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.
Mots-clés : relaxation, convex constraints, integral representation
@article{COCV_2010__16_1_37_0, author = {Anza Hafsa, Omar}, title = {On the integral representation of relaxed functionals with convex bounded constraints}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {37--57}, publisher = {EDP-Sciences}, volume = {16}, number = {1}, year = {2010}, doi = {10.1051/cocv:2008063}, mrnumber = {2598087}, zbl = {1183.49014}, language = {en}, url = {https://www.numdam.org/articles/10.1051/cocv:2008063/} }
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 PY - 2010 SP - 37 EP - 57 VL - 16 IS - 1 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv:2008063/ DO - 10.1051/cocv:2008063 LA - en ID - COCV_2010__16_1_37_0 ER -
%0 Journal Article %A Anza Hafsa, Omar %T On the integral representation of relaxed functionals with convex bounded constraints %J ESAIM: Control, Optimisation and Calculus of Variations %D 2010 %P 37-57 %V 16 %N 1 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv:2008063/ %R 10.1051/cocv:2008063 %G en %F COCV_2010__16_1_37_0
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. https://www.numdam.org/articles/10.1051/cocv:2008063/
[1] Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86 (1984) 125-145. | Zbl
and ,[2] Relaxation of variational problems in two-dimensional nonlinear elasticity. Ann. Mat. Pura Appl. 186 (2007) 187-198.
and ,[3] Relaxation theorems in nonlinear elasticity. Ann. Inst. H. Poincaré, Anal. Non Linéaire 25 (2008) 135-148. | Numdam | Zbl
and ,[4] W1,p-quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58 (1984) 225-253. | Zbl
and ,[5] Relaxation of singular functionals defined on Sobolev spaces. ESAIM: COCV 5 (2000) 71-85. | Numdam | Zbl
,[6] 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
and ,[7] Direct methods in the Calculus of Variations. Springer-Verlag (1989). | Zbl
,[8] General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases. Acta Math. 178 (1997) 1-37. | Zbl
and ,[9] Implicit partial differential equations, Progress in Nonlinear Differential Equations and their Applications 37. Birkhäuser Boston, Inc., Boston, MA, USA (1999). | Zbl
and ,[10] The lower quasiconvex envelope of the stored energy function for an elastic crystal. J. Math. Pures Appl. 67 (1988) 175-195. | Zbl
,[11] Characterizations of Young measures generated by gradients. Arch. Rational Mech. Anal. 115 (1991) 329-365. | Zbl
and ,[12] Quasiconvexity and lower semicontinuity of multiple integrals. Pacific J. Math. 2 (1952) 25-53. | Zbl
,[13] A sharp version of Zhang's theorem on truncating sequences of gradients. Trans. Amer. Math. Soc. 351 (1999) 4585-4597. | Zbl
,[14] Variational analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 317. Springer-Verlag, Berlin (1998). | Zbl
and ,[15] On the lower semicontinuous quasiconvex envelope for unbounded integrands. To appear on ESAIM: COCV (to appear). | Numdam | Zbl
,[16] Quasiconvex relaxation of multidimensional control problems. Adv. Math. Sci. Appl. (to appear). | Zbl
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- Γ-convergence of nonconvex unbounded integrals in strongly connected sets, Applicable Analysis, Volume 103 (2024) no. 9, p. 1704 | DOI:10.1080/00036811.2023.2261187
- Radial extension of
-limits, Bollettino dell'Unione Matematica Italiana, Volume 16 (2023) no. 4, p. 687 | DOI:10.1007/s40574-023-00356-w - Integral representation of unbounded variational functionals on Sobolev spaces, Ricerche di Matematica, Volume 72 (2023) no. 1, p. 193 | DOI:10.1007/s11587-021-00652-7
- Relaxation of nonconvex unbounded integrals with general growth conditions in Cheeger–Sobolev spaces, Bulletin des Sciences Mathématiques, Volume 142 (2018), p. 49 | DOI:10.1016/j.bulsci.2017.09.002
- Homogenization of unbounded integrals with quasiconvex growth, Annali di Matematica Pura ed Applicata (1923 -), Volume 194 (2015) no. 6, p. 1619 | DOI:10.1007/s10231-014-0437-z
- Radial representation of lower semicontinuous envelope, Bollettino dell'Unione Matematica Italiana, Volume 7 (2014) no. 1, p. 1 | DOI:10.1007/s40574-014-0001-1
- Homogenization of unbounded singular integrals in W 1,∞, Ricerche di Matematica, Volume 61 (2012) no. 2, p. 185 | DOI:10.1007/s11587-011-0124-y
- Homogenization of nonconvex integrals with convex growth, Journal de Mathématiques Pures et Appliquées, Volume 96 (2011) no. 2, p. 167 | DOI:10.1016/j.matpur.2011.03.003
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