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.
Mots-clés : quasiconvex functions, quasiconvex hull, homogeneous Young measure, quasiconvex exposed points, Straszewicz theorem
@article{COCV_2001__6__1_0,
author = {Zhang, Kewei},
title = {On the quasiconvex exposed points},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1--19},
publisher = {EDP-Sciences},
volume = {6},
year = {2001},
mrnumber = {1804495},
zbl = {0970.49013},
language = {en},
url = {http://www.numdam.org/item/COCV_2001__6__1_0/}
}
Zhang, Kewei. On the quasiconvex exposed points. ESAIM: Control, Optimisation and Calculus of Variations, Tome 6 (2001), pp. 1-19. http://www.numdam.org/item/COCV_2001__6__1_0/
[1] , Compact Convex Sets and Boundary Integrals. Springer-Verlag (1971). | MR | Zbl
[2] and, Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86 (1984) 125-145. | Zbl
[3] and, Intégrandes normales et mesures paramétrées en calcul des variations. Bull. Soc. Math. France 101 (1973) 129-184. | EuDML | Numdam | Zbl
[4] , Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977) 337-403. | Zbl
[5] , 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. | Zbl
[6] , Sets of gradients with no rank-one connections. J. Math. Pures Appl. 69 (1990) 241-259. | Zbl
[7] ,, and, Restrictions on Microstructures. Proc. Roy. Soc. Edinburgh Sect. A 124 (1994) 843-878. | Zbl
[8] and, Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal. 100 (1987) 13-52. | Zbl
[9] and, 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. | Zbl
[10] and, Lower semicontinuity and multiple integrals and the biting lemma. Proc. Roy. Soc. Edinburgh Sect. A 114 (1990) 367-379. | Zbl
[11] and, Equilibrium configurations of crystals. Arch. Rational Mech. Anal. 103 (1988) 237-277. | Zbl
[12] , Direct Methods in the Calculus of Variations. Springer-Verlag (1989). | MR | Zbl
[13] and, 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. | Zbl
[14] and, 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. | Zbl
[15] and, General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial case. Acta Math. 178 (1997) 1-37. | Zbl
[16] and, Cauchy-Dirichlet problem for first order nonlinear systems. J. Funct. Anal. 152 (1998) 404-446. | Zbl
[17] and, Implicit second order partial differential equations. Ann. Scuola. Norm. Sup. Pisa Cl. Sci. (4) 25 (1997) 299-328. | Numdam | Zbl
[18] , General Topology. van Nostrand (1955). | MR | Zbl
[19] and, Characterizations of Young measures generated by gradients. Arch. Rational Mech. Anal 115 (1991) 329-365. | Zbl
[20] , The relaxation of a double well energy. Cont. Mech. Therm. 3 (1991) 981-1000. | Zbl
[21] , Convex Sets and Their Applications. John Wiley & Sons (1982). | MR | Zbl
[22] , Multiple integrals in the calculus of variations. Springer (1966). | Zbl
[23] and, Attainment results for the two-well problem by convex integration. Preprint (1993). | MR | Zbl
[24] , Liouville's theorem on conformal mappings under minimal regularity assumptions. Siberian Math. J. 8 (1967) 631-653. | Zbl
[25] , Convex Analysis. Princeton University Press (1970). | MR | Zbl
[26] , Functional Analysis. McGraw-Hill (1973). | MR | Zbl
[27] , On regularity for the Monge-Ampère equations. Preprint.
[28] , New examples of quasiconvex functions. Arch. Rational Mech. Anal. 119 (1992) 293-330. | Zbl
[29] , 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. | Zbl
[30] , On Tartar's conjecture. Ann. Inst. H. Poincaré 10 (1993) 405-412. | Numdam | Zbl
[31] , Compensated compactness and applications to partial differential equations, in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, IV, edited by R.J. Knops. Pitman (1979). | MR | Zbl
[32] , A construction of quasiconvex functions with linear growth at infinity. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) XIX (1992) 313-326. | Numdam | Zbl
[33] , On connected subsets of without rank-one connections. Proc. Roy. Soc. Edinburgh Sect. A 127 (1997) 207-216. | Zbl
[34] , On various semiconvex hulls in the calculus of variations. Calc. Var. Partial Differential Equations 6 (1998) 143-160. | Zbl
[35] , On the structure of quasiconvex hulls. Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998) 663-686. | Numdam | Zbl
