Free Access
Issue
ESAIM: COCV
Volume 18, Number 2, April-June 2012
Page(s) 583 - 610
DOI https://doi.org/10.1051/cocv/2011107
Published online 26 August 2011
  1. D. Blanchard and A. Gaudiello, Homogenization of highly oscillating boundaries and reduction of dimension for a monotone problem. ESAIM : COCV 9 (2003) 449–460. [CrossRef] [EDP Sciences]
  2. D. Blanchard, A. Gaudiello and G. Griso, Junction of a periodic family of elastic rods with 3d plate. Part I. J. Math. Pures Appl. 88 (2007) 1–33 (Part I); 88 (2007) 149–190 (Part II). [CrossRef] [MathSciNet]
  3. D. Blanchard, A. Gaudiello and T.A. Mel’nyk, Boundary homogenization and reduction of dimention in a Kirchhoff-Love plate. SIAM J. Math. Anal. 39 (2008) 1764–1787. [CrossRef] [MathSciNet]
  4. G. Buttazzo, Γ-convergence and its applications to some problem in the calculus of variations, in School on Homogenization, ICTP, Trieste, 1993 (1994) 38–61.
  5. G. Buttazzo and G. Dal Maso, Γ-convergence and optimal control problems. J. Optim. Theory Appl. 38 (1982) 385–407. [CrossRef] [MathSciNet]
  6. G.A. Chechkin, T.P. Chechkina, C. D’Apice, U. De-Maio and T.A. Mel’nyk, Asymptotic analysis of a boundary value problem in a cascade thick junction with a random transmission zone. Appl. Anal. 88 (2009) 1543–1562. [CrossRef] [MathSciNet]
  7. U. De Maio, A. Gaudiello and C. Lefter, optimal control for a parabolic problem in a domain with highly oscillating boundary. Appl. Anal. 83 (2004) 1245–1264. [CrossRef] [MathSciNet]
  8. U. De Maio, T. Durante and T.A. Mel’nyk, Asymptotic approximation for the solution to the Robin problem in a thick multi-level junction. Math. Models Methods Appl. Sci. 15 (2005) 1897–1921. [CrossRef] [MathSciNet]
  9. Z. Denkowski and S. Mortola, Asymptotic behavior of optimal solutions to control problems for systems described by differential inclusions corresponding to partial differential equations. J. Optim. Theory Appl. 78 (1993) 365–391. [CrossRef] [MathSciNet]
  10. T. Durante and T.A. Mel’nyk, Asymptotic analysis of an optimal control problem involving a thick two-level junction with alternate type of controls. J. Optim. Theory Appl. 144 (2010) 205–225. [CrossRef] [MathSciNet]
  11. T. Durante, L. Faella and C. Perugia, Homogenization and behaviour of optimal controls for the wave equation in domains with oscillating boundary. Nonlinear Differ. Equ. Appl. 14 (2007) 455–489. [CrossRef]
  12. S. Kesavan and J. Saint Jean Paulin, Optimal control on perforated domains. J. Math. Anal. Appl. 229 (1999) 563–586. [CrossRef] [MathSciNet]
  13. Y.I. Lavrentovich, T.V. Knyzkova and V.V. Pidlisnyuk, The potential of application of new nanostructural materials for degradation of pesticides in water, in Proceedings of the 7th Int. HCH and Pesticides Forum Towards the establishment of an obsolete POPS/pecticides stockpile fund for Central and Eastern European countries and new independent states, Kyiv, Ukraine (2003) 167–169.
  14. M. Lenczner, Multiscale model for atomic force microscope array mechanical behavior. Appl. Phys. Lett. 90 (2007) 091908; doi : 10.1063/1.2710001. [CrossRef]
  15. J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag, Berlin (1971).
  16. S.E. Lyshevshi, Mems and Nems : Systems, Devices, and Structures. CRC Press, Boca Raton, FL (2002).
  17. T.A. Mel’nyk, Homogenization of the Poisson equation in a thick periodic junction. Z. f. Anal. Anwendungen 18 (1999) 953–975. [CrossRef]
  18. T.A. Mel’nyk, Homogenization of a perturbed parabolic problem in a thick periodic junction of type 3 : 2 : 1. Ukr. Math. J. 52 (2000) 1737–1749. [CrossRef]
  19. T.A. Mel’nyk, Homogenization of a boundary-value problem with a nonlinear boundary condition in a thick junction of type 3 : 2 : 1. Math. Models Meth. Appl. Sci. 31 (2008) 1005–1027. [CrossRef]
  20. T.A. Mel’nyk and G.A. Chechkin, Asymptotic analysis of boundary value problems in thick three-dimensional multi-level junctions. Math. Sb. 200 3 (2009) 49–74 (in Russian); English transl. : Sb. Math. 200 (2009) 357–383. [CrossRef]
  21. T.A. Mel’nyk and S.A. Nazarov, Asymptotic structure of the spectrum in the problem of harmonic oscillations of a hub with heavy spokes. Dokl. Akad. Nauk Russia 333 (1993) 13–15 (in Russian); English transl. : Russian Acad. Sci. Dokl. Math. 48 (1994) 28–32.
  22. T.A. Mel’nyk and S.A. Nazarov, Asymptotic structure of the spectrum of the Neumann problem in a thin comb-like domain. C.R. Acad Sci. Paris, Ser. 1 319 (1994) 1343–1348.
  23. T.A. Mel’nyk and S.A. Nazarov, Asymptotics of the Neumann spectral problem solution in a domain of thick comb type. Trudy Seminara imeni I.G. Petrovskogo 19 (1996) 138–173 (in Russian); English transl. : J. Math. Sci. 85 (1997) 2326–2346.
  24. T.A. Mel’nyk and D. Yu. Sadovyj, Homogenization of elliptic problems with alternating boundary conditions in a thick two-level junction of type 3 :2 :2. J. Math. Sci. 165 (2010) 67–90.
  25. T.A. Mel’nyk, Iu.A. Nakvasiuk and W.L. Wendland, Homogenization of the Signorini boundary-value problem in a thick junction and boundary integral equations for the homogenized problem. Math. Meth. Appl. Sci. 34 (2011) 758–775. [CrossRef]

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