Dimension reduction for functionals on solenoidal vector fields
ESAIM: Control, Optimisation and Calculus of Variations, Tome 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 : https://doi.org/10.1051/cocv/2010051
Classification : 49J45,  35E99
Mots clés : divergence-free fields, gamma-convergence, dimension reduction
@article{COCV_2012__18_1_259_0,
     author = {Kr\"omer, Stefan},
     title = {Dimension reduction for functionals on solenoidal vector fields},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {259--276},
     publisher = {EDP-Sciences},
     volume = {18},
     number = {1},
     year = {2012},
     doi = {10.1051/cocv/2010051},
     zbl = {1251.49018},
     mrnumber = {2887935},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2010051/}
}
TY  - JOUR
AU  - Krömer, Stefan
TI  - Dimension reduction for functionals on solenoidal vector fields
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2012
DA  - 2012///
SP  - 259
EP  - 276
VL  - 18
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2010051/
UR  - https://zbmath.org/?q=an%3A1251.49018
UR  - https://www.ams.org/mathscinet-getitem?mr=2887935
UR  - https://doi.org/10.1051/cocv/2010051
DO  - 10.1051/cocv/2010051
LA  - en
ID  - COCV_2012__18_1_259_0
ER  - 
Krömer, Stefan. Dimension reduction for functionals on solenoidal vector fields. ESAIM: Control, Optimisation and Calculus of Variations, Tome 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 2596547 | Zbl 1210.35242

[2] N. Ansini and A. Garroni, Γ-convergence of functionals on divergence-free fields. ESAIM : COCV 13 (2007) 809-828. | Numdam | MR 2351405 | Zbl 1127.49011

[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 1036070 | Zbl 0991.49500

[4] A. Braides, Γ-convergence for beginners, Oxford Lecture Series in Mathematics and its Applications 22. Oxford University Press, Oxford (2002). | MR 1968440 | Zbl 1198.49001

[5] A. Braides, I. Fonseca and G. Leoni, A-quasiconvexity : Relaxation and homogenization. ESAIM : COCV 5 (2000) 539-577. | Numdam | MR 1799330 | Zbl 0971.35010

[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 2610532 | Zbl 1193.49053

[7] G. Dal Maso, An introduction to Γ-convergence, Progress in Nonlinear Differential Equations and their Applications 8. Birkhäuser, Basel (1993). | MR 1201152 | Zbl 0816.49001

[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 2660690 | Zbl 1195.49019

[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 713808 | Zbl 0511.49007

[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 448194 | Zbl 0339.49005

[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 463994 | Zbl 0322.90046

[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 2648074 | Zbl 1198.49011

[13] I. Fonseca and G. Leoni, Modern methods in the calculus of variations. Lp spaces. Springer Monographs in Mathematics, New York, Springer (2007). | MR 2341508 | Zbl 1153.49001

[14] I. Fonseca and S. Müller, 𝒜-quasiconvexity, lower semicontinuity, and Young measures. SIAM J. Math. Anal. 30 (1999) 1355-1390. | MR 1718306 | Zbl 0940.49014

[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 2210909 | Zbl 1100.74039

[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 2067561 | Zbl 1108.74050

[17] E. Giusti, Direct methods in the calculus of variations. World Scientific, Singapore (2003). | MR 1962933 | Zbl 1028.49001

[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 1365259 | Zbl 0847.73025

[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 1375820 | Zbl 0844.73045

[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 1784962 | Zbl 0969.74040

[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 1230.49009

[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 1263.74035

[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 1219.74027

[24] S. Müller, Rank-one convexity implies quasiconvexity on diagonal matrices. Int. Math. Res. Not. 1999 (1999) 1087-1095. | MR 1728018 | Zbl 1055.49506

[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 1731640 | Zbl 0968.74050

[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 616901 | Zbl 0464.46034

[27] M. Palombaro, Rank-(n-1) convexity and quasiconvexity for divergence free fields. Adv. Calc. Var 3 (2010) 279-285. | MR 2660689 | Zbl 1195.49022

[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 2563624 | Zbl 1176.49022

[29] P. Pedregal, Parametrized measures and variational principles, Progress in Nonlinear Differential Equations and their Applications 30. Birkhäuser, Basel (1997). | MR 1452107 | Zbl 0879.49017

[30] R.T. Rockafellar, Convex analysis. Princeton University Press, Princeton, NJ (1970). | MR 274683 | Zbl 0932.90001

[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 584398 | Zbl 0437.35004

Cité par Sources :