Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-27T10:39:50.670Z Has data issue: false hasContentIssue false
  • English
  • Français

Anisotropic singular perturbations—the vectorial case

Published online by Cambridge University Press:  14 November 2011

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

Abstract

We obtain the Γ(L1(Ώ))-limit of the sequence

where Eε is the family of anisotropic perturbations

of the nonconvex functional of vector-valued functions

The proof relies on the blow-up argument introduced by Fonseca and Müller.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 3 , 1994 , pp. 527 - 571
Copyright
Copyright © Royal Society of Edinburgh 1994

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.)

References

1Ambrosio, L. and dal Maso, G.. On the relaxation in BV(Ω;ℝm) of quasi-convex integrals J. Fund. Anal. 109 (1992), 76–97.CrossRefGoogle Scholar
2Attouch, H.. Variational Convergence for Functions and Operators (London: Pitman, 1984).Google Scholar
3Baldo, S.. Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids. Ann. Inst. H. Poincaré- Anal. Non Linéaire 7 (1990), 37–65.CrossRefGoogle Scholar
4Bouchitté, G.. Singular perturbations of variational problems arising from a two-phase transition model. Appl. Math. Optim. 21 (1990), 289–314.Google Scholar
5Dacorogna, B.. Direct Methods in the Calculus of Variations (Berlin: Springer, 1989).Google Scholar
6De Giorgi, E.. Sulla convergenza di alcune successioni d'integrali del tipo dell'area. Rend. Mat. 8 (1975), 277–294.Google Scholar
7De Giorgi, E. and dal Maso, G.. Γ-convergence and the calculus of variations. In Mathematical Theories of Optimization, eds Cecconi, J. P. and Zolezzi, T., Springer Lecture Notes 979, 121–193 (Berlin: Springer, 1983).Google Scholar
8Dal Maso, G.. Integral representation on BV(Ω) of Γ-limits of variational integrals. Manuscripta Math. 30(1980), 387–416.Google Scholar
9Dal Maso, G. and Modica, L.. Nonlinear stochastic homogenization. Ann. Mat. Pura. Appl. (4) 144 (1986), 347–389.Google Scholar
10Evans, L. C. and Gariepy, R. F.. Lecture Notes on Measure Theory and Fine Properties of Functions (CRC Press, 1992).Google Scholar
11Federer, H.. Geometric Measure Theory (Berlin: Springer, 1969).Google Scholar
12Fonseca, I.. The Wulff theorem revisited. Proc. Roy. Soc. London Ser. A 432 (1991), 125–145.Google Scholar
13Fonseca, I.. Lower semicontinuity of surface energies. Proc. Roy. Soc. Edinburgh Sect. A 120 (1992), 99–115.Google Scholar
14Fonseca, I. and Müller, S.. Quasiconvex integrands and lower semiconductivity in L1. SIAM J. Math. Anal. 23 (1992), 1081–1098.Google Scholar
15Fonseca, I. and Müller, S.. Relaxation of quasiconvex functionals in BV(Ω;ℝp) Arch. Rat. Mech. Anal 123(1993), 1–49.CrossRefGoogle Scholar
16Fonseca, I. and Müller, S.. A uniqueness proof for the Wulff Theorem. Proc. Roy. Soc. Edinburgh Sect. A 119 (1991), 125–136.Google Scholar
17Fonseca, I. and Rybka, P.. Relaxation of multiple integrals in the space BV(Q; Rp). Proc. Roy. Soc. Edinburgh Sect. A 121 (1992), 321–48.Google Scholar
18Fonseca, I. and Tartar, L.. The gradient theory of phase transitions for systems with two potential wells. Proc. Roy. Soc. Edinburgh Sect. A 111 (1989), 89–102.CrossRefGoogle Scholar
19Giusti, E.. Minimal Surfaces and Functions of Bounded Variation (Boston: Birkhäuser, 1984).Google Scholar
20Goffman, C. and Serrin, J.. Sublinear functions of measures and variational integrals. Duke Math. J. 31 (1964). 159–178.CrossRefGoogle Scholar
21Gurtin, M. E.. Some remarks concerning phase transitions in RN (Department of Mathematics, Carnegie Mellon University, 1983).Google Scholar
22Gurtin, M. E.. Some results and conjectures in the gradient theory of phase transitions. In Metastability and Incompletely Posed Problems, eds Antman, S., Ericksen, J. L., Kinderlehrer, D. and Muller, I., 135–146 (Berlin: Springer, 1987).Google Scholar
23Kohn, R. and Sternberg, P.. Local minimizers and singular perturbations. Proc. Roy. Soc. Edinburgh Sect. A 111 (1989), 69.CrossRefGoogle Scholar
24Modica, L.. Gradient theory of phase transitions and minimal interface criterion. Arch. Rational Mech. Anal. 98 (1987), 123–142.Google Scholar
25Muller, S.. On quasiconvex functions which are homogeneous of degree one. Indiana Univ. Math. J. (to appear).Google Scholar
26Owen, N. C. and Sternberg, P.. Nonconvex variational problems with anisotropic perturbations. Nonlinear Anal. 78 (1991), 705–719.Google Scholar
27Reshetnyak, Y. 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).Google Scholar
28Stein, E. M.. Singular Integrals and Differentiability Properties of Functions (Princeton: Princeton University Press, 1970).Google Scholar
29Sternberg, P.. The effect of a singular perturbation on nonconvex variational problems. Arch. Rational Mech. Anal. 101 (1988), 209–260.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 (London: Pitman, 1979).Google Scholar
31Taylor, J.. Existence and structure of solutions to a class of nonelliptic variational problems. Sympos. Math. 14(1974), 499–508.Google Scholar
32Taylor, J.. Unique structure of solutions to a class of nonelliptic variational problems. Proc. Sympos. Pure Math. 27 (1975), 419–427.Google Scholar
33Wulff, G.. Zur Frage der Geschwindigkeit des Wachsturms und der Auflösung der Kristallflächen. Z. Krist. 34 (1901), 449–530.Google Scholar
34Ziemer, W.. Weakly Differentiable Functions (Berlin: Springer, 1989).Google Scholar