# American Institute of Mathematical Sciences

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 and 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. [2] O. Anza Hafsa and J. P. Mandallena, Interchange of infimum and integral, Calc. Var. Partial Differential Equations, 18 (2003), 433-449. [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. [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. [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. [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. [7] A. L. Bessoud, F. Krasucki and G. Michaille, Multi-materials with strong interface: Variational modelings, Asympto. Anal., 61 (2009), 1-19. [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. [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. [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. [11] C. Castaing and M. Valadier, "Convex Analysis and Measurable Multifunctions," Lect. Notes Math., 580, Springer-Verlag, Berlin-New York, 1977. [12] B. Dacorogna, "Direct Methods in the Calculus of Variations," Appl. Math. Sciences, 78, Springer-Verlag, Berlin, 1989. [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. [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. [15] E. Mascolo and L. Migliaccio, Relaxation methods in control theory, Appl. Math. Optim., 20 (1989), 97-103. doi: 10.1007/BF01447649. [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. [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. [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. [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.

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##### 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. [2] O. Anza Hafsa and J. P. Mandallena, Interchange of infimum and integral, Calc. Var. Partial Differential Equations, 18 (2003), 433-449. [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. [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. [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. [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. [7] A. L. Bessoud, F. Krasucki and G. Michaille, Multi-materials with strong interface: Variational modelings, Asympto. Anal., 61 (2009), 1-19. [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. [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. [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. [11] C. Castaing and M. Valadier, "Convex Analysis and Measurable Multifunctions," Lect. Notes Math., 580, Springer-Verlag, Berlin-New York, 1977. [12] B. Dacorogna, "Direct Methods in the Calculus of Variations," Appl. Math. Sciences, 78, Springer-Verlag, Berlin, 1989. [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. [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. [15] E. Mascolo and L. Migliaccio, Relaxation methods in control theory, Appl. Math. Optim., 20 (1989), 97-103. doi: 10.1007/BF01447649. [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. [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. [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. [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.
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