Hostname: page-component-848d4c4894-2pzkn Total loading time: 0 Render date: 2024-05-15T12:31:03.770Z Has data issue: false hasContentIssue false
  • English
  • Français

Relaxation of quasi-convex integrals of arbitrary order

Published online by Cambridge University Press:  14 November 2011

Micol Amar  and
Virginia De Cicco
Micol Amar
Affiliation:
Laboratoire d'Analyse Numerique Tour 55-65, Universite de Paris VI, 4, Place Jussieu, F-75252 Paris Cedex 05, France
Virginia De Cicco
Affiliation:
Laboratoire d'Analyse Numerique Tour 55-65, Universite de Paris VI, 4, Place Jussieu, F-75252 Paris Cedex 05, France
Get access Rights & Permissions [Opens in a new window]

Extract

An integral representation result is given for the lower semicontinuous envelope of the functional ʃΩf(∇ku) dx on the space BVk(Ω:ℝm) of the integrable functions, whose the f-th derivative in the sense of distributions is a Radon measure with bounded total variation.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 5 , 1994 , pp. 927 - 946
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

1Acerbi, A. and Fusco, N.. Semicontinuity problems in calculus of variations. Arch. Rational Mech. Anal. 86(1984), 125–145.CrossRefGoogle Scholar
2Adams, R. A.. Sobolev Spaces (New York: Academic Press, 1975).Google Scholar
3Alberti, G.. Rank one properties for derivatives of functions with bounded variation. Proc. Roy. Soc. Edinburgh Sect. A. (to appear).Google Scholar
4Ambrosio, L. and Dal, G. Maso. On the relaxation in BV(Ω;ℝm) of quasi-convex integrals. J. Fund. Anal, (to appear).Google Scholar
5Ball, J. M.. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63(1977), 337–403.Google Scholar
6Ball, J. M., Currie, J. C. and Olver, P. J.. Null lagrangians, weak continuity and variational problems of arbitrary order. J. Fund. Anal. 41 (1981), 135–174.Google Scholar
7Ball, J. M. and Murat, F.. W1.p quasiconvexity and variational problems for multiple integrals. J. Fund. Anal. 58 (1984), 225–253.CrossRefGoogle Scholar
8Buttazzo, G.. Semicontinuity, Relaxation and Integral Representation Problems in the Calculus of Variations, Pitman Research Notes in Mathematics (Harlow: Longman, 1989).Google Scholar
9Carriero, M. and Pascali, E.. Г-convergenza di integrali non negativi maggiorati da funzionali del tipo del'area. Ann. Uni. Ferrara Sez. VII (N.S.) 24 (1978), 51–64.Google Scholar
10Dacorogna, B.. A relaxation theorem and its applications to the equilibrium of gases. Arch. Rational Mech. Anal. 77 (1981), 359–386.CrossRefGoogle Scholar
11Dacorogna, B.. Direct Methods in the Calculus of Variations (Berlin: Springer, 1989).Google Scholar
12Dacorogna, B.. Weak Continuity and Weak Lower Semicontinuity of Nonlinear Functionals, Lecture Notes in Mathematics 922 (Berlin: Springer, 1982).CrossRefGoogle Scholar
13Maso, G. Dal. An Introduction to the Г-convergence (Birkhaiiser, Boston, 1993).Google Scholar
14Maso, G. Dal and Modica, L.. A general theory of variational functionals. In Topics in Functional Analysis 1980–81 (Pisa: Scuola Normale Superiore, 1981).Google Scholar
15Giorgi, E. De and Letta, G.. Une notion generate de convergence faible pour des fonctions croissantes d'ensemble. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), 61–99.Google Scholar
16Evans, L. C. and Gariepy, R. F.. Lecture Notes on Measure Theory and Fine Properties of Functions (in prep.).Google Scholar
17Federer, H.. Geometric Measure Theory (Berlin: Springer, 1969).Google Scholar
18Fonseca, I.. Lower semicontinuity of surface energies. Proc. Royal Soc. Edin., 120 A (1992), 9–115.Google Scholar
19Fonseca, I. and Muller, S.. Quasiconvex integrands and lower semicontinuity in Ll (Research Report, Carnegie Mellon University, Pittsburgh, 1991). To appear on Siam J. Math. Anal.Google Scholar
20Fusco, N.. Quasi-convessita e semicontinuita per integrali multipli di ordine superiore. Ricerche Mat. 29(1980), 307–323.Google Scholar
21Giusti, E.. Minimal Surfaces and Functions of Bounded Variations (Boston: Birkhauser, 1984).Google Scholar
22Luckhaus, S. and Modica, L.. The Gibbs-Thompson relation within the gradient theory of phase transitions. Arch. Rational Mech. Anal. 107(1989), 71–83.Google Scholar
23Maz'ya, V. G.. Sobolev Spaces (Berlin: Springer, 1985).Google Scholar
24Meyers, N. G.. Quasiconvexity and lower semicontinuity of multiple variational integrals of any order. Trans. Amer. Math. Soc. 119(1965), 125–149.Google Scholar
25Miranda, M.. Distribuzioni aventi derivate misure, insiemi di perimetro localmente finite Ann. Scuola Norm. Sup. Pisa (1964).Google Scholar
26Morrey, C. B.. Multiple Integrals in the Calculus of Variations (Berlin: Springer, 1966).Google Scholar
27Morrey, C. B.. Quasiconvexity and the semicontinuity of multiple integrals. Pacific J. Math. 2 (1952). 25–53.Google Scholar
28Müller, S.. Homogenization of nonconvex integral functionals and cellular elastic materials. Arch. Rational Mech. Anal. 99 (1987), 189–212.Google Scholar
29Reshetnyak, Y. G.. Weak convergence of completely additive vector functions on a set. Siberian Math. J. 9(1968), 1039–1045.CrossRefGoogle Scholar
30Šverák, V.. Quasiconvex functions with subquadratic growth. Proc. Royal Soc. London 433–A (1991)733–735.Google Scholar
31Vol'pert, A. I.. The space BV and quasilinear equations. Math. USSR Sb. 2 (1967), 225–267.Google Scholar
32Vol'pert, A. I. and Hudjaev, S. I.. Analysis in Classes of Discontinuous Functions and Equations of Mathematical Physics (Dordrecht: Martinus Nijhoff Publishers, 1985).Google Scholar
33Kewei, Zhang. A construction of quasiconvex functions with linear growth at infinity (to appear).Google Scholar
34Ziemer, W. P.. Weakly Differentiable Functions (Berlin: Springer, 1989).Google Scholar