Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T16:05:20.495Z Has data issue: false hasContentIssue false
  • English
  • Français

Dimension reduction for functionals on solenoidal vector fields

Published online by Cambridge University Press:  02 December 2010

Stefan Krömer
Stefan Krömer*
Affiliation:
Universität zu Köln, Köln, Germany. skroemer@math.uni-koeln.de
Get access

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.

Keywords

Divergence-free fieldsGamma-convergencedimension reduction
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 1 , January 2012 , pp. 259 - 276
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alama, A., Bronsard, L. and Galvão-Sousa, B., Thin film limits for Ginzburg-Landau for strong applied magnetic fields. SIAM J. Math. Anal. 42 (2010) 97–124. Google Scholar
Ansini, N. and Garroni, A., Γ-convergence of functionals on divergence-free fields. ESAIM : COCV 13 (2007) 809–828. Google Scholar
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.
A. Braides, Γ -convergence for beginners, Oxford Lecture Series in Mathematics and its Applications 22. Oxford University Press, Oxford (2002).
Braides, A., Fonseca, I. and Leoni, G., A-quasiconvexity : Relaxation and homogenization. ESAIM : COCV 5 (2000) 539–577. Google Scholar
Contreras, A. and Sternberg, P., Γ-convergence and the emergence of vortices for Ginzburg-Landau on thin shells and manifolds. Calc. Var. Partial Differ. Equ. 38 (2010) 243–274. Google Scholar
G. Dal Maso, An introduction to Γ -convergence, Progress in Nonlinear Differential Equations and their Applications 8. Birkhäuser, Basel (1993).
Dal Maso, G., Fonseca, I. and Leoni, G., Nonlocal character of the reduced theory of thin films with higher order perturbations. Adv. Calc. Var. 3 (2010) 287–319. Google Scholar
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.
De Giorgi, E. and Franzoni, T., Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 58 (1975) 842–850. Google Scholar
I. Ekeland and R. Temam, Convex analysis and variational problems, Studies in Mathematics and its Applications 1. North-Holland Publishing Company, Amsterdam, Oxford (1976).
Fonseca, I. and Krömer, S., Multiple integrals under differential constraints : two-scale convergence and homogenization. Indiana Univ. Math. J. 59 (2010) 427–457. Google Scholar
I. Fonseca and G. Leoni, Modern methods in the calculus of variations. L p spaces. Springer Monographs in Mathematics, New York, Springer (2007).
Fonseca, I. and Müller, S., 𝒜-quasiconvexity, lower semicontinuity, and Young measures. SIAM J. Math. Anal. 30 (1999) 1355–1390. Google Scholar
Friesecke, G., James, R.D. and Müller, S., A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence. Arch. Ration. Mech. Anal. 180 (2006) 183–236. Google Scholar
Garroni, A. and Nesi, V., Rigidity and lack of rigidity for solenoidal matrix fields. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 460 (2004) 1789–1806. Google Scholar
E. Giusti, Direct methods in the calculus of variations. World Scientific, Singapore (2003).
Le Dret, H. and Raoult, A., The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl., IX. Sér. 74 (1995) 549–578. Google Scholar
Le Dret, H. and Raoult, A., The membrane shell model in nonlinear elasticity : A variational asymptotic derivation. J. Nonlinear Sci. 6 (1996) 59–84. Google Scholar
Le Dret, H. and Raoult, A., Variational convergence for nonlinear shell models with directors and related semicontinuity and relaxation results. Arch. Ration. Mech. Anal. 154 (2000) 101–134. Google Scholar
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.
M. Lewicka and R. Pakzad, The infinite hierarchy of elastic shell models : some recent results and a conjecture. Fields Institute Communications (to appear).
M. Lewicka, L. Mahadevan and R. Pakzad, The Föppl-von Kármán equations for plates with incompatible strains. Proc. Roy. Soc. A (to appear).
Müller, S., Rank-one convexity implies quasiconvexity on diagonal matrices. Int. Math. Res. Not. 1999 (1999) 1087–1095. Google Scholar
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.
Murat, F., 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. Google Scholar
Palombaro, M., Rank-(n-1) convexity and quasiconvexity for divergence free fields. Adv. Calc. Var 3 (2010) 279–285. Google Scholar
Palombaro, M. and Smyshlyaev, V.P., Relaxation of three solenoidal wells and characterization of extremal three-phase H-measures. Arch. Ration. Mech. Anal. 194 (2009) 775–822. Google Scholar
P. Pedregal, Parametrized measures and variational principles, Progress in Nonlinear Differential Equations and their Applications 30. Birkhäuser, Basel (1997).
R.T. Rockafellar, Convex analysis. Princeton University Press, Princeton, NJ (1970).
Tartar, L., 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. Google Scholar