June  2012, 5(3): 399-417. doi: 10.3934/dcdss.2012.5.399

A variational convergence for bifunctionals. Application to a model of strong junction

1. 

IMATH, Université du Sud Toulon-Var, BP 20132 - 83957 La Garde Cedex, France

Received  August 2010 Revised  September 2010 Published  October 2011

We introduce a notion of variational convergence for bifunctionals in an abstract setting. Then we apply this convergence to the asymptotic analysis of a junction problem in order to capture the gradient oscillations in the joint by considering the energy functional as a bifunctional of Sobolev-function/Young measure arguments. The well known asymptotic model described in terms of Sobolev-functions is obtained by eliminating the Young-measure argument considered as an internal variable through a marginal map. Furthermore, the surface energy of the classical model can be considered as a relaxation of a Dirichlet condition.
Citation: Anne-Laure Bessoud. A variational convergence for bifunctionals. Application to a model of strong junction. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 399-417. doi: 10.3934/dcdss.2012.5.399
References:
[1]

E. Acerbi, G. Buttazzo and D. Percivale, Thin inclusions in linear elasticity: A variational approach,, J. Reine Angew. Math., 386 (1988), 99. doi: 10.1515/crll.1988.386.99. Google Scholar

[2]

O. Anza Hafsa and J. P. Mandallena, Interchange of infimum and integral,, Calc. Var. Partial Differential Equations, 18 (2003), 433. Google Scholar

[3]

H. Attouch, G. Buttazzo and G. Michaille, "Variational Analysis in Sobolev and BV Spaces. Application to PDEs and Optimization,", MPS/SIAM Series on Optimization, 6 (2006). Google Scholar

[4]

E. J. Balder, Lectures on Young measures theory and its applications in economics,, Workshop di Teoria della Misura e Analisi Reale (Grado, 31 (2000), 1. Google Scholar

[5]

J. M. Ball, A version of the fundamental theorem for Young measures,, in, 344 (1989), 207. Google Scholar

[6]

J. M. Ball and R. D. James, Fine phase mixtures as minimizers of energy,, Arch. Rat. Mech. Anal., 100 (1987), 13. doi: 10.1007/BF00281246. Google Scholar

[7]

A. L. Bessoud, F. Krasucki and G. Michaille, Multi-materials with strong interface: Variational modelings,, Asympto. Anal., 61 (2009), 1. Google Scholar

[8]

A. L. Bessoud, F. Krasucki and G. Michaille, A relaxation process for bifunctionals of displacement-Young measure state variables: A model of multi-material with micro-structured strong interface,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 447. Google Scholar

[9]

M. Bocea and I. Fonseca, A Young measure approach to a nonlinear membrane model involving the bending moment,, Poc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 845. doi: 10.1017/S0308210500003516. Google Scholar

[10]

C. Castaing, P. Raynaud de Fitte and M. Valadier, "Young Measure on Topological Spaces. With Applications in Control Theory and Probability Theory,", Mathematics and Its Applications, 571 (2004). Google Scholar

[11]

C. Castaing and M. Valadier, "Convex Analysis and Measurable Multifunctions,", Lect. Notes Math., 580 (1977). Google Scholar

[12]

B. Dacorogna, "Direct Methods in the Calculus of Variations,", Appl. Math. Sciences, 78 (1989). Google Scholar

[13]

I. Fonseca, S. Müller and P. Pedregal, Analysis of concentration and oscillation effects generated by gradients,, SIAM J. Math. Anal., 29 (1998), 736. doi: 10.1137/S0036141096306534. Google Scholar

[14]

D. Kinderlehrer and P. Pedregal, Characterization of Young measures generated by gradients,, Arch. Rational Mech. Anal., 115 (1991), 329. doi: 10.1007/BF00375279. Google Scholar

[15]

E. Mascolo and L. Migliaccio, Relaxation methods in control theory,, Appl. Math. Optim., 20 (1989), 97. doi: 10.1007/BF01447649. Google Scholar

[16]

C. Licht, G. Michaille and S. Pagano, A model of elastic adhesive bonded joints through oscillation-concentration measures,, J. Math. Pures Appl. (9), 87 (2007), 343. doi: 10.1016/j.matpur.2007.01.008. Google Scholar

[17]

L. Tartar, $H$-measures, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations,, Proceedings of the Royal Society of Edinburgh Sect. A, 115 (1990), 193. Google Scholar

[18]

M. Valadier, Young measures,, in, 1446 (1990), 152. Google Scholar

[19]

M. Valadier, A course on Young measures,, Workshop di Teoria della Misura e Analisi Reale, 26 (1994), 349. Google Scholar

show all references

References:
[1]

E. Acerbi, G. Buttazzo and D. Percivale, Thin inclusions in linear elasticity: A variational approach,, J. Reine Angew. Math., 386 (1988), 99. doi: 10.1515/crll.1988.386.99. Google Scholar

[2]

O. Anza Hafsa and J. P. Mandallena, Interchange of infimum and integral,, Calc. Var. Partial Differential Equations, 18 (2003), 433. Google Scholar

[3]

H. Attouch, G. Buttazzo and G. Michaille, "Variational Analysis in Sobolev and BV Spaces. Application to PDEs and Optimization,", MPS/SIAM Series on Optimization, 6 (2006). Google Scholar

[4]

E. J. Balder, Lectures on Young measures theory and its applications in economics,, Workshop di Teoria della Misura e Analisi Reale (Grado, 31 (2000), 1. Google Scholar

[5]

J. M. Ball, A version of the fundamental theorem for Young measures,, in, 344 (1989), 207. Google Scholar

[6]

J. M. Ball and R. D. James, Fine phase mixtures as minimizers of energy,, Arch. Rat. Mech. Anal., 100 (1987), 13. doi: 10.1007/BF00281246. Google Scholar

[7]

A. L. Bessoud, F. Krasucki and G. Michaille, Multi-materials with strong interface: Variational modelings,, Asympto. Anal., 61 (2009), 1. Google Scholar

[8]

A. L. Bessoud, F. Krasucki and G. Michaille, A relaxation process for bifunctionals of displacement-Young measure state variables: A model of multi-material with micro-structured strong interface,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 447. Google Scholar

[9]

M. Bocea and I. Fonseca, A Young measure approach to a nonlinear membrane model involving the bending moment,, Poc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 845. doi: 10.1017/S0308210500003516. Google Scholar

[10]

C. Castaing, P. Raynaud de Fitte and M. Valadier, "Young Measure on Topological Spaces. With Applications in Control Theory and Probability Theory,", Mathematics and Its Applications, 571 (2004). Google Scholar

[11]

C. Castaing and M. Valadier, "Convex Analysis and Measurable Multifunctions,", Lect. Notes Math., 580 (1977). Google Scholar

[12]

B. Dacorogna, "Direct Methods in the Calculus of Variations,", Appl. Math. Sciences, 78 (1989). Google Scholar

[13]

I. Fonseca, S. Müller and P. Pedregal, Analysis of concentration and oscillation effects generated by gradients,, SIAM J. Math. Anal., 29 (1998), 736. doi: 10.1137/S0036141096306534. Google Scholar

[14]

D. Kinderlehrer and P. Pedregal, Characterization of Young measures generated by gradients,, Arch. Rational Mech. Anal., 115 (1991), 329. doi: 10.1007/BF00375279. Google Scholar

[15]

E. Mascolo and L. Migliaccio, Relaxation methods in control theory,, Appl. Math. Optim., 20 (1989), 97. doi: 10.1007/BF01447649. Google Scholar

[16]

C. Licht, G. Michaille and S. Pagano, A model of elastic adhesive bonded joints through oscillation-concentration measures,, J. Math. Pures Appl. (9), 87 (2007), 343. doi: 10.1016/j.matpur.2007.01.008. Google Scholar

[17]

L. Tartar, $H$-measures, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations,, Proceedings of the Royal Society of Edinburgh Sect. A, 115 (1990), 193. Google Scholar

[18]

M. Valadier, Young measures,, in, 1446 (1990), 152. Google Scholar

[19]

M. Valadier, A course on Young measures,, Workshop di Teoria della Misura e Analisi Reale, 26 (1994), 349. Google Scholar

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