Correctors for the Neumann problem in thin domains with locally periodic oscillatory structure

Pereira, Marcone; Silva, Ricardo

doi:10.1090/qam/1388

Authors: Marcone C. Pereira and Ricardo P. Silva
Journal: Quart. Appl. Math. 73 (2015), 537-552
MSC (2010): Primary 35B25, 35B27, 74Kxx.
DOI: https://doi.org/10.1090/qam/1388
Published electronically: June 12, 2015
MathSciNet review: 3400758
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Abstract: In this paper we are concerned with convergence of solutions of the Poisson equation with Neumann boundary conditions in a two-dimensional thin domain exhibiting highly oscillatory behavior in part of its boundary. We deal with the resonant case in which the height, amplitude and period of the oscillations are all of the same order, which is given by a small parameter $\epsilon > 0$. Applying an appropriate corrector approach we get strong convergence when we replace the original solutions by a kind of first-order expansion through the Multiple-Scale Method.

References

Nadia Ansini and Andrea Braides, Homogenization of oscillating boundaries and applications to thin films, J. Anal. Math. 83 (2001), 151–182. MR 1828490, DOI https://doi.org/10.1007/BF02790260
Jose M. Arrieta, Spectral properties of Schroedinger operators under perturbations of the domain, ProQuest LLC, Ann Arbor, MI, 1991. Thesis (Ph.D.)–Georgia Institute of Technology. MR 2686891
José M. Arrieta and Simone M. Bruschi, Very rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a non uniformly Lipschitz deformation, Discrete Contin. Dyn. Syst. Ser. B 14 (2010), no. 2, 327–351. MR 2660861, DOI https://doi.org/10.3934/dcdsb.2010.14.327
José M. Arrieta, Alexandre N. Carvalho, Marcone C. Pereira, and Ricardo P. Silva, Semilinear parabolic problems in thin domains with a highly oscillatory boundary, Nonlinear Anal. 74 (2011), no. 15, 5111–5132. MR 2810693, DOI https://doi.org/10.1016/j.na.2011.05.006
José M. Arrieta and Marcone C. Pereira, Elliptic problems in thin domains with highly oscillating boundaries, Bol. Soc. Esp. Mat. Apl. SeMA 51 (2010), 17–24. MR 2675957, DOI https://doi.org/10.1007/bf03322549
José M. Arrieta and Marcone C. Pereira, Homogenization in a thin domain with an oscillatory boundary, J. Math. Pures Appl. (9) 96 (2011), no. 1, 29–57 (English, with English and French summaries). MR 2812711, DOI https://doi.org/10.1016/j.matpur.2011.02.003
José M. Arrieta and Marcone C. Pereira, The Neumann problem in thin domains with very highly oscillatory boundaries, J. Math. Anal. Appl. 404 (2013), no. 1, 86–104. MR 3061383, DOI https://doi.org/10.1016/j.jmaa.2013.02.061
Margarida Baía and Elvira Zappale, A note on the 3D-2D dimensional reduction of a micromagnetic thin film with nonhomogeneous profile, Appl. Anal. 86 (2007), no. 5, 555–575. MR 2332672, DOI https://doi.org/10.1080/00036810701233942
Alain Bensoussan, Jacques-Louis Lions, and George Papanicolaou, Asymptotic analysis for periodic structures, Studies in Mathematics and its Applications, vol. 5, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 503330
Dominique Blanchard, Antonio Gaudiello, and Georges Griso, Junction of a periodic family of elastic rods with a thin plate. II, J. Math. Pures Appl. (9) 88 (2007), no. 2, 149–190 (English, with English and French summaries). MR 2348767, DOI https://doi.org/10.1016/j.matpur.2007.04.004
Dominique Blanchard, Antonio Gaudiello, and Jacqueline Mossino, Highly oscillating boundaries and reduction of dimension: the critical case, Anal. Appl. (Singap.) 5 (2007), no. 2, 137–163. MR 2311706, DOI https://doi.org/10.1142/S0219530507000924
Mahdi Boukrouche and Ionel Ciuperca, Asymptotic behaviour of solutions of lubrication problem in a thin domain with a rough boundary and Tresca fluid-solid interface law, Quart. Appl. Math. 64 (2006), no. 3, 561–591. MR 2259055, DOI https://doi.org/10.1090/S0033-569X-06-01030-3
D. Caillerie, Thin elastic and periodic plates, Math. Methods Appl. Sci. 6 (1984), no. 2, 159–191. MR 751739, DOI https://doi.org/10.1002/mma.1670060112
J. Casado-Díaz, M. Luna-Laynez, and F. J. Suárez-Grau, Asymptotic behavior of the Navier-Stokes system in a thin domain with Navier condition on a slightly rough boundary, SIAM J. Math. Anal. 45 (2013), no. 3, 1641–1674. MR 3061467, DOI https://doi.org/10.1137/120873479
Gregory A. Chechkin, Avner Friedman, and Andrey L. Piatnitski, The boundary-value problem in domains with very rapidly oscillating boundary, J. Math. Anal. Appl. 231 (1999), no. 1, 213–234. MR 1676697, DOI https://doi.org/10.1006/jmaa.1998.6226
Gregory A. Chechkin and Andrey L. Piatnitski, Homogenization of boundary-value problem in a locally periodic perforated domain, Appl. Anal. 71 (1999), no. 1-4, 215–235. MR 1690100, DOI https://doi.org/10.1080/00036819908840714
Doina Cioranescu and Patrizia Donato, An introduction to homogenization, Oxford Lecture Series in Mathematics and its Applications, vol. 17, The Clarendon Press, Oxford University Press, New York, 1999. MR 1765047
Doina Cioranescu and Jeannine Saint Jean Paulin, Homogenization of reticulated structures, Applied Mathematical Sciences, vol. 136, Springer-Verlag, New York, 1999. MR 1676922
Gianni Dal Maso, An introduction to $\Gamma $-convergence, Progress in Nonlinear Differential Equations and their Applications, vol. 8, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1201152
Alain Damlamian and Klas Pettersson, Homogenization of oscillating boundaries, Discrete Contin. Dyn. Syst. 23 (2009), no. 1-2, 197–219. MR 2449075, DOI https://doi.org/10.3934/dcds.2009.23.197
J. Dieudonné, Foundations of modern analysis, Academic Press, New York-London, 1969. Enlarged and corrected printing; Pure and Applied Mathematics, Vol. 10-I. MR 0349288
Jack K. Hale and Geneviève Raugel, Reaction-diffusion equation on thin domains, J. Math. Pures Appl. (9) 71 (1992), no. 1, 33–95. MR 1151557
Dan Henry, Perturbation of the boundary in boundary-value problems of partial differential equations, London Mathematical Society Lecture Note Series, vol. 318, Cambridge University Press, Cambridge, 2005. With editorial assistance from Jack Hale and Antônio Luiz Pereira. MR 2160744
J.-L. Lions, Asymptotic expansions in perforated media with a periodic structure, Rocky Mountain J. Math. 10 (1980), no. 1, 125–140. MR 573867, DOI https://doi.org/10.1216/RMJ-1980-10-1-125
Alexandre L. Madureira and Frédéric Valentin, Asymptotics of the Poisson problem in domains with curved rough boundaries, SIAM J. Math. Anal. 38 (2006/07), no. 5, 1450–1473. MR 2286014, DOI https://doi.org/10.1137/050633895
T. A. Mel′nik and A. V. Popov, Asymptotic analysis of boundary value and spectral problems in thin perforated domains with rapidly changing thickness and different limiting dimensions, Mat. Sb. 203 (2012), no. 8, 97–124 (Russian, with Russian summary); English transl., Sb. Math. 203 (2012), no. 7-8, 1169–1195. MR 3024814, DOI https://doi.org/10.1070/SM2012v203n08ABEH004259

I. Pazanin and F. J. Suárez-Grau, Effects of rough boundary on the heat transfer in a thin-film flow, Comptes Rendus Mécanique 341 (8) (2012), 646-652.

Marcone C. Pereira and Ricardo P. Silva, Error estimates for a Neumann problem in highly oscillating thin domains, Discrete Contin. Dyn. Syst. 33 (2013), no. 2, 803–817. MR 2975135, DOI https://doi.org/10.3934/dcds.2013.33.803

Martino Prizzi and Krzysztof P. Rybakowski, The effect of domain squeezing upon the dynamics of reaction-diffusion equations, J. Differential Equations 173 (2001), no. 2, 271–320. MR 1834117, DOI https://doi.org/10.1006/jdeq.2000.3917

Geneviève Raugel, Dynamics of partial differential equations on thin domains, Dynamical systems (Montecatini Terme, 1994) Lecture Notes in Math., vol. 1609, Springer, Berlin, 1995, pp. 208–315. MR 1374110, DOI https://doi.org/10.1007/BFb0095241

Enrique Sánchez-Palencia, Nonhomogeneous media and vibration theory, Lecture Notes in Physics, vol. 127, Springer-Verlag, Berlin-New York, 1980. MR 578345

Luc Tartar, The general theory of homogenization, Lecture Notes of the Unione Matematica Italiana, vol. 7, Springer-Verlag, Berlin; UMI, Bologna, 2009. A personalized introduction. MR 2582099