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February  2013, 33(2): 803-817. doi: 10.3934/dcds.2013.33.803

Error estimates for a Neumann problem in highly oscillating thin domains

Marcone C. Pereira 1, and  Ricardo P. Silva 2,

1. 

Escola de Artes, Ciências e Humanidades, Universidade de São Paulo, São Paulo, SP, Brazil
2. 

Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, Rio Claro, SP, Brazil

Received  May 2011 Revised  July 2012 Published  September 2012

In this work we analyze the convergence of solutions of the Poisson equation with Neumann boundary conditions in a two-dimensional thin domain with highly oscillatory behavior. We consider the case where the height of the domain, amplitude and period of the oscillations are all of the same order, and given by a small parameter $\epsilon>0$. Using an appropriate corrector approach, we show strong convergence and give error estimates when we replace the original solutions by the first-order expansion through the Multiple-Scale Method.
Keywords: homogenization, correctors, error estimate., Thin domains.
Mathematics Subject Classification: Primary: 35B25, 35B27, 74Q0.
Citation: Marcone C. Pereira, Ricardo P. Silva. Error estimates for a Neumann problem in highly oscillating thin domains. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 803-817. doi: 10.3934/dcds.2013.33.803
References:
[1]

S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary value conditions I, Comm. Pure Appl. Math., 12 (1959), 623-727. doi: 10.1002/cpa.3160120405.  Google Scholar

[2]

Y. Amirat, O. Bodart, U. de Maio and A. Gaudiello, Asymptotic Approximation of the solution of the Laplace equation in a domain with highly oscillating boundary, SIAM J. Math. Anal., 35 (2004), 1598-1616. doi: 10.1137/S0036141003414877.  Google Scholar

[3]

J. M. Arrieta, "Spectral Properties of Schrödinger Operators Under Perturbations of the Domain,'' Ph. D. thesis, Georgia Inst. of Tech., 1991. Google Scholar

[4]

J. M. Arrieta, A. N. Carvalho, M. C. Pereira and R. P. Silva, Semilinear parabolic problems in thin domains with a highly oscillatory boundary, Nonlinear Analysis: Theory Methods and Appl., 74 (2011), 5111-5132.  Google Scholar

[5]

J. M. Arrieta and M. C. Pereira, Elliptic problems in thin domains with highly oscillating boundaries, Bol. Soc. Esp. Mat. Apl., 51 (2010), 17-25.  Google Scholar

[6]

J. M. Arrieta and M. C. Pereira, Homogenization in a thin domain with an oscillatory boundary, J. Math. Pures et Appl., 96 (2011), 29-57. doi: 10.1016/j.matpur.2011.02.003.  Google Scholar

[7]

A. Bensoussan, J. L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structures,'' North-Holland, 1978.  Google Scholar

[8]

R. Brizzi and J. P. Chalot, Boundary homogenization and Neumann boundary problem, Ricerce di Matematica XLVI, 2 (1997), 341-387.  Google Scholar

[9]

D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization, SIAM J. Math. Anal., 40 (2008), 1585-1620.  Google Scholar

[10]

D. Cioranescu and P. Donato, "An Introduction to Homogenization,'' Oxford lecture series in mathematics and its applications, 1999.  Google Scholar

[11]

D. Cioranescu and J. S. J. Paulin, Homogenization in open sets with holes, J. Math Anal. Appl., 71 (1979), 590-607. doi: 10.1016/0022-247X(79)90211-7.  Google Scholar

[12]

D. Cioranescu and J. S. J. Paulin, "Homogenization of Reticulated Structures,'' Springer-Verlag, 1980.  Google Scholar

[13]

A. Damlamian and K. Pettersson, Homogenization of oscillating boundaries, Discrete and Continuous Dynamical Systems, 23 (2009), 197-219.  Google Scholar

[14]

T. Elsken, Continuity of attractors for net-shaped thin domain, Topol. Meth. Nonlinear Analysis, 26 (2005), 315-354.  Google Scholar

[15]

J. K. Hale and G. Raugel, Reaction-diffusion equation on thin domains, J. Math. Pures et Appl., 9 (1992), 33-95.  Google Scholar

[16]

J. L. Lions, Asymptotic expansions in perforated media with a periodic structure, Rocky Mountain J. Math., 10 (1998), 125-140.  Google Scholar

[17]

D. N. Arnold and A. L. Madureira, Asymptotic estimates of hierarchical, Mathematical Models and Methods in Applied Sciences, 13 (2003), 1325-1350.  Google Scholar

[18]

A. L. Madureira and F. Valentin, Asymptotics of the Poisson Problem in domains with curved rough boundaries, SIAM Journal on Mathematical Analysis, 38 (2007), 1450-1473.  Google Scholar

[19]

T. A. Mel'nyk, Homogenization of the Poisson equation in a thick periodic junction, Z. Anal. Anwendungen, 18 (1999), 953-975.  Google Scholar

[20]

J. Nevard and J. B. Keller, Homogenization of rough boundaries and interfaces, SIAM J. Appl. Math., 57 (1997) 1660-1686.  Google Scholar

[21]

G. Panasenko, "Multi-scale Modelling for Structures and Composites,'' Springer-Verlag, Dordrecht, 2005.  Google Scholar

[22]

M. Prizzi and K. P. Rybakowski, The effect of domain squeezing upon the dynamics of reaction-diffusion equations, Journal of Diff. Equations, 173 (2001), 271-320.  Google Scholar

[23]

M. Prizzi and M. Rinaldi and K. P. Rybakowski, Curved thin domains and parabolic equations, Studia mathematica, 151 (2002), 109-140.  Google Scholar

[24]

G. Raugel, "Dynamics of Partial Differential Equations on Thin Domains," Lecture Notes in Math., Springer-Verlag, 1609, 1995.  Google Scholar

[25]

E. Sánchez-Palencia, "Non-Homogeneous Media and Vibration Theory,'' Lecture Notes in Phys., Springer-Verlag, 127, 1980.  Google Scholar

[26]

R. P. Silva, "Semicontinuidade Inferior de Atratores Para Problemas Parabólicos em Domínios Finos,'' Phd Thesis, Universidade de São Paulo, 2007. Google Scholar

[27]

L. Tartar, "The General Theory of Homogenization. A Personalized Introduction,'' Lecture Notes of the Un. Mat. Italiana, Springer-Verlag, Berlin, 7, 2009.  Google Scholar

show all references

References:
[1]

S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary value conditions I, Comm. Pure Appl. Math., 12 (1959), 623-727. doi: 10.1002/cpa.3160120405.  Google Scholar

[2]

Y. Amirat, O. Bodart, U. de Maio and A. Gaudiello, Asymptotic Approximation of the solution of the Laplace equation in a domain with highly oscillating boundary, SIAM J. Math. Anal., 35 (2004), 1598-1616. doi: 10.1137/S0036141003414877.  Google Scholar

[3]

J. M. Arrieta, "Spectral Properties of Schrödinger Operators Under Perturbations of the Domain,'' Ph. D. thesis, Georgia Inst. of Tech., 1991. Google Scholar

[4]

J. M. Arrieta, A. N. Carvalho, M. C. Pereira and R. P. Silva, Semilinear parabolic problems in thin domains with a highly oscillatory boundary, Nonlinear Analysis: Theory Methods and Appl., 74 (2011), 5111-5132.  Google Scholar

[5]

J. M. Arrieta and M. C. Pereira, Elliptic problems in thin domains with highly oscillating boundaries, Bol. Soc. Esp. Mat. Apl., 51 (2010), 17-25.  Google Scholar

[6]

J. M. Arrieta and M. C. Pereira, Homogenization in a thin domain with an oscillatory boundary, J. Math. Pures et Appl., 96 (2011), 29-57. doi: 10.1016/j.matpur.2011.02.003.  Google Scholar

[7]

A. Bensoussan, J. L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structures,'' North-Holland, 1978.  Google Scholar

[8]

R. Brizzi and J. P. Chalot, Boundary homogenization and Neumann boundary problem, Ricerce di Matematica XLVI, 2 (1997), 341-387.  Google Scholar

[9]

D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization, SIAM J. Math. Anal., 40 (2008), 1585-1620.  Google Scholar

[10]

D. Cioranescu and P. Donato, "An Introduction to Homogenization,'' Oxford lecture series in mathematics and its applications, 1999.  Google Scholar

[11]

D. Cioranescu and J. S. J. Paulin, Homogenization in open sets with holes, J. Math Anal. Appl., 71 (1979), 590-607. doi: 10.1016/0022-247X(79)90211-7.  Google Scholar

[12]

D. Cioranescu and J. S. J. Paulin, "Homogenization of Reticulated Structures,'' Springer-Verlag, 1980.  Google Scholar

[13]

A. Damlamian and K. Pettersson, Homogenization of oscillating boundaries, Discrete and Continuous Dynamical Systems, 23 (2009), 197-219.  Google Scholar

[14]

T. Elsken, Continuity of attractors for net-shaped thin domain, Topol. Meth. Nonlinear Analysis, 26 (2005), 315-354.  Google Scholar

[15]

J. K. Hale and G. Raugel, Reaction-diffusion equation on thin domains, J. Math. Pures et Appl., 9 (1992), 33-95.  Google Scholar

[16]

J. L. Lions, Asymptotic expansions in perforated media with a periodic structure, Rocky Mountain J. Math., 10 (1998), 125-140.  Google Scholar

[17]

D. N. Arnold and A. L. Madureira, Asymptotic estimates of hierarchical, Mathematical Models and Methods in Applied Sciences, 13 (2003), 1325-1350.  Google Scholar

[18]

A. L. Madureira and F. Valentin, Asymptotics of the Poisson Problem in domains with curved rough boundaries, SIAM Journal on Mathematical Analysis, 38 (2007), 1450-1473.  Google Scholar

[19]

T. A. Mel'nyk, Homogenization of the Poisson equation in a thick periodic junction, Z. Anal. Anwendungen, 18 (1999), 953-975.  Google Scholar

[20]

J. Nevard and J. B. Keller, Homogenization of rough boundaries and interfaces, SIAM J. Appl. Math., 57 (1997) 1660-1686.  Google Scholar

[21]

G. Panasenko, "Multi-scale Modelling for Structures and Composites,'' Springer-Verlag, Dordrecht, 2005.  Google Scholar

[22]

M. Prizzi and K. P. Rybakowski, The effect of domain squeezing upon the dynamics of reaction-diffusion equations, Journal of Diff. Equations, 173 (2001), 271-320.  Google Scholar

[23]

M. Prizzi and M. Rinaldi and K. P. Rybakowski, Curved thin domains and parabolic equations, Studia mathematica, 151 (2002), 109-140.  Google Scholar

[24]

G. Raugel, "Dynamics of Partial Differential Equations on Thin Domains," Lecture Notes in Math., Springer-Verlag, 1609, 1995.  Google Scholar

[25]

E. Sánchez-Palencia, "Non-Homogeneous Media and Vibration Theory,'' Lecture Notes in Phys., Springer-Verlag, 127, 1980.  Google Scholar

[26]

R. P. Silva, "Semicontinuidade Inferior de Atratores Para Problemas Parabólicos em Domínios Finos,'' Phd Thesis, Universidade de São Paulo, 2007. Google Scholar

[27]

L. Tartar, "The General Theory of Homogenization. A Personalized Introduction,'' Lecture Notes of the Un. Mat. Italiana, Springer-Verlag, Berlin, 7, 2009.  Google Scholar

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