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Relaxation of multiple integrals in the space BV(Ω, RP)

Published online by Cambridge University Press:  14 November 2011

Irene Fonseca  and
Piotr Rybka
Irene Fonseca
Affiliation:
Department of Mathematics, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A.
Piotr Rybka
Affiliation:
Department of Mathematics, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A.
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Synopsis

A characterisation of the surface energy density for the relaxation in V(Ω; Rp) of the functional

is obtained. A lemma of De Giorgi is used to modify a sequence near the boundary without increasing its total energy.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 121 , Issue 3-4 , 1992 , pp. 321 - 348
Copyright
Copyright © Royal Society of Edinburgh 1992

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References

1Ambrosio, L., Mortola, S. and Tortorelli, V. M.. Functional with linear growth defined on vector valued BV functions (to appear).Google Scholar
2Ambrosio, L. and Pallara, D.. Integral representation of relaxed functionals on BV(Rn, Rk) and polyhedral approximation (to appear).Google Scholar
3Aviles, P. and Giga, Y.. Variational integrals on mappings of bounded variation and their lower semicontinuity (to appear).Google Scholar
4Baldo, S.. Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids. Ann. Inst. H. Poincaré 7 (1990), 67–90.CrossRefGoogle Scholar
5Ball, J. M. and James, R. D.. Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal. 100 (1988), 13–52.CrossRefGoogle Scholar
6Bouchitte, G.. Singular perturbations of variational problems arising from a two phases transition model. Appl. Math. Optim. 21 (1990), 289–314.CrossRefGoogle Scholar
7Cahn, J. W.. Surface stress and the chemical equilibrium of small crystals. 1. The case of isotropic surface. Act. Metall. 28 (1980), 1333–1338.CrossRefGoogle Scholar
8Carr, J., Gurtin, M. E. and Slemrod, M.. Structured phase transitions on a finite interval. Arch. Rational Mech. Anal. 86 (1984), 317–351.CrossRefGoogle Scholar
9Clarke, F. H. and Vinter, R. B.. Regularity properties of solutions to the basic problems in the calculus of variations. Trans. Amer. Math. Soc. 291 (1985), 73–98.CrossRefGoogle Scholar
10Maso, G. Dal. Integral representation on BV(Ω) of T-limits of variational integrals. Manuscripta Math. 30 (1980), 387–416.CrossRefGoogle Scholar
11Evans, L. C. and Gariepy, R. F.. Lecture Notes on Measure Theory and Fine Properties of Functions (CRC. Press 1992).Google Scholar
12Federer, H.. Geometric Measure Theory (Berlin: Springer, 1969).Google Scholar
13Fonseca, I.. The Wulffs theorem revisited. Proc. R. Soc. London A 432 (1991), 125–145.Google Scholar
14Fonseca, I. and Muller, S.. An uniqueness proof for the Wulff Problem. Proc. R. Soc. Edin. A119 (1991), 125–136.CrossRefGoogle Scholar
15Fonseca, 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
16Fonseca, I. and Tartar, L.. The gradient theory of phase transitions in nonlinear elasticity (to appear).Google Scholar
17Giaquinta, M., Modica, G. and Souček, J.. Functionals with linear growth in the calculus of variations. Comment. Math. Univ. Carolinae 20 (1979), 143–172.Google Scholar
18Giusti, E.. Minimal Surfaces and Functions of Bounded Variation (Boston: 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
20Kohn, R. and Sternberg, P.. Local minimizers and singular perturbations. Proc. Roy. Soc. Edinburgh Sect. A 111 (1989), 69–84.CrossRefGoogle Scholar
21Modica, L.. The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal. 98 (1987), 123–142.CrossRefGoogle Scholar
22Owen, N. C.. Existence arid stability of necking deformations for nonlinear elastic rods. Arch. Rational Mech. Anal. 98 (1987), 357–383.CrossRefGoogle Scholar
23Owen, N. C.. Nonconvex variational problems with general singular perturbations. Trans. Amer. Math. Soc. (to appear).Google Scholar
24Owen, N. C. and Sternberg, P.. Nonconvex variational problems with anisotropic perturbations (to appear).Google Scholar
25Reshetnyak, 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
26Sternberg, P.. The effect of a singular perturbation on nonconvex variational problems. Arch. Rational Mech. Anal. 101 (1988), 109–160.CrossRefGoogle Scholar
27Taylor, J.. Existence and structure of solutions to a class of nonelliptic variational problems. Sympos. Math. 14 (1974), 499–508.Google Scholar
28Taylor, J.. Unique structure of solutions To a class of nonelliptic variational problems. Proc. Symp. Pure Math., A. M. S., 27 (1975), 419–427.CrossRefGoogle Scholar
29Wulff, G.. Zur Frage der Geschwindigkeit des Wachsturms und der Auflösung der Kristallflächen”. Z. Krist. 34 (1901), 449–530.Google Scholar
30Ziemer, W. P.. Weakly Differentiate Functions (Berlin: Springer, 1989).CrossRefGoogle Scholar