Download PDF - Effective energy integral functionals for thin films with three dimensional bending moment in the Orlicz-Sobolev space settingAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

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

Authors 1, 2

Affiliations

  1. School of Mathematics, West Pomeranian University of Technology, Al. Piastów 48, 70-311 Szczecin, Poland
  2. Institute of Mathematics, Szczecin University, ul. Wielkopolska 15, 70-451 Szczecin, Poland

Abstract

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.

Keywords

ENG
Γ-convergence, 3D-2D dimension reduction, quasiconvex relaxation, minimizers of variational integral functionals, thin films, elastic membranes, effective energy integral functional, bulk and surface energy, equilibrium states of the film, non-power-growth-type bulk energy density, reflexive Orlicz and Orlicz-Sobolev spaces

Bibliography

  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
  2. R.A. Adams and J.J.F. Fournier, Sobolev Spaces, 2 ed. (Academic Press, 2003).
  3. A. Alberico and A. Cianchi, Differentiability properties of Orlicz-Sobolev functions, Ark. Mat. 43 (2005), 1-28. doi: 10.1007/BF02383608
  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
  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.
  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
  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
  8. B. Dacorogna, Direct Methods in the Calculus of Variations, 2nd revised edition (Springer, Berlin, 2008).
  9. G. Dal Maso, An Introduction to Γ-Convergence (Birkhäuser, Boston, 1993). doi: 10.1007/978-1-4612-0327-8
  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
  11. N. Dunford and J.T. Schwartz, Linear Operators, Part I: General Theory (Interscience, New York, 1957).
  12. A. Fiorenza and M. Krbec, Indices of Orlicz spaces and some applications, Comment. Math. Univ. Carolinae 38 (1997), 433-451.
  13. M. Focardi, Semicontinuity of vectorial functionals in Orlicz-Sobolev spaces, Rend. Istit. Mat. Univ. Trieste 29 (1997), 141-161.
  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
  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.
  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
  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
  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
  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.
  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
  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
  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
  23. M.A. Krasnosel’skii and Ya.B. Rutickii, Convex Functions and Orlicz Spaces (P. Noordhoof LTD., Groningen, 1961).
  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.
  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.
  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.
  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.
  28. L. Maligranda, Indices and interpolation, Dissertationes Math. (Rozprawy Mat.) 234 (1985), 1-49.
  29. L. Maligranda, Orlicz Spaces and Interpolation, Seminars in Mathematics 5, Campinas SP, Univ. of Campinas (Brazil, 1989).
  30. V. Mazja, Sobolev Spaces (Springer, New York, 1985). doi: 10.1007/978-3-662-09922-3
  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
  32. C.B.Jr. Morrey, Multiple Integrals in the Calculus of Variations (Classics in Mathematics) (Springer, New York, 1966, Reprint in 2008).
  33. J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034 (Springer, Berlin, 1983).
  34. P. Pedregal, Parametrized Measures and Variational Principles (Birkhäuser, Basel, 1997). doi: 10.1007/978-3-0348-8886-8
  35. R. Płuciennik, S. Tian and Y. Wang, Non-convex integral functionals on MusielakOrlicz spaces, Comment. Math. Prace Mat. 30 (1990), 113-123.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
2016/36/1
Export to PBN, Opens in new tab
Pages:
7-31
Main language of publication
English
Received
2015-09-16
Accepted
2016-01-26
Published
2016

DOI: 10.7151/dmdico.1179

Exact and natural sciences