Two-scale homogenization for a general class of high contrast PDE systems with periodic coefficients

ABSTRACT

For two-scale homogenization of a general class of asymptotically degenerating strongly elliptic symmetric PDE systems with a critically scaled high contrast periodic coefficients of a small period $\epsilon$, we derive a two-scale limit resolvent problem under a single generic decomposition assumption for the ‘stiff’ part. We show that this key assumption does hold for a large number of examples with a high contrast, both studied before and some recent ones, including those in linear elasticity and electromagnetism. Following ideas of V. V. Zhikov, under very mild restrictions on the regularity of the domain $\mathrm{\Omega }$ we prove that the limit resolvent problem is well-posed and turns out to be a pseudo-resolvent problem for a well-defined non-negative self-adjoint two-scale limit operator. A key novel technical ingredient here is a proof that the linear span of product test functions in the functional spaces corresponding to the degeneracies is dense in associated two-scale energy space for a general coupling between the scales. As a result, we establish (both weak and strong) two-scale resolvent convergence, as well as some of its further implications for the spectral convergence and for convergence of parabolic and hyperbolic semigroups and of associated time-dependent initial boundary value problems. Some of the results of this work were announced in Kamotski IV, Smyshlyaev VP. Two-scale homogenization for a class of partially degenerating PDE systems. arXiv:1309.4579v1. 2013 (https://arxiv.org/abs/1309.4579v1).

KEYWORDS:

• High-contrast homogenization
• two-scale convergence
• operator convergence
• resolvent convergence
• convergence of semigroups
• homogenization of high-contrast evolution problems

AMS SUBJECT CLASSIFICATIONS:

• 35B27
• 47A10

Notes

No potential conflict of interest was reported by the authors.

1 Remark that, in the present context, the original weak formulation (2.8(2.8) $$\begin{array}{cc}& {\int }_{\mathrm{\Omega }}[a^{(1)}\left(\frac{x}{\epsilon }\right)\mathrm{\nabla }u·\mathrm{\nabla }\varphi (x)+{\epsilon }^2{a}^{(0)}\left(\frac{x}{\epsilon }\right)\mathrm{\nabla }u·\mathrm{\nabla }\varphi (x)\hfill \\ \hfill & +\lambda \rho \left(\frac{x}{\epsilon }\right)u·\varphi (x)]\mathrm{d}x={\int }_{\mathrm{\Omega }}\rho \left(\frac{x}{\epsilon }\right)f^{\epsilon }(x)·\varphi (x)\mathrm{d}x,\forall \varphi \in {\left(H_0^1(\mathrm{\Omega })\right)}^n.\hfill \end{array}$$(2.8) ) can be equally stated in a real Hilbert space with associated bilinear form $B_{\epsilon }(u,v)$, and in its standard complexification with the same form $B_{\epsilon }$ now viewed as a sesquilinear one. The latter formulation is more appropriate from the spectral-theoretic point of view.

2 The extension theorems, see e.g. [33,39], are normally formulated for Euclidean domains rather than for a periodic torus as needed here. However the result of e.g. Theorem 5 of §VI.3 of Stein [39] can be used to deduce the desired statement. For example, consider an extension from $Q_s\subset {\mathbb{R}}^d$ which is regarded as an infinite (periodic) set. Then it satisfies all the conditions from the above theorem of Stein. Take an infinitely periodic $w\in H_{\#}^1(Q_s)\subset H_{\mathit{loc}}^1({\mathbb{R}}^d)$ and multiply it by a smooth cut-off function ${\chi }_R(y)$ such that ${\chi }_R=1$ for $|y|\le R$, ${\chi }_R=0$ for $|y|\ge R+1$ and $|\mathrm{\nabla }{\chi }_R(y)|\le C$. Apply the Stein’s theorem to ${\chi }_{R}w$, denote the relevant extension by ${\tilde{w}}_R(y)$ and consider its (normalized) ‘periodization’ $w_R(y):={|B_R|}^{-1}{\sum }_{k\in {\mathbb{Z}}^d}{\tilde{w}}_R(y+k)$. One readily checks that $\Vert w_R{-w\Vert }_{H^1(Q_s)}\to 0$ as $R\to \infty$, and then by continuity of the Stein’s extension that $w_R$ has a limit $\mathcal{S}w$ in $H_{\#}^1(Q)$ with ${\Vert \mathcal{S}w\Vert }_{H_{\#}^1(Q)}\le C{\Vert w\Vert }_{H_{\#}^1(Q_s)}$ and so can be taken as the desired extension.

3 Notice that assumption (2.7(2.7) $$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^d}[a^{(1)}\left(y\right)\mathrm{\nabla }w(y)·\mathrm{\nabla }w(y)+a^{(0)}\left(y\right)\mathrm{\nabla }w(y)·\mathrm{\nabla }w(y)]\mathrm{d}y\ge \nu {\Vert \mathrm{\nabla }w\Vert }_{{(L^2({\mathbb{R}}^d))}^{n\times d}}^2,\forall w\in {\left(H^1({\mathbb{R}}^d)\right)}^n,\end{array}$$(2.7) ) implies similar inequality in ${\left(H_{\#}^1(Q)\right)}^n$ with the integral over ${\mathbb{R}}^d$ replaced by integral over Q, $u·u(y)$ added to the integrand on the left, and the norm on the right replaced by the $H_{\#}^1(Q)$-norm, with some constant $C>0$. This can be seen by e.g. multiplying an infinitely periodic $u\in {\left(H_{\#}^1(Q)\right)}^n$ by a smooth cut-off function ${\chi }_R(y)$ such that ${\chi }_R=1$ for $|y|\le R$, ${\chi }_R=0$ for $|y|>R+1$ and $|\mathrm{\nabla }{\chi }_R(y)|\le C$, and taking R large enough.

4 As clarified e.g. in [21,24], for $\mathrm{\Omega }={\mathbb{R}}^d$ this corresponds to non-vanishing contributions to the limit Floquet-Bloch spectrum as $\epsilon \to 0$ from the quasi-periodicity parameter (quasi-momentum) $\theta \ne 0$, for which the present two-scale description restricted to periodic functions ($\theta =0$) appears insufficient.

5 According to Definition 4.3 of [30], for a bounded sequence $v_{\epsilon }\in L^2(0,T;H_{\epsilon })$ we say that $v^{\epsilon }(x,t)\stackrel{2}{\rightharpoonup }v(x,y,t)\in L^2(0,T;H)$ if for any $z^{\epsilon }(x)\stackrel{2}{\to }z(x,y)$ and any $\phi (t)\in L^2(0,T)$,$${\int }_0^T{\left(v^{\epsilon }(x,t),z^{\epsilon }(x)\right)}_{H_{\epsilon }}\phi (t)\mathrm{d}t\to {\int }_0^T{\left(v(x,y,t),z(x,y,t)\right)}_H\phi (t)\mathrm{d}t.$$