Abstract
We investigate Lamé systems in periodically perforated domains, and establish quantitative homogenization results in the setting where the domain is clamped at the boundary of the holes. Our method is based on layer potentials and it provides a unified proof for various regimes of hole-cell ratios (the ratio between the size of the holes and the size of the periodic cells), and, more importantly, it yields natural correctors that facilitate error estimates. A key ingredient is the asymptotic analysis for the rescaled cell problems, and this is studied by exploring the convergence of the periodic layer potentials for the Lamé system to those in the whole space when the period tends to infinity.
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References
Allaire, G.: Homogenization of the Navier–Stokes equations in open sets perforated with tiny holes. I. Abstract framework, a volume distribution of holes. Arch. Ration. Mech. Anal. 113(3), 209–259 (1990)
Allaire, G.: Homogenization of the Navier–Stokes equations in open sets perforated with tiny holes. II. Noncritical sizes of the holes for a volume distribution and a surface distribution of holes. Arch. Ration. Mech. Anal. 113(3), 261–298 (1990)
Allaire, G.: Continuity of the Darcy’s law in the low-volume fraction limit. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 18(4), 475–499 (1991)
Ammari, H., Garapon, P., Kang, H., Lee, H.: Effective viscosity properties of dilute suspensions of arbitrarily shaped particles. Asymptot. Anal. 80(3–4), 189–211 (2012)
Ammari, H., Garnier, J., Giovangigli, L., Jing, W., Seo, J.-K.: Spectroscopic imaging of a dilute cell suspension. J. Math. Pures Appl. (9) 105(5), 603–661 (2016)
Ammari, H., Kang, H.: Polarization and Moment Tensors, with Applications to Inverse Problems and Effective Medium Theory. Applied Mathematical Sciences, vol. 162. Springer, New York (2007)
Bensoussan, A., Lions, J.-L., Papanicolaou, G.C.: Boundary layers and homogenization of transport processes. Publ. Res. Inst. Math. Sci. 15(1), 53–157 (1979)
Cioranescu, D., Murat, F.: Un terme étrange venu d’ailleurs. In: Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, vol. 2 (Paris, 1979/1980), volume 60 of Res. Notes in Math., pp. 98–138, 389–390. Pitman, Boston, London (1982)
Cioranescu, D., Paulin, J.S.J.: Homogenization in open sets with holes. J. Math. Anal. Appl. 71(2), 590��607 (1979)
Coifman, R.R., McIntosh, A., Meyer, Y.: L’intégrale de Cauchy définit un opérateur borné sur \(L^{2}\) pour les courbes lipschitziennes. Ann. Math. (2) 116(2), 361–387 (1982)
Dahlberg, B.E.J., Kenig, C.E., Verchota, G.C.: Boundary value problems for the systems of elastostatics in Lipschitz domains. Duke Math. J. 57(3), 795–818 (1988)
Fabes, E.B., Kenig, C.E., Verchota, G.C.: The Dirichlet problem for the Stokes system on Lipschitz domains. Duke Math. J. 57(3), 769–793 (1988)
Feppon, F.: High order homogenization of the Poisson equation in a perforated periodic domain. Working paper or preprint (2020)
Feppon, F.: High order homogenization of the Stokes system in a periodic porous medium. Working paper or preprint (2020)
Gerard-Varet, D.: A simple justification of effective models for conducting or fluid media with dilute spherical inclusions. arXiv:1909.11931 (2019)
Giunti, A., Höfer, R., Velázquez, J.J.L.: Homogenization for the Poisson equation in randomly perforated domains under minimal assumptions on the size of the holes. Commun. Partial Differ. Equ. 43(9), 1377–1412 (2018)
Jing, W.: Homogenization of randomly deformed conductivity resistant membranes. Commun. Math. Sci. 14(5), 1237–1268 (2016)
Jing, W.: A unified homogenization approach for the dirichlet problem in perforated domains. SIAM J. Math. Anal. 52(2), 1192–1220 (2020)
Kacimi, H., Murat, F.: Estimation de l’erreur dans des problèmes de Dirichlet où apparait un terme étrange. In: Partial Differential Equations and the Calculus of Variations, vol. 2, volume 2 of Progress Nonlinear Differential Equations Applications, pp 661–696. Birkhäuser, Boston (1989)
Lu, Y.: Homogenization of stokes equations in perforated domains: a unified approach. arXiv:1908.08259 (2019)
Sánchez-Palencia, E.: Nonhomogeneous Media and Vibration Theory. Lecture Notes in Physics, vol. 127. Springer, Berlin, New York (1980)
Tartar, L.: Incompressible fluid flow in a porous media-convergence of the homogenization process. Appendix to [21], pp. 368–377 (1980)
Verchota, G.: Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains. J. Funct. Anal. 59(3), 572–611 (1984)
Acknowledgements
The author would like to thank Xin Fu for helpful discussions on layer potentials for Lamé systems. This work is partially supported by the NSF of China under Grants Nos. 11701314 and 11871300.
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Appendix A: Some useful lemmas
Appendix A: Some useful lemmas
The following results are very helpful and have been used in the main parts of the paper.
Theorem A.1
(A Poincaré inequality). Let \(d\ge 2\). Let r, R be two positive real numbers and \(r < R\). Then there exists a constant \(C > 0\) that depends only on the dimension d, such that for any \(u \in H^1(B_R(0))\) satisfying \(u = 0\) in \(B_r(0)\), we have
We refer to [2, Lemma 3.4.1] or [18, Theorem A.1] for the proof. This inequality accounts for the various asymptotic regimes for (1.1) depending on the relative smallness of \(\eta \) with respect to \(\varepsilon \). Clearly, if we change one or both of the balls to cubes, the above inequality still holds. In particular, it can be applied on the \(\varepsilon \)-cubes, \(\varepsilon (z+\overline{Y}_f)\), \(z\in \mathbb {Z}^d\), which form \(\varepsilon \mathbb {R}^d_f\) and \(D^\varepsilon \).
Lemma A.2
Suppose H is a Hilbert space and \(\mathcal {T} : H\rightarrow H\) is a bounded linear operator on H and \(\mathcal {T}^*\) is the adjoint operator. Suppose \(\mathcal {T}\) has closed range, \(\ker (\mathcal {T})\) has finite dimension k, and, moreover, \(\mathcal {T} - \mathcal {T}^*\) is compact. Then \(\dim \ker (\mathcal {T}^*) = k\) as well.
This is rephrased from Lemma 2.3 of [12]. It can be proved directly, or, by using the fact that \(\mathcal {T}\) is semi-Fredholm and that semi-Fredholmness and the index of such an operator are preserved by compact perturbations.
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Jing, W. Layer potentials for Lamé systems and homogenization of perforated elastic medium with clamped holes. Calc. Var. 60, 2 (2021). https://doi.org/10.1007/s00526-020-01862-x
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DOI: https://doi.org/10.1007/s00526-020-01862-x