Hostname: page-component-7dd5485656-jtdwj Total loading time: 0 Render date: 2025-10-30T12:40:25.265Z Has data issue: false hasContentIssue false
  • English
  • Français

Lower semicontinuity of surface energies

Published online by Cambridge University Press:  14 November 2011

Irene Fonseca
Irene Fonseca
Affiliation:
Department of Mathematics, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

Using the theory of indicator measures, a lower semicontinuity result for quasiconvex functions in W1,1 and assuming only L1 convergence is obtained.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 120 , Issue 1-2 , 1992 , pp. 99 - 115
Copyright
Copyright © Royal Society of Edinburgh 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1Acerbi, E. and Fusco, N.. Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86 (1984), 125–145.CrossRefGoogle Scholar
2Allard, W.. On the first variation of a varifold. Ann. of Math. 95 (1972), 417–491.CrossRefGoogle Scholar
3Almgren, F. J. Jr, Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure. Ann. of Math. 87 (1968), 321–391.CrossRefGoogle Scholar
4Ball, J. M. and James, R.. Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal. 100 (1987), 13–52.CrossRefGoogle Scholar
5Ball, J. M. and Murat, F.. Wl,p quasiconvexity and variational problems for multiple integrals. J. Fund. Anal. 58 (1984), 225–253.CrossRefGoogle Scholar
6Besicovitch, A. S.. A general form of the covering principle and relative differentiation of additive functions. Proc. Cambridge Philos. Soc. 41 (1945), 103–110.CrossRefGoogle Scholar
7Chipot, M. and Kinderlehrer, D.. Equilibrium configurations of crystals. Arch. Rational Mech. Anal. 103 (1988), 237–277.CrossRefGoogle Scholar
8Dacorogna, B.. A relaxation theorem and its applications to the equilibrium of gases. Arch. Rational Mech. Anal. 77 (1981), 359–386.CrossRefGoogle Scholar
9Dacorogna, B.. Direct Methods in the Calculus of Variations (Berlin: Springer, 1989).CrossRefGoogle Scholar
10De Giorgi, E.. Convergence problems for functionals and operators. In Proc. Int. Meeting on Recent Methods in Nonlinear Analysis, eds De Giorgi et, E. al., pp. 223–244 (Bologna: Pitagora, 1979).Google Scholar
11Ericksen, J. L.. Special topics in elastostatics. Adv. in Appl. Mech., ed. C.-S. Yih, 17 (1977), 189–243.CrossRefGoogle Scholar
12Ericksen, J. L.. Twinning of crystals I, IMA Vol. Math. Appl. 3, eds. Antman, S., Ericksen, J. L., Kinderlehrer, D. and Müller, I., 77–94 (Berlin: Springer, 1987).Google Scholar
13Evans, L. C.. Weak convergence methods for nonlinear partial differential equations (to appear).Google Scholar
14Evans, L. C. and Gariepy, R. F.. Lecture Notes on Measure Theory and Fine Properties of Functions (Kentucky EPSCoR Preprint Series).Google Scholar
15Fonseca, I.. Variational methods for elastic crystals. Arch. Rational Mech. Anal. 97 (1987), 189–220.CrossRefGoogle Scholar
16Fonseca, I.. Interfacial energy and the Maxwell rule. Arch. Rational Mech. Anal. 106 (1989), 63–95.CrossRefGoogle Scholar
17Fonseca, I.. Phase transitions of elastic solid materials. Arch. Rational Mech. Anal. 107 (1989), 195–223.CrossRefGoogle Scholar
18Giusti, E.. Minimal Surfaces and Functions of Bounded Variation (Basel: Birkhauser, 1984).CrossRefGoogle Scholar
19Goffman, C. and Serrin, J.. Sublinear functions of measures and variational integrals. Duke Math. J. 31 (1964), 159–178.CrossRefGoogle Scholar
20Gurtin, M. E.. On phase transitions with bulk, interfacial, and boundary energy. Arch. Rational Mech. Anal. 96 (1986), 243–264.CrossRefGoogle Scholar
21Herring, C.. Some theorems on the free energies of crystal surfaces. Phys. Rev. 82 (1951), 87–93.CrossRefGoogle Scholar
22James, R. and Kinderlehrer, D.. Theory of diffusionless phase transitions (IMA Preprint Series #492, 1989).CrossRefGoogle Scholar
23Kinderlehrer, D.. Twinning of crystals II. IMA Vol. Math. Appl. 3, eds Antman, S., Ericksen, J. L., Kinderlehrer, D. and Miiller, I., 185–211 (Berlin: Springer 1987).Google Scholar
24Kinderlehrer, D. and Pedregal, P.. Analysis of Young measures (to appear).Google Scholar
25Kinderlehrer, D. and Vergara-Caffarelli, G.. The relaxation of functionals with surface energies Asym. Anal. 2 (1989), 279–298.Google Scholar
26Marcellini, P.. Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals. Manuscripta Math. 51 (1985), 1–28.CrossRefGoogle Scholar
27Morrey, C. B.. Multiple Integrals in the Calculus of Variations (Berlin: Springer, 1986).Google Scholar
28Reshetnyak, Yu. G.. Weak convergence of completely additive vector functions on a set. Siberian Math. J. 9 (1968), 1039–1045 (translation of: Sibirsk. Mat. Z. 9 (1968), 1386–1394).CrossRefGoogle Scholar
29Sverak, V.. Ouasiconvex functions with subquadratic growth (manuscript, 1990).Google Scholar
30Tartar, L.. Compensated compactness and applications to partial differential equations. In Nonlinear analysis and mechanics: Heriot-Watt Symposium, vol. IV, Research Notes in Mathematics 39, 136–212 (New York: Pitman, 1979).Google Scholar
31Young, L. C.. Generalized surfaces in the calculus of variations I, II. Ann. of Math. 43 (1952), 84–103; 530–544.CrossRefGoogle Scholar