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September  2014, 3(3): 363-372. doi: 10.3934/eect.2014.3.363

Lower semicontinuity for polyconvex integrals without coercivity assumptions

Micol Amar 1, and  Virginia De Cicco 1,

1. 

Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza - Università di Roma, Via A. Scarpa 16, 00161 Roma, Italy, Italy

Received  May 2013 Revised  November 2013 Published  August 2014

We prove a lower semicontinuity theorem for a polyconvex functional of integral form, related to maps $u:\Omega \subset \mathbb{R}^{n}\rightarrow \mathbb{R}^{m}$ in $W^{1,n}(\Omega ;\mathbb{R}^{m})$ with $n\geq m\geq 2$, with respect to the weak $W^{1,p}$-convergence for $p>m-1$, without assuming any coercivity condition.
Keywords: lower semicontinuity., Vector-valued maps, Jacobian determinants, polyconvex integrals.
Mathematics Subject Classification: Primary: 49J45; Secondary: 49K2.
Citation: Micol Amar, Virginia De Cicco. Lower semicontinuity for polyconvex integrals without coercivity assumptions. Evolution Equations & Control Theory, 2014, 3 (3) : 363-372. doi: 10.3934/eect.2014.3.363
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal., 63 (1977), 337-403. doi: 10.1007/BF00279992.  Google Scholar

[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.  Google Scholar

[8]

P. G. Ciarlet, Three Dimensional Elasticity, Vol. 1, Studies in mathematics and its applications, Elsevier Science, 1988.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

N. Fusco, M. Gori and F. Maggi, A remark on Serrin's Theorem, NoDEA Nonlinear Differential Equations Appl., 28 (2007), 427-447. Google Scholar

[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.  Google Scholar

[20]

W. Gangbo, On the weak lower semicontinuty of energies with polyconvex integrands, J. Math. Pures Appl., 73 (1994), 455-469.  Google Scholar

[21]

J. Maly, Weak lower semicontinuity of polyconvex integrals, Proc. Edinb. Math. Soc., 123 (1993), 681-691. doi: 10.1017/S0308210500030900.  Google Scholar

[22]

P. Marcellini, Approximation of quasiconvex functions and lower semicontinuity of multiple integrals, Manuscripta Math., 51 (1985), 1-28. doi: 10.1007/BF01168345.  Google Scholar

[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.  Google Scholar

[24]

C. B. Morrey, Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin, 1966.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal., 63 (1977), 337-403. doi: 10.1007/BF00279992.  Google Scholar

[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.  Google Scholar

[8]

P. G. Ciarlet, Three Dimensional Elasticity, Vol. 1, Studies in mathematics and its applications, Elsevier Science, 1988.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

N. Fusco, M. Gori and F. Maggi, A remark on Serrin's Theorem, NoDEA Nonlinear Differential Equations Appl., 28 (2007), 427-447. Google Scholar

[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.  Google Scholar

[20]

W. Gangbo, On the weak lower semicontinuty of energies with polyconvex integrands, J. Math. Pures Appl., 73 (1994), 455-469.  Google Scholar

[21]

J. Maly, Weak lower semicontinuity of polyconvex integrals, Proc. Edinb. Math. Soc., 123 (1993), 681-691. doi: 10.1017/S0308210500030900.  Google Scholar

[22]

P. Marcellini, Approximation of quasiconvex functions and lower semicontinuity of multiple integrals, Manuscripta Math., 51 (1985), 1-28. doi: 10.1007/BF01168345.  Google Scholar

[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.  Google Scholar

[24]

C. B. Morrey, Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin, 1966.  Google Scholar

[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.  Google Scholar

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