Dimension reduction for functionals on solenoidal vector fields

Stefan Krömer

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 1, page 259-276
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Krömer, Stefan. "Dimension reduction for functionals on solenoidal vector fields." ESAIM: Control, Optimisation and Calculus of Variations 18.1 (2012): 259-276. <http://eudml.org/doc/221928>.

@article{Krömer2012,
abstract = {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. },
author = {Krömer, Stefan},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Divergence-free fields; Gamma-convergence; dimension reduction; divergence-free fields; -convergence},
language = {eng},
month = {2},
number = {1},
pages = {259-276},
publisher = {EDP Sciences},
title = {Dimension reduction for functionals on solenoidal vector fields},
url = {http://eudml.org/doc/221928},
volume = {18},
year = {2012},
}

TY - JOUR
AU - Krömer, Stefan
TI - Dimension reduction for functionals on solenoidal vector fields
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2012/2//
PB - EDP Sciences
VL - 18
IS - 1
SP - 259
EP - 276
AB - 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.
LA - eng
KW - Divergence-free fields; Gamma-convergence; dimension reduction; divergence-free fields; -convergence
UR - http://eudml.org/doc/221928
ER -

References

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  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.  Zbl1210.35242
  2. N. Ansini and A. Garroni, Γ-convergence of functionals on divergence-free fields. ESAIM : COCV13 (2007) 809–828.  Zbl1127.49011
  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.  
  4. A. Braides, Γ-convergence for beginners, Oxford Lecture Series in Mathematics and its Applications22. Oxford University Press, Oxford (2002).  
  5. A. Braides, I. Fonseca and G. Leoni, A-quasiconvexity : Relaxation and homogenization. ESAIM : COCV5 (2000) 539–577.  Zbl0971.35010
  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.  Zbl1193.49053
  7. G. Dal Maso, An introduction toΓ-convergence, Progress in Nonlinear Differential Equations and their Applications8. Birkhäuser, Basel (1993).  
  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.  Zbl1195.49019
  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.  
  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.  
  11. I. Ekeland and R. Temam, Convex analysis and variational problems, Studies in Mathematics and its Applications1. North-Holland Publishing Company, Amsterdam, Oxford (1976).  Zbl0322.90046
  12. I. Fonseca and S. Krömer, Multiple integrals under differential constraints : two-scale convergence and homogenization. Indiana Univ. Math. J.59 (2010) 427–457.  Zbl1198.49011
  13. I. Fonseca and G. Leoni, Modern methods in the calculus of variations.Lp spaces. Springer Monographs in Mathematics, New York, Springer (2007).  Zbl1153.49001
  14. I. Fonseca and S. Müller, 𝒜-quasiconvexity, lower semicontinuity, and Young measures. SIAM J. Math. Anal.30 (1999) 1355–1390.  Zbl0940.49014
  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.  Zbl1100.74039
  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.  Zbl1108.74050
  17. E. Giusti, Direct methods in the calculus of variations. World Scientific, Singapore (2003).  Zbl1028.49001
  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.  Zbl0847.73025
  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.  Zbl0844.73045
  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.  Zbl0969.74040
  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.  Zbl1230.49009
  22. M. Lewicka and R. Pakzad, The infinite hierarchy of elastic shell models : some recent results and a conjecture. Fields Institute Communications (to appear).  Zbl1263.74035
  23. M. Lewicka, L. Mahadevan and R. Pakzad, The Föppl-von Kármán equations for plates with incompatible strains. Proc. Roy. Soc. A (to appear).  Zbl1219.74027
  24. S. Müller, Rank-one convexity implies quasiconvexity on diagonal matrices. Int. Math. Res. Not.1999 (1999) 1087–1095.  Zbl1055.49506
  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.  
  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.  Zbl0464.46034
  27. M. Palombaro, Rank-(n-1) convexity and quasiconvexity for divergence free fields. Adv. Calc. Var3 (2010) 279–285.  Zbl1195.49022
  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.  Zbl1176.49022
  29. P. Pedregal, Parametrized measures and variational principles, Progress in Nonlinear Differential Equations and their Applications30. Birkhäuser, Basel (1997).  
  30. R.T. Rockafellar, Convex analysis. Princeton University Press, Princeton, NJ (1970).  Zbl0193.18401
  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.  

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