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), 99-115. 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), 433-449.  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, SIAM, MPS, Philadelphia, PA, 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, 1997), Rend. Istit. Mat. Univ. Trieste, 31 (2000), 1-69.  Google Scholar

[5]

J. M. Ball, A version of the fundamental theorem for Young measures, in "PDEs and Continuum Models of Phase Transitions" (Nice, 1988) (eds. M. Rascle, D. Serre and M. Slemrod) Lecture Notes in Physics, 344, Springer Verlag, Berlin, (1989), 207-215.  Google Scholar

[6]

J. M. Ball and R. D. James, Fine phase mixtures as minimizers of energy, Arch. Rat. Mech. Anal., 100 (1987), 13-52. 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), 1-19.  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), 447-469.  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), 845-883. 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, Kluwer Academic Publisher, Dordrecht, 2004.  Google Scholar

[11]

C. Castaing and M. Valadier, "Convex Analysis and Measurable Multifunctions," Lect. Notes Math., 580, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[12]

B. Dacorogna, "Direct Methods in the Calculus of Variations," Appl. Math. Sciences, 78, Springer-Verlag, Berlin, 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), 736-756. 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), 329-365. doi: 10.1007/BF00375279.  Google Scholar

[15]

E. Mascolo and L. Migliaccio, Relaxation methods in control theory, Appl. Math. Optim., 20 (1989), 97-103. 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), 343-365. 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), 193-230.  Google Scholar

[18]

M. Valadier, Young measures, in "Methods of Nonconvex Analysis" (Varenna, 1989) (ed. A. Cellina), Lecture Notes in Math., 1446, Springer, Berlin, (1990), 152-188.  Google Scholar

[19]

M. Valadier, A course on Young measures, Workshop di Teoria della Misura e Analisi Reale, (Grado, 1993), Rend. Istit. Mat. Univ. Trieste, 26 (1994), 349-394.  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), 99-115. 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), 433-449.  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, SIAM, MPS, Philadelphia, PA, 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, 1997), Rend. Istit. Mat. Univ. Trieste, 31 (2000), 1-69.  Google Scholar

[5]

J. M. Ball, A version of the fundamental theorem for Young measures, in "PDEs and Continuum Models of Phase Transitions" (Nice, 1988) (eds. M. Rascle, D. Serre and M. Slemrod) Lecture Notes in Physics, 344, Springer Verlag, Berlin, (1989), 207-215.  Google Scholar

[6]

J. M. Ball and R. D. James, Fine phase mixtures as minimizers of energy, Arch. Rat. Mech. Anal., 100 (1987), 13-52. 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), 1-19.  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), 447-469.  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), 845-883. 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, Kluwer Academic Publisher, Dordrecht, 2004.  Google Scholar

[11]

C. Castaing and M. Valadier, "Convex Analysis and Measurable Multifunctions," Lect. Notes Math., 580, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[12]

B. Dacorogna, "Direct Methods in the Calculus of Variations," Appl. Math. Sciences, 78, Springer-Verlag, Berlin, 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), 736-756. 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), 329-365. doi: 10.1007/BF00375279.  Google Scholar

[15]

E. Mascolo and L. Migliaccio, Relaxation methods in control theory, Appl. Math. Optim., 20 (1989), 97-103. 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), 343-365. 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), 193-230.  Google Scholar

[18]

M. Valadier, Young measures, in "Methods of Nonconvex Analysis" (Varenna, 1989) (ed. A. Cellina), Lecture Notes in Math., 1446, Springer, Berlin, (1990), 152-188.  Google Scholar

[19]

M. Valadier, A course on Young measures, Workshop di Teoria della Misura e Analisi Reale, (Grado, 1993), Rend. Istit. Mat. Univ. Trieste, 26 (1994), 349-394.  Google Scholar

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