Effective energy integral functionals for thin films in the Orlicz-Sobolev space setting

Laskowski, W.; Nguyen, H. T.

Artykuł - szczegóły

Tytuł artykułu

Effective energy integral functionals for thin films in the Orlicz-Sobolev space setting

Autorzy

Laskowski W. , Nguyen H. T.

Wybrane pełne teksty z tego czasopisma

http://www.degruyter.com/view/j/dema

Identyfikatory

Warianty tytułu

Języki publikacji

Abstrakty

We consider an elastic thin film as a bounded open subset ω of R2. First, the effective energy functional for the thin film ω is obtained, by Γ-convergence and 3D-2D dimension reduction techniques applied to the sequence of re-scaled total energy integral functionals of the elastic cylinders (…) as the thickness ε goes to 0. Then we prove the existence of minimizers of the film energy functional. These results are proved in the case when the energy density function for the elastic cylinders 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 (…) and (…) (that is equivalent to the reflexivity of Orlicz and Orlicz–Sobolev spaces generated by M). These results extend results of H. Le Dret and A. Raoult for the case M(t) = (…) for some (…).

Słowa kluczowe

Γ-convergence 3D-2D dimension reduction quasiconvex relaxation minimizers of variational integral functionals thin films elastic membrane effective energy integral functional bulk energy surface energy equilibrium state non-power-growth-type bulk energy density Orlicz space Orlicz-Sobolev space

gamma-zbieżność Γ-zbieżność zmniejszenie wymiaru 3D-2D cienkie warstwy membrana elastyczna masa powierzchniowa energia powierzchniowa stan równowagi przestrzeń Orlicza przestrzeń Orlicza-Soboleva

Wydawca

De Gruyter

Czasopismo

Demonstratio Mathematica

Rocznik

2013

Tom

Vol. 46, nr 3

Strony

585--604

Opis fizyczny

Bibliogr. 34 poz.

Twórcy

autor

Laskowski W.

wlaskowski@zut.edu.pl

School of Mathematics, West Pomeranian University of Technology, Al. Piastów 48, 70-311 Szczecin, Poland

autor

Nguyen H. T.

nguyenhthaimathuspl@yahoo.com

Institute of Mathematics, Szczecin University, ul. Wielkopolska 15, 70-451 Szczecin, Poland

Bibliografia

[1] E. Acerbi, N. Fusco, Semicontinuity problems in the calculus of variations, Arch. Ration. Mech. Anal. 86 (1984), 125–145.
[2] R. A. Adams, J. J. F. Fournier, Sobolev Spaces, 2 ed., Academic Press, New York, 2003.
[3] J. Appell, H. T. Nguyêñ, P. P. Zabrejko, Multivalued superposition operators in ideal spaces of vector functions II, Indag. Math. (N.S.) 2 (1991), 397–409.
[4] H. Attouch, G. Buttazzo, G. Michaille, Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization, SIAM, Philadelphia, 2006.
[5] G. Bouchitté, I. Fonseca, M. L. Mascarenhas, Bending moment in membrane theory, J. Elasticity 73 (2004), 75–99.
[6] A. Braides, A. Defranceschi, Homogenization of Multiple Integrals, Oxford University Press, Oxford, 1998.
[7] C. Castaing, M. Valadier, Convex Analysis and Measurable Mulifunctions, Lecture Notes in Math., vol. 580, Springer, Berlin, 1977.
[8] P. G. Ciarlet, Theory of Plates, Mathematical Elasticity, Elsevier, Amsterdam, 1997.
[9] B. Dacorogna, Direct Methods in the Calculus of Variations, Springer, Berlin, 2008 (2nd revised edition).
[10] G. Dal Maso, An Introduction to Γ-Convergence, Birkhäuser, Boston, 1993.
[11] T. K. Donaldson, N. S. Trudinger, Orlicz–Sobolev spaces and imbedding theorems, J. Funct. Anal. 8 (1971), 52–75.
[12] A. Fiorenza, 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, G. Leoni, Modern Methods in the Calculus of Variations: Lp Spaces, Springer, New York, 2007.
[15] A. Fougères, Théoremès de trace et de prolongement dans les espaces de Sobolev et Sobolev-Orlicz, C. R. Acad. Sci. Paris SÃľr. A-B 274 (1972), A181–A184.
[16] G. Friesecke, R. D. James, S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity, Comm. Pure Appl. Math. 55 (2002), 1461–1506.
[17] A. Fryszkowski, Fixed Point Theory for Decomposable Sets, Kluwer, Dordrecht, 2004.
[18] M. García-Huidobro, V. K. Le, R. Manásevich, K. Schmitt, On principal eigenvalues for quasilinear elliptic differential operators: an Orlicz–Sobolev space setting, NoDEA Nonlinear Differential Equations Appl. 6 (1999), 207–225.
[19] J.-P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Amer. Math. Soc. 190 (1974), 163–205.
[20] H. Hudzik, The problems of separability, duality, reflexivity and of comparison for generalized Orlicz–Sobolev spaces WkM(...), Comment. Math. Prace Mat. 21 (1980), 315–324.
[21] A. Kamińska, B. Turett, Type and cotype in Musielak–Orlicz spaces, in: Geometry of Banach Spaces, London Mathematical Society Lecture Note Series 158, Cambridge University Press (1991), pp. 165–180.
[22] L. V. Kantorovich, G. P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982.
[23] V. S. Klimov, On imbedding theorems for anisotropic classes of functions, Mathematics of the USSR-Sbornik 55 (1986), 195–205.
[24] M. A. Krasnosel’ski ̆ı, Ya. B. Ruticki ̆ı, Convex Functions and Orlicz Spaces, P. Noordhoof LTD., Groningen, 1961.
[25] H. Le Dret, A. Raoult, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl. 74 (1995), 549–578.
[26] A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, Cambridge Univ. Press, 4th ed., 1927.
[27] L. Maligranda, Indices and interpolation, Dissertationes Math. (Rozprawy Mat.) 234 (1985), 1–49.
[28] L. Maligranda, Orlicz Spaces and Interpolation, Seminars in Mathematics, vol. 5, Campinas SP, Univ. of Campinas, Brazil, 1989.
[29] V. Mazja, Sobolev Spaces, Springer, New York, 1985.
[30] C. B. Jr. Morrey, Multiple Integrals in the Calculus of Variations (Classics in Mathematics), Springer, New York, 1966 (Reprint in 2008).
[31] J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math., vol. 1034, Springer, Berlin, 1983.
[32] H. T. Nguyêñ, Semicontinuity and continuous selections for the multivalued superposition operator without assuming growth-type conditions, Studia Math. 163 (2004), 1–19.
[33] M. M. Rao, Z. D. Ren, Theory of Orlicz Spaces, Marcel Dekker, New York, 1991.
[34] M. M. Rao, Z. D. Ren, Applications of Orlicz Spaces, Marcel Dekker, New York, 2002.

Typ dokumentu

Bibliografia

Identyfikator YADDA

bwmeta1.element.baztech-dfb022f7-0e9a-492c-a8ca-2d5f899011a8