On the integral representation of relaxed functionals with convex bounded constraints
ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 1, pp. 37-57.

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.

DOI : 10.1051/cocv:2008063
Classification : 49J45
Mots-clés : relaxation, convex constraints, integral representation 
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     title = {On the integral representation of relaxed functionals with convex bounded constraints},
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     pages = {37--57},
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Anza Hafsa, Omar. On the integral representation of relaxed functionals with convex bounded constraints. ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 1, pp. 37-57. doi : 10.1051/cocv:2008063. http://www.numdam.org/articles/10.1051/cocv:2008063/

[1] E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86 (1984) 125-145. | Zbl

[2] O. Anza Hafsa and J.-P. Mandallena, Relaxation of variational problems in two-dimensional nonlinear elasticity. Ann. Mat. Pura Appl. 186 (2007) 187-198.

[3] O. Anza Hafsa and J.-P. Mandallena, Relaxation theorems in nonlinear elasticity. Ann. Inst. H. Poincaré, Anal. Non Linéaire 25 (2008) 135-148. | Numdam | Zbl

[4] J.M. Ball and F. Murat, W1,p-quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58 (1984) 225-253. | Zbl

[5] H. Ben Belgacem, Relaxation of singular functionals defined on Sobolev spaces. ESAIM: COCV 5 (2000) 71-85. | Numdam | Zbl

[6] 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). | Zbl

[7] B. Dacorogna, Direct methods in the Calculus of Variations. Springer-Verlag (1989). | Zbl

[8] B. Dacorogna and P. Marcellini, General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases. Acta Math. 178 (1997) 1-37. | Zbl

[9] 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). | Zbl

[10] I. Fonseca, The lower quasiconvex envelope of the stored energy function for an elastic crystal. J. Math. Pures Appl. 67 (1988) 175-195. | Zbl

[11] D. Kinderlehrer and P. Pedregal, Characterizations of Young measures generated by gradients. Arch. Rational Mech. Anal. 115 (1991) 329-365. | Zbl

[12] C.B. Morrey, Quasiconvexity and lower semicontinuity of multiple integrals. Pacific J. Math. 2 (1952) 25-53. | Zbl

[13] S. Müller, A sharp version of Zhang's theorem on truncating sequences of gradients. Trans. Amer. Math. Soc. 351 (1999) 4585-4597. | Zbl

[14] T.R. Rockafellar and J.-B. Wets, Variational analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 317. Springer-Verlag, Berlin (1998). | Zbl

[15] M. Wagner, On the lower semicontinuous quasiconvex envelope for unbounded integrands. To appear on ESAIM: COCV (to appear). | Numdam | Zbl

[16] M. Wagner, Quasiconvex relaxation of multidimensional control problems. Adv. Math. Sci. Appl. (to appear). | Zbl

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