# On the quasiconvex exposed points

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 6, page 1-19
- ISSN: 1292-8119

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topZhang, Kewei. "On the quasiconvex exposed points." ESAIM: Control, Optimisation and Calculus of Variations 6 (2010): 1-19. <http://eudml.org/doc/197361>.

@article{Zhang2010,

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.
},

author = {Zhang, Kewei},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Quasiconvex functions; quasiconvex hull; homogeneous Young measure; quasiconvex exposed
points; Straszewicz theorem.; quasiconvexity; extreme points; material microstructures},

language = {eng},

month = {3},

pages = {1-19},

publisher = {EDP Sciences},

title = {On the quasiconvex exposed points},

url = {http://eudml.org/doc/197361},

volume = {6},

year = {2010},

}

TY - JOUR

AU - Zhang, Kewei

TI - On the quasiconvex exposed points

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 6

SP - 1

EP - 19

AB -
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.

LA - eng

KW - Quasiconvex functions; quasiconvex hull; homogeneous Young measure; quasiconvex exposed
points; Straszewicz theorem.; quasiconvexity; extreme points; material microstructures

UR - http://eudml.org/doc/197361

ER -

## References

top- E.M. Alfsen, Compact Convex Sets and Boundary Integrals. Springer-Verlag (1971).
- E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal.86 (1984) 125-145.
- H. Berliocchi and J.M. Lasry, Intégrandes normales et mesures paramétrées en calcul des variations. Bull. Soc. Math. France101 (1973) 129-184.
- J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal.63 (1977) 337-403.
- 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.
- J.M. Ball, Sets of gradients with no rank-one connections. J. Math. Pures Appl.69 (1990) 241-259.
- K. Bhattacharya, N.B. Firoozye, R.D. James and R.V. Kohn, Restrictions on Microstructures. Proc. Roy. Soc. Edinburgh Sect. A124 (1994) 843-878.
- J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal.100 (1987) 13-52.
- J.M. Ball and R.D. James, Proposed experimental tests of a theory of fine microstructures and the two-well problem. Philos. Trans. Roy. Soc. London Ser. A338 (1992) 389-450.
- J.M. Ball and K.-W. Zhang, Lower semicontinuity and multiple integrals and the biting lemma. Proc. Roy. Soc. Edinburgh Sect. A114 (1990) 367-379.
- M. Chipot and D. Kinderlehrer, Equilibrium configurations of crystals. Arch. Rational Mech. Anal.103 (1988) 237-277.
- B. Dacorogna, Direct Methods in the Calculus of Variations. Springer-Verlag (1989).
- B. Dacorogna and P. Marcellini, 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.
- B. Dacorogna and P. Marcellini, 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.
- B. Dacorogna and P. Marcellini, General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial case. Acta Math.178 (1997) 1-37.
- B. Dacorogna and P. Marcellini, Cauchy-Dirichlet problem for first order nonlinear systems. J. Funct. Anal.152 (1998) 404-446.
- B. Dacorogna and P. Marcellini, 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).
- D. Kinderlehrer and P. Pedregal, Characterizations of Young measures generated by gradients. Arch. Rational Mech. Anal115 (1991) 329-365.
- R.V. Kohn, The relaxation of a double well energy. Cont. Mech. Therm.3 (1991) 981-1000.
- 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).
- Yu.G. Reshetnak, Liouville's theorem on conformal mappings under minimal regularity assumptions. Siberian Math. J.8 (1967) 631-653.
- 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.
- V. Sverák, New examples of quasiconvex functions. Arch. Rational Mech. Anal.119 (1992) 293-330.
- 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.
- V. Sverák, On Tartar's conjecture. Ann. Inst. H. Poincaré10 (1993) 405-412.
- 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.
- K.-W. Zhang, On connected subsets of ${M}^{2\times 2}$ without rank-one connections. Proc. Roy. Soc. Edinburgh Sect. A127 (1997) 207-216.
- K.-W. Zhang, On various semiconvex hulls in the calculus of variations. Calc. Var. Partial Differential Equations6 (1998) 143-160.
- K.-W. Zhang, On the structure of quasiconvex hulls. Ann. Inst. H. Poincaré Anal. Non Linéaire15 (1998) 663-686.

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