Dimension reduction for functionals on solenoidal vector fields

 Access to full text

1. [1] A. Alama, L. Bronsard and B. Galvão-Sousa, Thin film limits for Ginzburg-Landau for strong applied magnetic fields. SIAM J. Math. Anal.42 (2010) 97–124. Zbl1210.35242MR2596547
2. [2] N. Ansini and A. Garroni, Γ-convergence of functionals on divergence-free fields. ESAIM : COCV 13 (2007) 809–828. Zbl1127.49011MR2351405
3. [3] J.M. Ball, A version of the fundamental theorem for young measures, in PDEs and continuum models of phase transitions – Proceedings of an NSF-CNRS joint seminar held in Nice, France, January 18–22, 1988, Lect. Notes Phys. 344, M. Rascle, D. Serre and M. Slemrod Eds., Springer, Berlin etc. (1989) 207–215. Zbl0991.49500MR1036070
4. [4] A. Braides, Γ-convergence for beginners, Oxford Lecture Series in Mathematics and its Applications 22. Oxford University Press, Oxford (2002). Zbl1198.49001MR1968440
5. [5] A. Braides, I. Fonseca and G. Leoni, A-quasiconvexity : Relaxation and homogenization. ESAIM : COCV 5 (2000) 539–577. Zbl0971.35010MR1799330
6. [6] A. Contreras and P. Sternberg, Γ-convergence and the emergence of vortices for Ginzburg-Landau on thin shells and manifolds. Calc. Var. Partial Differ. Equ.38 (2010) 243–274. Zbl1193.49053MR2610532
7. [7] G. Dal Maso, An introduction to Γ-convergence, Progress in Nonlinear Differential Equations and their Applications 8. Birkhäuser, Basel (1993). Zbl0816.49001MR1201152
8. [8] G. Dal Maso, I. Fonseca and G. Leoni, Nonlocal character of the reduced theory of thin films with higher order perturbations. Adv. Calc. Var.3 (2010) 287–319. Zbl1195.49019MR2660690
9. [9] E. De Giorgi and G. Dal Maso, Gamma-convergence and calculus of variations, in Mathematical theories of optimization, Proc. Conf., Genova, 1981, Lect. Notes Math. 979 (1983) 121–143. Zbl0511.49007MR713808
10. [10] E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 58 (1975) 842–850. Zbl0339.49005MR448194
11. [11] I. Ekeland and R. Temam, Convex analysis and variational problems, Studies in Mathematics and its Applications 1. North-Holland Publishing Company, Amsterdam, Oxford (1976). Zbl0322.90046MR463994
12. [12] I. Fonseca and S. Krömer, Multiple integrals under differential constraints : two-scale convergence and homogenization. Indiana Univ. Math. J.59 (2010) 427–457. Zbl1198.49011MR2648074
13. [13] I. Fonseca and G. Leoni, Modern methods in the calculus of variations. Lp spaces. Springer Monographs in Mathematics, New York, Springer (2007). Zbl1153.49001MR2341508
14. [14] I. Fonseca and S. Müller, 𝒜-quasiconvexity, lower semicontinuity, and Young measures. SIAM J. Math. Anal. 30 (1999) 1355–1390. Zbl0940.49014MR1718306
15. [15] G. Friesecke, R.D. James and S. Müller, A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence. Arch. Ration. Mech. Anal.180 (2006) 183–236. Zbl1100.74039MR2210909
16. [16] A. Garroni and V. Nesi, Rigidity and lack of rigidity for solenoidal matrix fields. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 460 (2004) 1789–1806. Zbl1108.74050MR2067561
17. [17] E. Giusti, Direct methods in the calculus of variations. World Scientific, Singapore (2003). Zbl1028.49001MR1962933
18. [18] H. Le Dret and A. Raoult, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl., IX. Sér. 74 (1995) 549–578. Zbl0847.73025MR1365259
19. [19] H. Le Dret and A. Raoult, The membrane shell model in nonlinear elasticity : A variational asymptotic derivation. J. Nonlinear Sci.6 (1996) 59–84. Zbl0844.73045MR1375820
20. [20] H. Le Dret and A. Raoult, Variational convergence for nonlinear shell models with directors and related semicontinuity and relaxation results. Arch. Ration. Mech. Anal.154 (2000) 101–134. Zbl0969.74040MR1784962
21. [21] J. Lee, P.F.X. Müller and S. Müller, Compensated compactness, separately convex functions and interpolatory estimates between Riesz transforms and Haar projections. Preprint MPI-MIS 7/2008. Zbl1230.49009
22. [22] M. Lewicka and R. Pakzad, The infinite hierarchy of elastic shell models : some recent results and a conjecture. Fields Institute Communications (to appear). Zbl1263.74035
23. [23] M. Lewicka, L. Mahadevan and R. Pakzad, The Föppl-von Kármán equations for plates with incompatible strains. Proc. Roy. Soc. A (to appear). Zbl1219.74027
24. [24] S. Müller, Rank-one convexity implies quasiconvexity on diagonal matrices. Int. Math. Res. Not.1999 (1999) 1087–1095. Zbl1055.49506MR1728018
25. [25] S. Müller, Variational models for microstructure and phase transisions, in Calculus of variations and geometric evolution problems – Lectures given at the 2nd session of the Centro Internazionale Matematico Estivo (CIME), Cetraro, Italy, June 15–22, 1996, Lect. Notes Math. 1713, S. Hildebrandt Ed., Springer, Berlin (1999) 85–210. Zbl0968.74050MR1731640
26. [26] F. Murat, Compacité par compensation : condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 8 (1981) 69–102. Zbl0464.46034MR616901
27. [27] M. Palombaro, Rank-(n-1) convexity and quasiconvexity for divergence free fields. Adv. Calc. Var3 (2010) 279–285. Zbl1195.49022MR2660689
28. [28] M. Palombaro and V.P. Smyshlyaev, Relaxation of three solenoidal wells and characterization of extremal three-phase H-measures. Arch. Ration. Mech. Anal.194 (2009) 775–822. Zbl1176.49022MR2563624
29. [29] P. Pedregal, Parametrized measures and variational principles, Progress in Nonlinear Differential Equations and their Applications 30. Birkhäuser, Basel (1997). Zbl0879.49017MR1452107
30. [30] R.T. Rockafellar, Convex analysis. Princeton University Press, Princeton, NJ (1970). Zbl0932.90001MR274683
31. [31] L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear analysis and mechanics : Heriot-Watt Symp. 4, Edinburgh, Res. Notes Math. 39 (1979) 136–212. Zbl0437.35004MR584398