Effective energy integral functionals for thin films with three dimensional bending moment in the Orlicz-Sobolev space setting

Włodzimierz Laskowski ; Hong Thai Nguyen

Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 36 (2016), p. 7-31 / Harvested from The Polish Digital Mathematics Library

Résumé

In this paper we consider an elastic thin film ω ⊂ ℝ² with the bending moment depending also on the third thickness variable. The effective energy functional defined on the Orlicz-Sobolev space over ω is described by Γ-convergence and 3D-2D dimension reduction techniques. Then we prove the existence of minimizers of the film energy functional. These results are proved in the case when the energy density function has the growth prescribed by an Orlicz convex function M. Here M is assumed to be non-power-growth-type and to satisfy the conditions Δ₂ and ∇₂.

Détail

Publié le : 2016-01-01

Zbl 1307.49014

EUDML-ID : urn:eudml:doc:286877

Bibliographie

[000] [1] E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations, Arch. Ration. Mech. Anal. 86 (1984), 125-145. doi: 10.1007/BF00275731 | Zbl 0565.49010

[001] [2] R.A. Adams and J.J.F. Fournier, Sobolev Spaces, 2 ed. (Academic Press, 2003). | Zbl 1098.46001

[002] [3] A. Alberico and A. Cianchi, Differentiability properties of Orlicz-Sobolev functions, Ark. Mat. 43 (2005), 1-28. doi: 10.1007/BF02383608 | Zbl 1119.46030

[003] [4] G. Bouchitté, I. Fonseca and M.L. Mascarenhas, Bending moment in membrane theory, J. Elasticity 73 (2004), 75-99. doi: 10.1023/B:ELAS.0000029996.20973.92 | Zbl 1059.74034

[004] [5] G. Bouchitté, I. Fonseca and M.L. Mascarenhas, The Cosserat Vector In Membrane Theory: A Variational Approach, J. Convex Anal. 16 (2009), 351-365. | Zbl 1179.35015

[005] [6] A. Braides, I. Fonseca and G. Francfort, 3D-2D-asymptotic analysis for inhomogeneous thin films, Indiana Univ. Math. J. 49 (2000), 1367-1404. doi: 10.1512/iumj.2000.49.1822 | Zbl 0987.35020

[006] [7] D. Breit, B. Stroffolini and A. Verde, A general regularity theorem for functionals with ϕ-growth, J. Math. Anal. Appl. 383 (2011), 226-233. doi: 10.1016/j.jmaa.2011.05.012 | Zbl 1218.49043

[007] [8] B. Dacorogna, Direct Methods in the Calculus of Variations, 2nd revised edition (Springer, Berlin, 2008). | Zbl 1140.49001

[008] [9] G. Dal Maso, An Introduction to Γ-Convergence (Birkhäuser, Boston, 1993). doi: 10.1007/978-1-4612-0327-8

[009] [10] T.K. Donaldson and N.S. Trudinger, Orlicz-Sobolev spaces and imbedding theorems, J. Funct. Anal. 8 (1971), 52-75. doi: 10.1016/0022-1236(71)90018-8 | Zbl 0216.15702

[010] [11] N. Dunford and J.T. Schwartz, Linear Operators, Part I: General Theory (Interscience, New York, 1957). | Zbl 0084.10402

[011] [12] A. Fiorenza and M. Krbec, Indices of Orlicz spaces and some applications, Comment. Math. Univ. Carolinae 38 (1997), 433-451. | Zbl 0937.46023

[012] [13] M. Focardi, Semicontinuity of vectorial functionals in Orlicz-Sobolev spaces, Rend. Istit. Mat. Univ. Trieste 29 (1997), 141-161. | Zbl 0924.49011

[013] [14] I. Fonseca, S. Müller and P.Pedregal, Analysis of concentration and oscillation effects generated by gradients, S, IAM J. Math. Anal. 29 (1998), 736-756. doi: 10.1137/S0036141096306534 | Zbl 0920.49009

[014] [15] A. Fougères, Théoremès de trace et de prolongement dans les espaces de Sobolev et Sobolev-Orlicz, C.R. Acad. Sci. Paris Sér. A-B 274 (1972), A181-A184. | Zbl 0226.46036

[015] [16] 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. doi: 10.1007/s00205-005-0400-7 | Zbl 1100.74039

[016] [17] M. García-Huidobro, V.K. Le, R. Manásevich and K. Schmitt, On principal eigenvalues for quasilinear elliptic differential operators: an Orlicz-Sobolev space setting, NoDEA Nonlinear Differential Equations Appl. 6 (1999), 207-225. doi: 10.1007/s000300050073 | Zbl 0936.35067

[017] [18] J.-P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coeffcients, Trans. Am. Math. Soc. 190 (1974), 163-205. doi: 10.1090/S0002-9947-1974-0342854-2 | Zbl 0239.35045

[018] [19] H. Hudzik, The problems of separability, duality, reflexivity and of comparison for generalized Orlicz-Sobolev spaces W_k^M(Ω), Comment. Math. Prace Mat. 21 (1980), 315-324.

[019] [20] A. Kamińska and B. Turett, Type and cotype in Musielak-Orlicz spaces, in: Geometry of Banach Spaces, London Mathematical Society Lecture Note Series 158 (1991) Cambridge University Press, 165-180. doi: 10.1017/CBO9780511662317.015 | Zbl 0770.46009

[020] [21] A. Kałamajska and M. Krbec, Traces of Orlicz-Sobolev functions under general growth restrictions, Mathematische Nachrichten 286 (2013), 730-742. doi: 10.1002/mana.201100185 | Zbl 1276.46026

[021] [22] V.S. Klimov, On imbedding theorems for anisotropic classes of functions, Mathematics of the USSR-Sbornik 55 (1986), 195-205. doi: 10.1070/SM1986v055n01ABEH002999 | Zbl 0603.46039

[022] [23] M.A. Krasnosel’skii and Ya.B. Rutickii, Convex Functions and Orlicz Spaces (P. Noordhoof LTD., Groningen, 1961).

[023] [24] W. Laskowski and H.T. Nguyen, Effective energy integral functionals for thin films in the Orlicz-Sobolev space setting, Demonstratio Math. 46 (2013), 589-608. | Zbl 1288.49005

[024] [25] W. Laskowski and H.T. Nguyen, Effective energy integral functionals for thin films with bending moment in the Orlicz-Sobolev space setting, Banach Center Publ., Polish Acad. Sci., Warsaw 102 (2014), 143-167. | Zbl 1307.49014

[025] [26] H. Le Dret and A. Raoult, Le modèle de membrane non linéaire comme limite variationnelle de l’élasticité non linéaire tridimensionnelle, C.R. Acad. Sci. Paris Sér. I Math. 317 (1993), 221-226.

[026] [27] H. Le Dret and A. Raoult, The nonlinear membrane model as variational limit of nonlinear threedimensional elasticity, J. Math. Pures Appl. 74 (1995), 549-578. | Zbl 0847.73025

[027] [28] L. Maligranda, Indices and interpolation, Dissertationes Math. (Rozprawy Mat.) 234 (1985), 1-49.

[028] [29] L. Maligranda, Orlicz Spaces and Interpolation, Seminars in Mathematics 5, Campinas SP, Univ. of Campinas (Brazil, 1989).

[029] [30] V. Mazja, Sobolev Spaces (Springer, New York, 1985). doi: 10.1007/978-3-662-09922-3

[030] [31] M.G. Mora and L. Scardia, Convergence of equilibria of thin elastic plates under physical growth conditions for the energy density, J. Diff. Equ. 252 (2012), 35-55. doi: 10.1016/j.jde.2011.09.009 | Zbl 1291.74128

[031] [32] C.B.Jr. Morrey, Multiple Integrals in the Calculus of Variations (Classics in Mathematics) (Springer, New York, 1966, Reprint in 2008). | Zbl 0142.38701

[032] [33] J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034 (Springer, Berlin, 1983).

[033] [34] P. Pedregal, Parametrized Measures and Variational Principles (Birkhäuser, Basel, 1997). doi: 10.1007/978-3-0348-8886-8 | Zbl 0879.49017

[034] [35] R. Płuciennik, S. Tian and Y. Wang, Non-convex integral functionals on MusielakOrlicz spaces, Comment. Math. Prace Mat. 30 (1990), 113-123. | Zbl 0762.46016