (1999), 1–13.CrossRefGoogle Scholar
15Lay, S. R.. Convex sets and their applications (Wiley, 1982).Google Scholar
16Dret, H. Le and Raoult, A.. Enveloppe quasi-convexe de la densite d'énergie de Saint Venant-Kirchhoff. C. R. Acad. Sci. Paris, Série I 318 (1994), 93–98.Google Scholar
17Mayer, J.. Algebraic topology (Englewood Cliffs, NJ: Prentice-Hall, 1972).Google Scholar
18Morrey, C. B. Jr.Multiple integrals in the calculus of variations (Springer, 1966).Google Scholar
19Rockafellar, R. T.. Convex analysis (Princeton, NJ: Princeton University Press, 1970).CrossRefGoogle Scholar
20Šverák, V.. On the problem of two wells. In Micro structure and phase transition (ed. Kinderlehrer, D. and others) (Springer, 1992).Google Scholar
21Šverák, V.. Rank one convexity does not imply quasiconvexity. Proc. R. Soc. Edinb. A 120 (1992), 185–189.CrossRefGoogle Scholar
22Zhang, K.-W.. A construction of quasiconvex functions with linear growth at infinity. Ann. Sci. Norm. Sup. Pisa Serie IV 19 (1992), 313–326.Google Scholar
23Zhang, K.-W.. On non-negative quasiconvex functions with unbounded zero sets. Proc. R. Soc. Edinb. A 127 (1997), 411–422.Google Scholar
24Zhang, K.-W.. Quasiconvex functions, SO(n) and two elastic wells. Ann. Inst. H. Poincare Analyse Non Linéare 14 (1997), 759–785.CrossRefGoogle Scholar
25Zhang, K.-W.. On some quasiconvex functions with linear growth. J. Convex Analysis 5 (1998), 133–146.Google Scholar