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On the quasiconvex exposed points

Published online by Cambridge University Press:  15 August 2002

Kewei Zhang
Kewei Zhang*
Affiliation:
School of Mathematical Sciences, University of Sussex, Falmer, Brighton BN1 9QH, U.K.; [email protected].
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Abstract

The notion of quasiconvex exposed points is introduced for compact sets of matrices, motivated from the variational approach to material microstructures. We apply the notion to give geometric descriptions of the quasiconvex extreme points for a compact set. A weak version of Straszewicz type density theorem in convex analysis is established for quasiconvex extreme points. Some examples are examined by using known explicit quasiconvex functions.

Keywords

Quasiconvex functionsquasiconvex hullhomogeneous Young measurequasiconvex exposed pointsStraszewicz theorem.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 6 , 2001 , pp. 1 - 19
Copyright
© EDP Sciences, SMAI, 2001

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References

E.M. Alfsen, Compact Convex Sets and Boundary Integrals. Springer-Verlag (1971).
Acerbi, E. and Fusco, N., Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86 (1984) 125-145. CrossRef
Berliocchi, H. and Lasry, J.M., Intégrandes normales et mesures paramétrées en calcul des variations. Bull. Soc. Math. France 101 (1973) 129-184. CrossRef
Ball, J.M., Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977) 337-403. CrossRef
J.M. Ball, A version of the fundamental theorem of Young measures, in Partial Differential Equations and Continuum Models of Phase Transitions, edited by M. Rascle, D. Serre and M. Slemrod. Springer-Verlag (1989) 207-215.
Ball, J.M., Sets of gradients with no rank-one connections. J. Math. Pures Appl. 69 (1990) 241-259.
Bhattacharya, K., Firoozye, N.B., James, R.D. and Kohn, R.V., Restrictions on Microstructures. Proc. Roy. Soc. Edinburgh Sect. A 124 (1994) 843-878. CrossRef
Ball, J.M. and James, R.D., Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal. 100 (1987) 13-52. CrossRef
Ball, J.M. and James, R.D., Proposed experimental tests of a theory of fine microstructures and the two-well problem. Philos. Trans. Roy. Soc. London Ser. A 338 (1992) 389-450. CrossRef
Ball, J.M. and Zhang, K.-W., Lower semicontinuity and multiple integrals and the biting lemma. Proc. Roy. Soc. Edinburgh Sect. A 114 (1990) 367-379. CrossRef
Chipot, M. and Kinderlehrer, D., Equilibrium configurations of crystals. Arch. Rational Mech. Anal. 103 (1988) 237-277. CrossRef
B. Dacorogna, Direct Methods in the Calculus of Variations. Springer-Verlag (1989).
Dacorogna, B. and Marcellini, P., Théorème d'existence dans le cas scalaire et vectoriel pour les équations de Hamilton-Jacobi. C. R. Acad. Sci. Paris Sér. I Math. 322 (1996) 237-240.
Dacorogna, B. and Marcellini, P., Sur le problème de Cauchy-Dirichlet pour les systèmes d'équations non linéaires du premier ordre. C. R. Acad. Sci. Paris Sér. I Math. 323 (1996) 599-602.
Dacorogna, B. and Marcellini, P., General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial case. Acta Math. 178 (1997) 1-37. CrossRef
Dacorogna, B. and Marcellini, P., Cauchy-Dirichlet problem for first order nonlinear systems. J. Funct. Anal. 152 (1998) 404-446. CrossRef
Dacorogna, B. and Marcellini, P., Implicit second order partial differential equations. Ann. Scuola. Norm. Sup. Pisa Cl. Sci. (4) 25 (1997) 299-328.
J.L. Kelly, General Topology. van Nostrand (1955).
Kinderlehrer, D. and Pedregal, P., Characterizations of Young measures generated by gradients. Arch. Rational Mech. Anal 115 (1991) 329-365. CrossRef
Kohn, R.V., The relaxation of a double well energy. Cont. Mech. Therm. 3 (1991) 981-1000. CrossRef
S.R. Lay, Convex Sets and Their Applications. John Wiley & Sons (1982).
C.B. Morrey Jr., Multiple integrals in the calculus of variations. Springer (1966).
S. Müller and V. Sverák, Attainment results for the two-well problem by convex integration. Preprint (1993).
Reshetnak, Yu.G., Liouville's theorem on conformal mappings under minimal regularity assumptions. Siberian Math. J. 8 (1967) 631-653. CrossRef
R.T. Rockafellar, Convex Analysis. Princeton University Press (1970).
W. Rudin, Functional Analysis. McGraw-Hill (1973).
V. Sverák, On regularity for the Monge-Ampère equations. Preprint.
Sverák, V., New examples of quasiconvex functions. Arch. Rational Mech. Anal. 119 (1992) 293-330. CrossRef
V. Sverák, On the problem of two wells, in Microstructure and phase transitions, edited by D. Kinderlehrer, R.D. James, M. Luskin and J. Ericksen. Springer, IMA J. Appl. Math. 54 (1993) 183-189.
Sverák, V., Tartar's, On conjecture. Ann. Inst. H. Poincaré 10 (1993) 405-412. CrossRef
L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, IV, edited by R.J. Knops. Pitman (1979).
K.-W. Zhang, A construction of quasiconvex functions with linear growth at infinity. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) XIX (1992) 313-326.
Zhang, K.-W., On connected subsets of $M^{2\times 2}$ without rank-one connections. Proc. Roy. Soc. Edinburgh Sect. A 127 (1997) 207-216. CrossRef
Zhang, K.-W., On various semiconvex hulls in the calculus of variations. Calc. Var. Partial Differential Equations 6 (1998) 143-160. CrossRef
Zhang, K.-W., On the structure of quasiconvex hulls. Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998) 663-686. CrossRef