# On the integral representation of relaxed functionals with convex bounded constraints

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 16, Issue: 1, page 37-57
- ISSN: 1292-8119

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topAnza Hafsa, Omar. "On the integral representation of relaxed functionals with convex bounded constraints." ESAIM: Control, Optimisation and Calculus of Variations 16.1 (2010): 37-57. <http://eudml.org/doc/250725>.

@article{AnzaHafsa2010,

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.
},

author = {Anza Hafsa, Omar},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Relaxation; convex constraints; integral representation; relaxation},

language = {eng},

month = {1},

number = {1},

pages = {37-57},

publisher = {EDP Sciences},

title = {On the integral representation of relaxed functionals with convex bounded constraints},

url = {http://eudml.org/doc/250725},

volume = {16},

year = {2010},

}

TY - JOUR

AU - Anza Hafsa, Omar

TI - On the integral representation of relaxed functionals with convex bounded constraints

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/1//

PB - EDP Sciences

VL - 16

IS - 1

SP - 37

EP - 57

AB -
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.

LA - eng

KW - Relaxation; convex constraints; integral representation; relaxation

UR - http://eudml.org/doc/250725

ER -

## References

top- E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal.86 (1984) 125–145.
- O. Anza Hafsa and J.-P. Mandallena, 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 Linéaire25 (2008) 135–148.
- J.M. Ball and F. Murat, W1,p-quasiconvexity and variational problems for multiple integrals. J. Funct. Anal.58 (1984) 225–253.
- H. Ben Belgacem, Relaxation of singular functionals defined on Sobolev spaces. ESAIM: COCV5 (2000) 71–85.
- L. Carbone and R. De Arcangelis, Unbounded functionals in the calculus of variations, Representation, relaxation, and homogenization, Monographs and Surveys in Pure and Applied Mathematics125. Chapman & Hall/CRC, Boca Raton, FL, USA (2002).
- B. Dacorogna, Direct methods in the Calculus of Variations. Springer-Verlag (1989).
- B. Dacorogna and P. Marcellini, General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases. Acta Math.178 (1997) 1–37.
- B. Dacorogna and P. Marcellini, Implicit partial differential equations, Progress in Nonlinear Differential Equations and their Applications37. Birkhäuser Boston, Inc., Boston, MA, USA (1999).
- I. Fonseca, The lower quasiconvex envelope of the stored energy function for an elastic crystal. J. Math. Pures Appl.67 (1988) 175–195.
- D. Kinderlehrer and P. Pedregal, Characterizations of Young measures generated by gradients. Arch. Rational Mech. Anal.115 (1991) 329–365.
- C.B. Morrey, Quasiconvexity and lower semicontinuity of multiple integrals. Pacific J. Math.2 (1952) 25–53.
- S. Müller, A sharp version of Zhang's theorem on truncating sequences of gradients. Trans. Amer. Math. Soc.351 (1999) 4585–4597.
- 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).

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