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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.   Google Scholar

[2]

E. Acerbi and G. Dal Maso, New lower semicontinuity results for polyconvex integrals,, Calc. Var. Partial Differential Equations, 2 (1994), 329.  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.  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, 14 (2008), 456.  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.  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.  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, 11 (1994), 661.   Google Scholar

[8]

P. G. Ciarlet, Three Dimensional Elasticity,, Vol. 1, (1988).   Google Scholar

[9]

B. Dacorogna, Direct Methods in the Calculus of Variations,, Appl. Math. Sci., (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.   Google Scholar

[11]

G. Dal Maso, Integral representation on $BV(\Omega)$ of $\Gamma$-limits of variational integrals,, Manuscripta Math., 30 (1980), 387.  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.  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.   Google Scholar

[14]

M. Fabrizio, Sulla convessità dei potenziali termodinamici per materiali con memoria,, Ann. Mat. Pura e Appl., 101 (1974), 33.  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.   Google Scholar

[16]

I. Fonseca and G. Leoni, Some remarks on lower semicontinuity,, Indiana Univ. Math. J., 49 (2000), 617.  doi: 10.1512/iumj.2000.49.1791.  Google Scholar

[17]

I. Fonseca and G. Leoni, On lower semicontinuity and relaxation,, Proc. Royal Soc. Edinb., 131 (2001), 519.  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.   Google Scholar

[19]

N. Fusco and J. E. Hutchinson, A direct proof for lower semicontinuity of polyconvex functionals,, Manuscripta Math., 87 (1995), 35.  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.   Google Scholar

[21]

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

[22]

P. Marcellini, Approximation of quasiconvex functions and lower semicontinuity of multiple integrals,, Manuscripta Math., 51 (1985), 1.  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.   Google Scholar

[24]

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

[25]

J. Serrin, On the definition and properties of certain variational integrals,, Trans. Amer. Math. Soc., 101 (1961), 139.  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.   Google Scholar

[2]

E. Acerbi and G. Dal Maso, New lower semicontinuity results for polyconvex integrals,, Calc. Var. Partial Differential Equations, 2 (1994), 329.  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.  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, 14 (2008), 456.  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.  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.  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, 11 (1994), 661.   Google Scholar

[8]

P. G. Ciarlet, Three Dimensional Elasticity,, Vol. 1, (1988).   Google Scholar

[9]

B. Dacorogna, Direct Methods in the Calculus of Variations,, Appl. Math. Sci., (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.   Google Scholar

[11]

G. Dal Maso, Integral representation on $BV(\Omega)$ of $\Gamma$-limits of variational integrals,, Manuscripta Math., 30 (1980), 387.  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.  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.   Google Scholar

[14]

M. Fabrizio, Sulla convessità dei potenziali termodinamici per materiali con memoria,, Ann. Mat. Pura e Appl., 101 (1974), 33.  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.   Google Scholar

[16]

I. Fonseca and G. Leoni, Some remarks on lower semicontinuity,, Indiana Univ. Math. J., 49 (2000), 617.  doi: 10.1512/iumj.2000.49.1791.  Google Scholar

[17]

I. Fonseca and G. Leoni, On lower semicontinuity and relaxation,, Proc. Royal Soc. Edinb., 131 (2001), 519.  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.   Google Scholar

[19]

N. Fusco and J. E. Hutchinson, A direct proof for lower semicontinuity of polyconvex functionals,, Manuscripta Math., 87 (1995), 35.  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.   Google Scholar

[21]

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

[22]

P. Marcellini, Approximation of quasiconvex functions and lower semicontinuity of multiple integrals,, Manuscripta Math., 51 (1985), 1.  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.   Google Scholar

[24]

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

[25]

J. Serrin, On the definition and properties of certain variational integrals,, Trans. Amer. Math. Soc., 101 (1961), 139.  doi: 10.1090/S0002-9947-1961-0138018-9.  Google Scholar

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