Pointwise estimate for elliptic equations in periodic perforated domains

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September 2015, 14(5): 1961-1986. doi: 10.3934/cpaa.2015.14.1961

Pointwise estimate for elliptic equations in periodic perforated domains

Li-Ming Yeh ^1,

1.	Department of Applied Mathematics, National Chiao Tung University, Hsinchu, 30050, Taiwan

Received November 2014 Revised February 2015 Published June 2015

Abstract
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Pointwise estimate for the solutions of elliptic equations in periodic perforated domains is concerned. Let $\epsilon$ denote the size ratio of the period of a periodic perforated domain to the whole domain. It is known that even if the given functions of the elliptic equations are bounded uniformly in $\epsilon$, the $C^{1,\alpha}$ norm and the $W^{2,p}$ norm of the elliptic solutions may not be bounded uniformly in $\epsilon$. It is also known that when $\epsilon$ closes to $0$, the elliptic solutions in the periodic perforated domains approach a solution of some homogenized elliptic equation. In this work, the Hölder uniform bound in $\epsilon$ and the Lipschitz uniform bound in $\epsilon$ for the elliptic solutions in perforated domains are proved. The $L^\infty$ and the Lipschitz convergence estimates for the difference between the elliptic solutions in the perforated domains and the solution of the homogenized elliptic equation are derived.

Keywords: Periodic perforated domains, homogenized elliptic equation, two-phase media..

Mathematics Subject Classification: Primary: 35J05, 35J15, 35J2.

Citation: Li-Ming Yeh. Pointwise estimate for elliptic equations in periodic perforated domains. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1961-1986. doi: 10.3934/cpaa.2015.14.1961

References:

[1]	E. Acerbi, V. Chiado Piat, G. Dal Maso and D. Percivale, An extension theorem from connected sets, and homogenization in general periodic domains, Nonlinear Analysis, 18 (1992), 481-496. doi: 10.1016/0362-546X(92)90015-7.
[2]	R. A. Adams, Sobolev Spaces, Academic Press, 1975.
[3]	Gregoire Allaire, Homogenization and two-scale convergence, SIAM I. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084.
[4]	R. C. Morgan and I. Babuska, An approach for constructing families of homogenized equations for periodic media. I: An integral representation and its consequences, SIAM J. Math. Anal., 22 (1991), 1-15. doi: 10.1137/0522001.
[5]	N. Bakhvalov and G. Panasenko, Homogenisation: Averaging Processes in Periodic Media: Mathematical Problems in the Mechanics of Composite Materials, Kluwer Academic Publishers, 1989. doi: 10.1007/978-94-009-2247-1.
[6]	B. Berkowitz, et al., Physical pictures of transport in heterogeneous media: Advection-dispersion, random walk, and fractional derivative formulations, Water resources Research, 38 (2002), 1191-1194.
[7]	Susanne C. Brenner and L. Ridgway Scott, The Mathematical Theory of Finite Element Methods, Springer, 2008. doi: 10.1007/978-0-387-75934-0.
[8]	Alain Bensoussan, Jacques-Louis Lions and George Papanicolaou, Asymptotic Analysis for Periodic Structures, Elsevier North-Holland, 1978.
[9]	Li-Qun Cao, Asymptotic expansions and numerical algorithms of eigenvalues and eigenfunctions of the Dirichlet problem for second order elliptic equations in perforated domains, Numerische Mathematik, 103 (2006), 11-45. doi: 10.1007/s00211-005-0668-4.
[10]	Philippe G. Ciarlet, The Finite Element Method for Elliptic Problems, Amsterdam: North-Holland, 1978.
[11]	D. Cioranescu, A. Damlamian, G. Grisoa and D. Onofrei, The periodic unfolding method for perforated domains and Neumann sieve models, J. Math. Pures Appl., 89 (2008), 248-277. doi: 10.1016/j.matpur.2007.12.008.
[12]	M. Giaquinta, Multiple integrals in the calculus of variations, Study 105, Annals of Math. Studies, Princeton Univ. Press., 1983.
[13]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, second edition, 1983. doi: 10.1007/978-3-642-61798-0.
[14]	Thomas Y. Hou, Xiao-Hui Wu and Zhiqiang Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comp., 68 (1999), 913-943. doi: 10.1090/S0025-5718-99-01077-7.
[15]	V.V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functions, Springer-Verlag, 1994. doi: 10.1007/978-3-642-84659-5.
[16]	Viviane Klein and Malgorzata Peszynska, Adaptive Double-Diffusion Model and Comparison to a Highly Heterogeneous Micro-Model, Journal of Applied Mathematics, (2012) (2012), 26 pages.
[17]	J. L. Lions, Asymptotic expansions in perforated media with a periodic structure, The Rocky Mountain J. Math., 10 (1980), 125-144. doi: 10.1216/RMJ-1980-10-1-125.
[18]	N. Neuss, W. Jäger and G. Wittum, Homogenization and multigrid, Computing, 66 (2001), 1-26. doi: 10.1007/s006070170036.
[19]	O. A. Oleinik, A. S. Shamaev and G. A. Tosifan, Mathematical Problems in Elasticity and Homogenization, North-Holland, Amsterdam, 1992.

show all references

References:

[1]	E. Acerbi, V. Chiado Piat, G. Dal Maso and D. Percivale, An extension theorem from connected sets, and homogenization in general periodic domains, Nonlinear Analysis, 18 (1992), 481-496. doi: 10.1016/0362-546X(92)90015-7.
[2]	R. A. Adams, Sobolev Spaces, Academic Press, 1975.
[3]	Gregoire Allaire, Homogenization and two-scale convergence, SIAM I. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084.
[4]	R. C. Morgan and I. Babuska, An approach for constructing families of homogenized equations for periodic media. I: An integral representation and its consequences, SIAM J. Math. Anal., 22 (1991), 1-15. doi: 10.1137/0522001.
[5]	N. Bakhvalov and G. Panasenko, Homogenisation: Averaging Processes in Periodic Media: Mathematical Problems in the Mechanics of Composite Materials, Kluwer Academic Publishers, 1989. doi: 10.1007/978-94-009-2247-1.
[6]	B. Berkowitz, et al., Physical pictures of transport in heterogeneous media: Advection-dispersion, random walk, and fractional derivative formulations, Water resources Research, 38 (2002), 1191-1194.
[7]	Susanne C. Brenner and L. Ridgway Scott, The Mathematical Theory of Finite Element Methods, Springer, 2008. doi: 10.1007/978-0-387-75934-0.
[8]	Alain Bensoussan, Jacques-Louis Lions and George Papanicolaou, Asymptotic Analysis for Periodic Structures, Elsevier North-Holland, 1978.
[9]	Li-Qun Cao, Asymptotic expansions and numerical algorithms of eigenvalues and eigenfunctions of the Dirichlet problem for second order elliptic equations in perforated domains, Numerische Mathematik, 103 (2006), 11-45. doi: 10.1007/s00211-005-0668-4.
[10]	Philippe G. Ciarlet, The Finite Element Method for Elliptic Problems, Amsterdam: North-Holland, 1978.
[11]	D. Cioranescu, A. Damlamian, G. Grisoa and D. Onofrei, The periodic unfolding method for perforated domains and Neumann sieve models, J. Math. Pures Appl., 89 (2008), 248-277. doi: 10.1016/j.matpur.2007.12.008.
[12]	M. Giaquinta, Multiple integrals in the calculus of variations, Study 105, Annals of Math. Studies, Princeton Univ. Press., 1983.
[13]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, second edition, 1983. doi: 10.1007/978-3-642-61798-0.
[14]	Thomas Y. Hou, Xiao-Hui Wu and Zhiqiang Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comp., 68 (1999), 913-943. doi: 10.1090/S0025-5718-99-01077-7.
[15]	V.V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functions, Springer-Verlag, 1994. doi: 10.1007/978-3-642-84659-5.
[16]	Viviane Klein and Malgorzata Peszynska, Adaptive Double-Diffusion Model and Comparison to a Highly Heterogeneous Micro-Model, Journal of Applied Mathematics, (2012) (2012), 26 pages.
[17]	J. L. Lions, Asymptotic expansions in perforated media with a periodic structure, The Rocky Mountain J. Math., 10 (1980), 125-144. doi: 10.1216/RMJ-1980-10-1-125.
[18]	N. Neuss, W. Jäger and G. Wittum, Homogenization and multigrid, Computing, 66 (2001), 1-26. doi: 10.1007/s006070170036.
[19]	O. A. Oleinik, A. S. Shamaev and G. A. Tosifan, Mathematical Problems in Elasticity and Homogenization, North-Holland, Amsterdam, 1992.

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