Publication Cover
Applicable Analysis
An International Journal
Volume 98, 2019 - Issue 16
228
Views
6
CrossRef citations to date
0
Altmetric
Articles

Biharmonic equation in a highly oscillating domain and homogenization of an associated control problem

&
Pages 2783-2801 | Received 28 Apr 2017, Accepted 22 Apr 2018, Published online: 31 May 2018

References

  • Amirat Y, Bodart O, De Maio U, et al. Asymptotic approximation of the solution of the Laplace equation in a domain with highly oscillating boundary. SIAM J Math Anal. 2004;35:1598–1616. doi: 10.1137/S0036141003414877
  • Nandakumaran AK, Prakash R, Sardar BC. Periodic controls in an oscillating domain: controls via unfolding and homogenization. SIAM J Control Optim. 2015;53(5):3245–3269. doi: 10.1137/140994575
  • Nandakumaran AK, Prakash R, Sardar BC. Asymptotic analysis of Neumann periodic optimal boundary control problem. Math Methods Appl Sci. 2016;39(15):4354–4374. doi: 10.1002/mma.3865
  • Cioranescu D, Damlamian A, Griso G. Periodic unfolding and homogenization. C R Math Acad Sci Paris. 2002;335(1):99–104. doi: 10.1016/S1631-073X(02)02429-9
  • Blanchard D, Gaudiello A, Griso G. Junction of a periodic family of elastic rods with a 3d plate, Part I. J Math Pures Appl. 2007;88(1):1–33. doi: 10.1016/j.matpur.2007.04.005
  • Blanchard D, Gaudiello A, Griso G. Junction of a periodic family of elastic rods with a thin plate. Part II. J Math Pures Appl. 2007;88(2):149–190. doi: 10.1016/j.matpur.2007.04.004
  • Aiyappan S, Nandakumaran AK. Optimal control problem in a domain with branched structure and homogenization. Math Methods Appl Sci. 2017;40:3173–3189. doi: 10.1002/mma.4231
  • Cioranescu D, Damlamian A, Griso G. The periodic unfolding method in homogenization. SIAM J Math Anal. 2008;40(4):1585–1620. doi: 10.1137/080713148
  • Cioranescu D, Damlamian A, Donato P, et al. The periodic unfolding method in domains with holes. SIAM J Math Anal. 2012;44(2):718–760. doi: 10.1137/100817942
  • Cioranescu D, Donato P, Zaki R. The periodic unfolding method in perforated domains and applications to Robin problems. Multiple scales problems in biomathematics, mechanics, physics and numerics; Gakktosho, Tokyo; 2009. p. 37–66 (GAKUTO Internat. Ser. Math. Sci. Appl.; 31).
  • Damlamian A, Pettersson K. Homogenization of oscillating boundaries. Discr Contin Dyn Syst. 2009;23(1–2):197–219.
  • Nandakumaran AK, Prakash R, Sardar BC. Homogenization of an optimal control problem in a domain with highly oscillating boundary using periodic unfolding method. Math Eng Sci Aerosp. 2013;4(3):281–303.
  • Durante T, Faella L, Perugia C. Homogenization and behaviour of optimal controls for the wave equation in domains with oscillating boundary. NoDEA Nonlinear Differ Equ Appl. 2007;14(5-6):455–489. doi: 10.1007/s00030-007-3043-6
  • De Maio U, Faella L, Perugia C. Optimal control for second-order linear evolution problem in a domain with oscillating boundary. Complex Var Elliptic Equ. 2015;60(10):1392–1410. doi: 10.1080/17476933.2015.1022169
  • De Maio U, Gaudiello A, Lefter C. Optimal control for a parabolic problem in a domain with highly oscillating boundary. Appl Anal. 2004;83(12):1245–1264. doi: 10.1080/00036810410001724670
  • Brizzi R, Chalot J-P. Boundary homogenization and Neumann boundary value problem. Ricerche Mat. 1997;46(2):341–387.
  • Gaudiello A. Asymptotic behaviour of non-homogeneous Neumann problems in domains with oscillating boundary. Ricerche Mat. 1994;43(2):239–292.
  • Gaudiello A, Hadiji R, Picard C. Homogenization of the Ginzburg-Landau equation in a domain with oscillating boundary. Commun Appl Anal. 2003;7(2-3):209–223.
  • Blanchard D, Gaudiello A. Homogenization of highly oscillating boundaries and reduction of dimension for a monotone problem. ESAIM. 2003;9:449–460. doi: 10.1051/cocv:2003022
  • Amirat Y, Bodart O, De Maio U, et al. Effective boundary condition for Stokes flow over a very rough surface. J Differ Equ. 2013;254(8):3395–3430. doi: 10.1016/j.jde.2013.01.024
  • D'Apice C, De Maio U, Kogut PI. Gap phenomenon in the homogenization of parabolic optimal control problems. IMA J Math Control Inform. 2008;25(4):461–489. doi: 10.1093/imamci/dnn010
  • Blanchard D, Gaudiello A, Mel'nyk TA. Boundary homogenization and reduction of dimension in a Kirchhoff-Love plate. SIAM J Math Anal. 2008;39(6):1764–1787. doi: 10.1137/070685919
  • Gaudiello A, Sili A. Homogenization of highly oscillating boundaries with strongly contrasting diffusivity. SIAM J Math Anal. 2015;47:1671–1692. doi: 10.1137/140987225
  • Chowdhury S, Gudi T, Nandakumaran AK. Error bounds for a Dirichlet boundary control problem based on energy spaces. Math Comput. 2017;86(305):1103–1126. doi: 10.1090/mcom/3125
  • Grisvard P. Elliptic problems in nonsmooth domains. Boston: Pitman; 1985.
  • Lions J-L. Optimal control of systems governed by partial differential equations. Die Grundlehren der mathematischen Wissenschaften Band. Vol. 170. New York: Springer-Verlag; 1971. Translated from the French by S. K. Mitter.
  • Tröltzsch F. Optimal control of partial differential equations. Graduate studies in mathematics. Vol. 112. Providence (RI): American Mathematical Society; 2010.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.