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Layer potentials for Lamé systems and homogenization of perforated elastic medium with clamped holes

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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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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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  1. Yau Mathematical Sciences Center, Tsinghua University, No. 1 Tsinghua Yuan, Beijing, 100084, People’s Republic of China

    Wenjia Jing

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  1. Wenjia Jing
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Correspondence to Wenjia Jing.

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Communicated by F.-H. Lin.

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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

$$\begin{aligned} \Vert u\Vert _{L^2(B_R)} \le {\left\{ \begin{array}{ll} CR(\frac{r}{R})^{-\frac{d-2}{2}} \Vert \nabla u\Vert _{L^2(B_R)}, &{}\qquad d\ge 3,\\ CR|\log (\frac{r}{R})|^\frac{1}{2} \Vert \nabla u\Vert _{L^2(B_R)}, &{}\qquad d=2. \end{array}\right. } \end{aligned}$$
(A.1)

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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