Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T19:38:22.784Z Has data issue: false hasContentIssue false
  • English
  • Français

On the integral representation of relaxed functionalswith convex bounded constraints

Published online by Cambridge University Press:  21 October 2008

Omar Anza Hafsa
Omar Anza Hafsa*
Affiliation:
Université Montpellier II, UMR-CNRS 5508, LMGC, Place Eugène Bataillon, 34090 Montpellier, France. Omar.Anza-Hafsa@univ-montp2.fr
Get access

Abstract

We study the integral representation of relaxed functionals in the multi-dimensional calculus of variations, for integrands which are finite in a convex bounded set with nonempty interior and infinite elsewhere.

Keywords

Relaxationconvex constraintsintegral representation
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 1 , January 2010 , pp. 37 - 57
Copyright
© EDP Sciences, SMAI, 2008

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

Acerbi, E. and Fusco, N., Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86 (1984) 125–145. CrossRef
Anza Hafsa, O. and Mandallena, J.-P., Relaxation of variational problems in two-dimensional nonlinear elasticity. Ann. Mat. Pura Appl. 186 (2007) 187–198.
O. Anza Hafsa and J.-P. Mandallena, Relaxation theorems in nonlinear elasticity. Ann. Inst. H. Poincaré, Anal. Non Linéaire 25 (2008) 135–148.
Ball, J.M. and Murat, F., W 1,p -quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58 (1984) 225–253. CrossRef
Ben Belgacem, H., Relaxation of singular functionals defined on Sobolev spaces. ESAIM: COCV 5 (2000) 71–85. CrossRef
L. Carbone and R. De Arcangelis, Unbounded functionals in the calculus of variations, Representation, relaxation, and homogenization, Monographs and Surveys in Pure and Applied Mathematics 125. Chapman & Hall/CRC, Boca Raton, FL, USA (2002).
B. Dacorogna, Direct methods in the Calculus of Variations. Springer-Verlag (1989).
Dacorogna, B. and Marcellini, P., General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases. Acta Math. 178 (1997) 1–37. CrossRef
B. Dacorogna and P. Marcellini, Implicit partial differential equations, Progress in Nonlinear Differential Equations and their Applications 37. Birkhäuser Boston, Inc., Boston, MA, USA (1999).
Fonseca, I., The lower quasiconvex envelope of the stored energy function for an elastic crystal. J. Math. Pures Appl. 67 (1988) 175–195.
Kinderlehrer, D. and Pedregal, P., Characterizations of Young measures generated by gradients. Arch. Rational Mech. Anal. 115 (1991) 329–365. CrossRef
Morrey, C.B., Quasiconvexity and lower semicontinuity of multiple integrals. Pacific J. Math. 2 (1952) 25–53. CrossRef
Müller, S., A sharp version of Zhang's theorem on truncating sequences of gradients. Trans. Amer. Math. Soc. 351 (1999) 4585–4597. CrossRef
T.R. Rockafellar and J.-B. Wets, Variational analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 317. Springer-Verlag, Berlin (1998).
M. Wagner, On the lower semicontinuous quasiconvex envelope for unbounded integrands. To appear on ESAIM: COCV (to appear).
M. Wagner, Quasiconvex relaxation of multidimensional control problems. Adv. Math. Sci. Appl. (to appear).