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Lower semicontinuity for polyconvex integrals without coercivity assumptions
1. | Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza - Università di Roma, Via A. Scarpa 16, 00161 Roma, Italy, Italy |
References:
[1] |
E. Acerbi, G. Buttazzo and N. Fusco, Semicontinuity and relaxation for integrals depending on vector valued functions, J. Math. Pures Appl., 62 (1983), 371-387. |
[2] |
E. Acerbi and G. Dal Maso, New lower semicontinuity results for polyconvex integrals, Calc. Var. Partial Differential Equations, 2 (1994), 329-371.
doi: 10.1007/BF01235534. |
[3] |
M. Amar, V. De Cicco, P. Marcellini and E. Mascolo, Weak lower semicontinuity for non coercive polyconvex integrals, Adv. Calc. Var., I (2008), 171-191.
doi: 10.1515/ACV.2008.006. |
[4] |
M. Amar, V. De Cicco and N. Fusco, Lower semicontinuity and relaxation results in BV for integral functionals with BV integrands, ESAIM: Control, Optimization and Calculus of Variations, 14 (2008), 456-477.
doi: 10.1051/cocv:2007061. |
[5] |
M. Amar, V. De Cicco and N. Fusco, Lower semicontinuity results for free discontinuity energies, Mathematical Models and Methods in Applied Sciences, 20 (2010), 707-730.
doi: 10.1142/S0218202510004416. |
[6] |
J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal., 63 (1977), 337-403.
doi: 10.1007/BF00279992. |
[7] |
P. Celada and G. Dal Maso, Further remarks on the lower semicontinuity of polyconvex integrals, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 11 (1994), 661-691. |
[8] |
P. G. Ciarlet, Three Dimensional Elasticity, Vol. 1, Studies in mathematics and its applications, Elsevier Science, 1988. |
[9] |
B. Dacorogna, Direct Methods in the Calculus of Variations, Appl. Math. Sci., 78, Springer-Verlag, Berlin, 1989.
doi: 10.1007/978-3-642-51440-1. |
[10] |
B. Dacorogna and P. Marcellini, Semicontinuité pour des intégrandes polyconvexes sans continuité des déterminants, C.R. Math. Acad. Sci. Paris, 311 (1990), 393-396. |
[11] |
G. Dal Maso, Integral representation on $BV(\Omega)$ of $\Gamma$-limits of variational integrals, Manuscripta Math., 30 (1980), 387-416.
doi: 10.1007/BF01301259. |
[12] |
G. Dal Maso and C. Sbordone, Weak lower semicontinuity of polyconvex integrals: a borderline case, Math. Z., 218 (1995), 603-609.
doi: 10.1007/BF02571927. |
[13] |
E. De Giorgi, G. Buttazzo and G. Dal Maso, On the lower semicontinuity of certain integral functions, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 74 (1983), 274-282. |
[14] |
M. Fabrizio, Sulla convessità dei potenziali termodinamici per materiali con memoria, Ann. Mat. Pura e Appl., 101 (1974), 33-48.
doi: 10.1007/BF02417097. |
[15] |
M. Focardi, N. Fusco, C. Leone, P. Marcellini, E. Mascolo and A. Verde, Weak lower semicontinuity for polyconvex integrals in the limit case, Calc. Var. Partial Differential Equations, (2013), 1-23. |
[16] |
I. Fonseca and G. Leoni, Some remarks on lower semicontinuity, Indiana Univ. Math. J., 49 (2000), 617-636.
doi: 10.1512/iumj.2000.49.1791. |
[17] |
I. Fonseca and G. Leoni, On lower semicontinuity and relaxation, Proc. Royal Soc. Edinb., Sect. A, Math., 131 (2001), 519-565.
doi: 10.1017/S0308210500000998. |
[18] |
N. Fusco, M. Gori and F. Maggi, A remark on Serrin's Theorem, NoDEA Nonlinear Differential Equations Appl., 28 (2007), 427-447. |
[19] |
N. Fusco and J. E. Hutchinson, A direct proof for lower semicontinuity of polyconvex functionals, Manuscripta Math., 87 (1995), 35-50.
doi: 10.1007/BF02570460. |
[20] |
W. Gangbo, On the weak lower semicontinuty of energies with polyconvex integrands, J. Math. Pures Appl., 73 (1994), 455-469. |
[21] |
J. Maly, Weak lower semicontinuity of polyconvex integrals, Proc. Edinb. Math. Soc., 123 (1993), 681-691.
doi: 10.1017/S0308210500030900. |
[22] |
P. Marcellini, Approximation of quasiconvex functions and lower semicontinuity of multiple integrals, Manuscripta Math., 51 (1985), 1-28.
doi: 10.1007/BF01168345. |
[23] |
P. Marcellini, On the definition and the lower semicontinuity of certain quasiconvex integrals, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 391-409. |
[24] |
C. B. Morrey, Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin, 1966. |
[25] |
J. Serrin, On the definition and properties of certain variational integrals, Trans. Amer. Math. Soc., 101 (1961), 139-167.
doi: 10.1090/S0002-9947-1961-0138018-9. |
show all references
References:
[1] |
E. Acerbi, G. Buttazzo and N. Fusco, Semicontinuity and relaxation for integrals depending on vector valued functions, J. Math. Pures Appl., 62 (1983), 371-387. |
[2] |
E. Acerbi and G. Dal Maso, New lower semicontinuity results for polyconvex integrals, Calc. Var. Partial Differential Equations, 2 (1994), 329-371.
doi: 10.1007/BF01235534. |
[3] |
M. Amar, V. De Cicco, P. Marcellini and E. Mascolo, Weak lower semicontinuity for non coercive polyconvex integrals, Adv. Calc. Var., I (2008), 171-191.
doi: 10.1515/ACV.2008.006. |
[4] |
M. Amar, V. De Cicco and N. Fusco, Lower semicontinuity and relaxation results in BV for integral functionals with BV integrands, ESAIM: Control, Optimization and Calculus of Variations, 14 (2008), 456-477.
doi: 10.1051/cocv:2007061. |
[5] |
M. Amar, V. De Cicco and N. Fusco, Lower semicontinuity results for free discontinuity energies, Mathematical Models and Methods in Applied Sciences, 20 (2010), 707-730.
doi: 10.1142/S0218202510004416. |
[6] |
J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal., 63 (1977), 337-403.
doi: 10.1007/BF00279992. |
[7] |
P. Celada and G. Dal Maso, Further remarks on the lower semicontinuity of polyconvex integrals, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 11 (1994), 661-691. |
[8] |
P. G. Ciarlet, Three Dimensional Elasticity, Vol. 1, Studies in mathematics and its applications, Elsevier Science, 1988. |
[9] |
B. Dacorogna, Direct Methods in the Calculus of Variations, Appl. Math. Sci., 78, Springer-Verlag, Berlin, 1989.
doi: 10.1007/978-3-642-51440-1. |
[10] |
B. Dacorogna and P. Marcellini, Semicontinuité pour des intégrandes polyconvexes sans continuité des déterminants, C.R. Math. Acad. Sci. Paris, 311 (1990), 393-396. |
[11] |
G. Dal Maso, Integral representation on $BV(\Omega)$ of $\Gamma$-limits of variational integrals, Manuscripta Math., 30 (1980), 387-416.
doi: 10.1007/BF01301259. |
[12] |
G. Dal Maso and C. Sbordone, Weak lower semicontinuity of polyconvex integrals: a borderline case, Math. Z., 218 (1995), 603-609.
doi: 10.1007/BF02571927. |
[13] |
E. De Giorgi, G. Buttazzo and G. Dal Maso, On the lower semicontinuity of certain integral functions, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 74 (1983), 274-282. |
[14] |
M. Fabrizio, Sulla convessità dei potenziali termodinamici per materiali con memoria, Ann. Mat. Pura e Appl., 101 (1974), 33-48.
doi: 10.1007/BF02417097. |
[15] |
M. Focardi, N. Fusco, C. Leone, P. Marcellini, E. Mascolo and A. Verde, Weak lower semicontinuity for polyconvex integrals in the limit case, Calc. Var. Partial Differential Equations, (2013), 1-23. |
[16] |
I. Fonseca and G. Leoni, Some remarks on lower semicontinuity, Indiana Univ. Math. J., 49 (2000), 617-636.
doi: 10.1512/iumj.2000.49.1791. |
[17] |
I. Fonseca and G. Leoni, On lower semicontinuity and relaxation, Proc. Royal Soc. Edinb., Sect. A, Math., 131 (2001), 519-565.
doi: 10.1017/S0308210500000998. |
[18] |
N. Fusco, M. Gori and F. Maggi, A remark on Serrin's Theorem, NoDEA Nonlinear Differential Equations Appl., 28 (2007), 427-447. |
[19] |
N. Fusco and J. E. Hutchinson, A direct proof for lower semicontinuity of polyconvex functionals, Manuscripta Math., 87 (1995), 35-50.
doi: 10.1007/BF02570460. |
[20] |
W. Gangbo, On the weak lower semicontinuty of energies with polyconvex integrands, J. Math. Pures Appl., 73 (1994), 455-469. |
[21] |
J. Maly, Weak lower semicontinuity of polyconvex integrals, Proc. Edinb. Math. Soc., 123 (1993), 681-691.
doi: 10.1017/S0308210500030900. |
[22] |
P. Marcellini, Approximation of quasiconvex functions and lower semicontinuity of multiple integrals, Manuscripta Math., 51 (1985), 1-28.
doi: 10.1007/BF01168345. |
[23] |
P. Marcellini, On the definition and the lower semicontinuity of certain quasiconvex integrals, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 391-409. |
[24] |
C. B. Morrey, Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin, 1966. |
[25] |
J. Serrin, On the definition and properties of certain variational integrals, Trans. Amer. Math. Soc., 101 (1961), 139-167.
doi: 10.1090/S0002-9947-1961-0138018-9. |
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