Lower semicontinuity for polyconvex integrals without coercivity assumptions

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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

Abstract
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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 and 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.
[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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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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