American Institute of Mathematical Sciences

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

Pointwise estimate for elliptic equations in periodic perforated domains

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

Received  November 2014 Revised  February 2015 Published  June 2015

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.
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.

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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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