Dimension reduction for functionals on solenoidal vector fields
ESAIM: Control, Optimisation and Calculus of Variations, Volume 18 (2012) no. 1, pp. 259-276.

We study integral functionals constrained to divergence-free vector fields in Lp on a thin domain, under standard p-growth and coercivity assumptions, 1 < p < ∞. We prove that as the thickness of the domain goes to zero, the Gamma-limit with respect to weak convergence in Lp is always given by the associated functional with convexified energy density wherever it is finite. Remarkably, this happens despite the fact that relaxation of nonconvex functionals subject to the limiting constraint can give rise to a nonlocal functional as illustrated in an example.

DOI: 10.1051/cocv/2010051
Classification: 49J45,  35E99
Keywords: divergence-free fields, gamma-convergence, dimension reduction
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title = {Dimension reduction for functionals on solenoidal vector fields},
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Krömer, Stefan. Dimension reduction for functionals on solenoidal vector fields. ESAIM: Control, Optimisation and Calculus of Variations, Volume 18 (2012) no. 1, pp. 259-276. doi : 10.1051/cocv/2010051. http://www.numdam.org/articles/10.1051/cocv/2010051/

[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. | MR | Zbl

[2] N. Ansini and A. Garroni, Γ-convergence of functionals on divergence-free fields. ESAIM : COCV 13 (2007) 809-828. | Numdam | MR | Zbl

[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. | MR | Zbl

[4] A. Braides, Γ-convergence for beginners, Oxford Lecture Series in Mathematics and its Applications 22. Oxford University Press, Oxford (2002). | MR | Zbl

[5] A. Braides, I. Fonseca and G. Leoni, A-quasiconvexity : Relaxation and homogenization. ESAIM : COCV 5 (2000) 539-577. | Numdam | MR | Zbl

[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. | MR | Zbl

[7] G. Dal Maso, An introduction to Γ-convergence, Progress in Nonlinear Differential Equations and their Applications 8. Birkhäuser, Basel (1993). | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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). | MR | Zbl

[12] I. Fonseca and S. Krömer, Multiple integrals under differential constraints : two-scale convergence and homogenization. Indiana Univ. Math. J. 59 (2010) 427-457. | MR | Zbl

[13] I. Fonseca and G. Leoni, Modern methods in the calculus of variations. Lp spaces. Springer Monographs in Mathematics, New York, Springer (2007). | MR | Zbl

[14] I. Fonseca and S. Müller, &#x1d49c;-quasiconvexity, lower semicontinuity, and Young measures. SIAM J. Math. Anal. 30 (1999) 1355-1390. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[17] E. Giusti, Direct methods in the calculus of variations. World Scientific, Singapore (2003). | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | Zbl

[22] M. Lewicka and R. Pakzad, The infinite hierarchy of elastic shell models : some recent results and a conjecture. Fields Institute Communications (to appear). | Zbl

[23] M. Lewicka, L. Mahadevan and R. Pakzad, The Föppl-von Kármán equations for plates with incompatible strains. Proc. Roy. Soc. A (to appear). | Zbl

[24] S. Müller, Rank-one convexity implies quasiconvexity on diagonal matrices. Int. Math. Res. Not. 1999 (1999) 1087-1095. | MR | Zbl

[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. | MR | Zbl

[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. | Numdam | MR | Zbl

[27] M. Palombaro, Rank-(n-1) convexity and quasiconvexity for divergence free fields. Adv. Calc. Var 3 (2010) 279-285. | MR | Zbl

[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. | MR | Zbl

[29] P. Pedregal, Parametrized measures and variational principles, Progress in Nonlinear Differential Equations and their Applications 30. Birkhäuser, Basel (1997). | MR | Zbl

[30] R.T. Rockafellar, Convex analysis. Princeton University Press, Princeton, NJ (1970). | MR | Zbl

[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. | MR | Zbl

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