2.1 Function spaces

Let $\mathcal {R}(Q_2)$ denote the space of rigid body displacements in $Q_2$, i.e. $\mathbf {u} = \mathbf {A}\mathbf {y} + \mathbf {b}$ with constant skew-symmetric matrix $A$ and constant vector $\mathbf {b}$ in $Q_2$. We start with the space of admissible functions for the velocity

(2.7)\begin{align} H(Q) & := \left\{ \mathbf{v}: \mathbf{v}\in H^{1}(Q_1\cup Q_2)^{n} \bigg\vert \; \text{div}_{\mathbf{y}} \mathbf{v} = 0,\; \mathbf{v} \cdot \mathbf{n} = 0 \text{ in } H^{-({1}/{2})}(\Gamma),\right. \nonumber\\ & \quad \left.{} [\kern-1pt[ \mathbf{v} ]\kern-1pt]_{\Gamma} = {{\mathbf 0}},\; (\mathbf{v},\mathbf{\eta})_{H^{1}(Q_2)}=0, \forall \mathbf{\eta}\in \mathcal{R}(Q_2),\,\mathbf{v} \text{ is } Q \text{-periodic} \right\} \end{align}

where $\mathbf {n}$ is the outward unit normal of $\partial Q_2$. $H (Q)$ is endowed with the inner product

(2.8)\begin{equation} (\mathbf{u},\mathbf{v})_{Q} = \int_{Q} 2\mu_1e(\mathbf{u}) : \overline{e(\mathbf{v})}\,{\rm d}\mathbf{y}. \end{equation}

The induced norm is denoted by $\left \lVert {\mathbf {u}}\right \rVert _{Q}^{2} := (\mathbf {u},\mathbf {u})_Q$. Note that we have $H(Q)\cap \mathcal {R}(Q)=\{\mathbf {0}\}$ because $\mathbf {A} = 0$ due to the $Q$-periodicity and $\mathbf {u} \cdot \mathbf {n} = 0$ implies $\mathbf {b} = {{\mathbf 0}}$. We observe that if $\mathbf {u} \in H(Q)$ then $\mathbf {u} \in H^{1}(Q)^{n}$ by the following argument. Obviously, $\mathbf {u} \in L^{2}(Q)^{n}$. To prove ${\partial u_i}/{\partial y_j} \in L^{2}(Q)$ for $i,j = 1,2,3$, let $\phi$ be any $C^{\infty }$ test function compactly supported in $Q$ and $h$ be the $i$-th component $u_i$ for any $i$. Then

\[ \int_{Q} h \nabla \phi\,{\rm d}\mathbf{y} ={-} \left(\int_{Q_1\cap \text{Supp}(\phi)} \phi\nabla h\,{\rm d}\mathbf{y} + \int_{Q_2\cap \text{Supp}(\phi)} \phi\nabla h\,{\rm d}\mathbf{y}\right) \]

here we used $[\kern-1pt[ h ]\kern-1pt] = {{\mathbf 0}}$. Now we can define a candidate function $\mathbf {g}$ such that

(2.9)\begin{equation} \mathbf{g}\vert_{Q_i} := \nabla h\vert_{Q_i},\quad i=1,2 \end{equation}

then clearly $\mathbf {g}\in L^{2}(Q)^{n}$ and $\left < h,\nabla \phi \right > = -\left < g,\phi \right >$, where $\left <\cdot \right >$ denotes the usual $L^{2}$ inner product. Therefore ${h} \in H^{1}(Q)$ and hence $u_i \in H^{1}(Q), \ i=1,\ldots,n$.

Next, we show that $\| \cdot \|_Q$ is equivalent to the usual $H^{1}$ norm, i.e., there exist constants $B_1$ and $B_2$ such that

(2.10)\begin{equation} B_1 \left\lVert{\mathbf{u}}\right\rVert_{H^{1}(Q)} \leq \left\lVert{\mathbf{u}}\right\rVert_{Q} \leq B_2 \left\lVert{\mathbf{u}}\right\rVert_{H^{1}(Q)} \end{equation}

Because $H^{1}(Q)\cap \mathcal {R}(Q)=\{0\}$, by theorem 2.5 in [Reference Oleinik, Shamaev and Yosifian34], there exists a Korn's constant $C_1$ such that

(2.11)\begin{equation} C_1 \left\lVert{\mathbf{u}}\right\rVert_{H^{1}(Q)} \leq \frac{1}{\sqrt{2\mu_1}} \left\lVert{\mathbf{u}}\right\rVert_{Q} \end{equation}

where $C_1$ depends only on $Q$. Therefore, we can take $B_1=\sqrt {2\mu _1}C_1$. To emphasize the dependence on $Q$, we will write it as $B_1(Q)$. On the other hand, according to the orthogonal decomposition that $\nabla \mathbf {u} = e(\mathbf {u}) + \tilde {e}(\mathbf {u})$, see (1.6),

\[ \left\lVert{\mathbf{u}}\right\rVert_{H^{1}(Q)}^{2} \geq \left\lVert{\nabla \mathbf{u}}\right\rVert_{L^{2}(Q)}^{2} = \left\lVert{e(\mathbf{u})}\right\rVert_{L^{2}(Q)}^{2} + \left\lVert{\tilde{e}(\mathbf{u})}\right\rVert^{2} \geq \left\lVert{e(\mathbf{u})}\right\rVert_{L^{2}(Q)}^{2} =\frac{1}{2\mu_1} \|\mathbf{u} \|_Q^{2} \]

therefore $B_2 = \sqrt {2\mu _1}$. The reason for introducing the $H(Q)$-norm is that the self-permeability in (2.5) can be represented in terms of the inner product. More specifically, using (2.5), (2.6) and the fact that $\overline {\mathbf{e} _i}=\mathbf{e} _i$, by a calculation similar to (1.5) and taking into account the interface condition $\mathbf {u}\cdot \mathbf {n}=0$ and the jump conditions in (2.6), (2.5) can be expressed in the following form

(2.12)\begin{equation} K_{ij}(z) = \int_{Q}2 {\mu}(\mathbf{y};\overline{z}) \overline{e( \mathbf{u}^{{i}}(z))} : {e(\mathbf{u}^{{j}}(z))}\,{\rm d}\mathbf{y} \end{equation}

and its conjugate transpose $\boldsymbol K^{*}:=\overline {\boldsymbol K^{T}}$ is

(2.13)\begin{equation} {(K^{*})_{ij}}(z) = \int_{Q}2 {\mu}(\mathbf{y};{z}) {e( \mathbf{u}^{{j}}(z))} : \overline{e(\mathbf{u}^{{i}}(z))}\,{\rm d}\mathbf{y} \end{equation}

2.2 Weak solution of the cell problem (2.6)

The weak formulation of the cell problem (2.6) is

(2.14)\begin{equation} \int_{Q_1\cup Q_2} 2\mu(\mathbf{y};z) e( \mathbf{u}^{k}) : \overline{e(\mathbf{v})}\,{\rm d}\mathbf{y} = \int_{Q_1\cup Q_2} \mathbf{e}_k\cdot \bar{\mathbf{v}}\,{\rm d}\mathbf{y} ,\quad \forall \mathbf{v} \in H(Q) \end{equation}

From this, we see that the solutions satisfy the following symmetry

\[ \mathbf{u}^{k}(\mathbf{y};\overline{z})=\overline{\mathbf{u}^{k}(\mathbf{y};z)} \]

Define the sesquilinear form on $H(Q)$

(2.15)\begin{equation} a(\mathbf{u},\mathbf{v}) = \int_{Q_1\cup Q_2}2\mu(\mathbf{y};z) e( \mathbf{u}^{k}) : \overline{e(\mathbf{v})}\,{\rm d}\mathbf{y} \end{equation}

It is clear that $a(\mathbf {u},\mathbf {v})$ is bounded in $H(Q)$. To check the coercivity, assume $\mathbf {u}^{k} \neq 0$ and define the parameter

(2.16)\begin{equation} \lambda := \frac{\int_{Q}2 \mu_1\chi_2 e( \mathbf{u}^{k}) : \overline{e(\mathbf{u}^{k})}\,{\rm d}\mathbf{y}}{\int_{Q}2 \mu_1 e( \mathbf{u}^{k}) : \overline{e(\mathbf{u}^{k})}\,{\rm d}\mathbf{y}} \end{equation}

then $0\leq \lambda \leq 1$. We note that

(2.17)\begin{equation} \frac{a(\mathbf{u}^{k},\mathbf{u}^{k})}{\int_{Q}2\mu_1 e( \mathbf{u}^{k}) : \overline{e(\mathbf{u}^{k})}\,{\rm d}\mathbf{y}}= \lambda z+(1-\lambda)\cdot 1 \end{equation}

and hence as long as 0 is not on the line segment joining $z$ and 1, there exist $\alpha (z):=\min _{0\le \lambda \le 1} |\lambda z+1-\lambda |>0$ such that

(2.18)\begin{equation} \left|a(\mathbf{u}^{k},\mathbf{u}^{k})\right|\ge \alpha(z) \int_{Q}2\mu_1 e( \mathbf{u}^{k}) : \overline{e(\mathbf{u}^{k})}\,{\rm d}\mathbf{y}=\alpha\| \mathbf{u}^{k}\|_Q^{2} \end{equation}

Therefore for $z\in \mathbb {C}\backslash \left \lbrace \Re {z}\leq 0\right \rbrace$, by the Lax–Milgram lemma [Reference Brenner and Scott13, chapter 2], there exists a unique weak solution $\mathbf {u}^{k}\in H(Q)$ to the cell problem (2.6) and with the solution $\mathbf {u}^{k}$, we can construct $p^{k} \in L^{2}(Q)/\mathbb {C}$.

Since $\alpha (z)$ is a continuous function in $z$, the coercivity of the sesquilinear form can be applied to conclude that $\mathbf {u}^{k}$ is analytic in $z$ and its $m$-th derivative, $m\ge 1$, satisfies the following recursive equation

(2.19)\begin{align} & \int_{Q_1\cup Q_2} 2\mu(\mathbf{y};z) e\left( \frac{{\rm d}^{m}\mathbf{u}^{k}}{{\rm d}z^{m}}\right) : \overline{e(\mathbf{v})}\,{\rm d}\mathbf{y}\nonumber\\ & \quad ={-}\int_{Q_2} 2 {m}\mu_1 e\left(\frac{{\rm d}^{m-1}\mathbf{u}^{k}}{{\rm d}z^{m-1}}\right):e(\overline{\mathbf{v}})\,{\rm d}\mathbf{y},\quad\forall \mathbf{v} \in H(Q) \end{align}

As a result, $\boldsymbol K(z)$ is also analytic for $z\in \mathbb {C}\backslash \left \{ \Re {z}\leq 0\right \}$. To relate the two-fluid problem with $\boldsymbol K^{(D)}$, we adapt the method used in [Reference Bruno and Leo16] to study the behaviour of $\boldsymbol K(z)$ near $z=\infty$ in the following section.

2.3 Analyticity of the solution for large $|z|$

Let $w:=\frac {1}{z}$ and consider $Q$-periodic solution in the series form near $w=0$

(2.20)\begin{equation} \mathbf{u}_\infty(\mathbf{y};\mathbf{e},w) := \sum_{k = 0}^{\infty} \mathbf{u}_k(\mathbf{y};\mathbf{e}) w^{k} \text{ and } p_\infty(\mathbf{y};\mathbf{e},w) := \sum_{k = 0}^{\infty} p_k(\mathbf{y};\mathbf{e}) w^{k} \end{equation}

where the $\mathbf {e}$ is an arbitrary constant unit vector. To set up the notation, we denote the restrictions of $\mathbf{u} _k$, $p_k$ in $Q_2$ (inclusion) and $Q_1$ as $\mathbf {u}^{in}_k$, $p^{in}_k$ and $\mathbf {u}^{out}_k$, $p^{out}_k$ respectively and define

(2.21)\begin{equation} \mathbf{u}_\infty^{in}(\mathbf{y};\mathbf{e},w): = \sum_{k = 0}^{\infty} \mathbf{u}^{in}_k(\mathbf{y},\mathbf{e}) w^{k},\quad \mathbf{u}_\infty^{out}(\mathbf{y};\mathbf{e},w): = \sum_{k = 0}^{\infty} \mathbf{u}^{out}_k(\mathbf{y},\mathbf{e}) w^{k} \end{equation}

By substituting (2.20) into (2.6) with the viscosity defined in (2.2), taking into account the additional two interface conditions $\mathbf {u}\cdot \mathbf{n} =0$ and $[\kern-1pt[ \mathbf {u} ]\kern-1pt] =0$, followed by equating terms of the same order with respect to $w$, we arrive in the following equations in $Q_1$:

(2.22)\begin{align} & O(w^{0}):\quad \text{div}_{\mathbf{y}} \left( 2\mu_1 e(\mathbf{u}_0^{out}) - p_0^{out}\mathbf{I} \right) ={-}\mathbf{e} \end{align}

(2.23)\begin{align} & O(w^{k}):\quad\text{div}_{\mathbf{y}} \left( 2\mu_1 e(\mathbf{u}_k^{out}) - p_k^{out}\mathbf{I} \right) = {{\mathbf 0}} \text{ for } k\geq 1 \end{align}

and in $Q_2$:

(2.24)\begin{align} & O(w^{{-}1}):\quad \text{div}_{\mathbf{y}} \left( 2\mu_1 e(\mathbf{u}_0^{in}) \right) = {{\mathbf 0}} \end{align}

(2.25)\begin{align} & O(w^{0}):\quad \text{div}_{\mathbf{y}} \left( 2\mu_1 e(\mathbf{u}_1^{in}) - p_0^{in}\mathbf{I} \right) ={-}\mathbf{e} \end{align}

(2.26)\begin{align} & O(w^{k}):\quad\text{div}_{\mathbf{y}} \left( 2\mu_1 e(\mathbf{u}_{k+1}^{in}) - p_k^{in}\mathbf{I} \right) = {{\mathbf 0}} \text{ for } k\geq 1 \end{align}

and the following interface conditions on $\Gamma$

(2.27)\begin{align} & O(w^{{-}1}):2\mu_1(e(\mathbf{u}_0^{in})\cdot \mathbf{n})|_\Gamma = C(\mathbf{y}) \mathbf{n} \text{ for some function }C(\mathbf{y}) \end{align}

(2.28)\begin{align} & O(w^{k}),\, k\ge 0:\left( \left( 2\mu_1 e(\mathbf{u}_k^{out}) - p_k^{out}\mathbf{I} \right) - \left( 2\mu_1 e(\mathbf{u}_{k+1}^{in}) - p_k^{in}\mathbf{I} \right) \right) \mathbf{n}\nonumber\\ & \qquad\qquad\qquad\;\;= \left\{ \left[\left( \left( 2\mu_1 e(\mathbf{u}_k^{out}) - p_k^{out}\mathbf{I} \right) - \left( 2\mu_1 e(\mathbf{u}_k^{in}) - p_k^{in}\mathbf{I} \right) \right) \mathbf{n} \right]\cdot \mathbf{n}\right\} \mathbf{n}, \end{align}

(2.29)\begin{align} & \mathbf{u}^{in}_k \cdot \mathbf{n} = \mathbf{u}^{out}_k \cdot \mathbf{n} = 0 \text{ and }\mathbf{u}^{in}_k = \mathbf{u}^{out}_k \end{align}

We introduce the following spaces, $i=1,2$

\begin{align*} H(Q_1) & = \big\{ \mathbf{v}: \mathbf{v}\in H^{1}(Q_1)^{n}\big\vert \; \text{div}_{\mathbf{y}} \mathbf{v} = 0, \mathbf{v} \cdot \mathbf{n} = 0 \text{ on } \Gamma, Q \text{-periodic} \big\}\\ H(Q_2) & = \big\{ \mathbf{v}: \mathbf{v}\in H^{1}(Q_2)^{n}\big\vert \; \text{div}_{\mathbf{y}} \mathbf{v} = 0, \mathbf{v} \cdot \mathbf{n} = 0 \text{ on } \Gamma, \, (\mathbf{v},\mathcal{R}(Q_2))_{H^{1}(Q_2)}=0, Q \text{-periodic} \big\}\\ \mathring{H}(Q_i)& = \big\{ \mathbf{v}: \mathbf{v}\in H^{1}(Q_i)^{n}\big\vert \; \text{div}_{\mathbf{y}} \mathbf{v} = 0, \mathbf{v}\vert_{\Gamma} = {{\mathbf 0}}, Q \text{-periodic} \big\} \subset H(Q_i),\\ L(Q_i)/ \mathbb{C} & = \bigg\{ p: p\in L^{2}(Q_i), \int_{Q_i} p(\mathbf{y})\,{\rm d}\mathbf{y} = 0, Q \text{-periodic},\bigg\} \end{align*}

Note that $H(Q_1)\cap \mathcal {R}(Q_1)=\{{{\mathbf 0}}\}$ because $\partial Q \subset \partial Q_1$. For $H(Q_2)$, the boundary condition $\mathbf {u}\cdot \mathbf{n} =0$ implies $H(Q_2)\cap \mathcal {R}(Q_2)=\{{{\mathbf 0}}\}$ because of the extra condition $(\mathbf {v},\mathcal {R}(Q_2))_{H^{1}(Q_2)}=0$ [Reference Oleinik, Shamaev and Yosifian34]. Therefore, $H(Q_i)$ and $\mathring {H}(Q_i)$ are equipped with inner product $(\mathbf {u},\mathbf {v})_{Q_i} = \int _{Q_i} 2\mu _1e(\mathbf {u}) : \overline {e(\mathbf {v})}\,{\rm d}\mathbf {y}$ and Korn's inequalities are valid in $H(Q_i)$, $i=1,2$.

Lemma 2.1 Let $Q_2$ be a connected, open bounded set such that $\partial Q_2 \cap \partial Q=\emptyset$ and $\partial Q_2$ is in $\mathcal {C}^{k,\sigma }$, $k,\sigma \ge 0$, $k+\sigma \ge 2$. For any vector field $\mathbf {u}^{in} \in H(Q_2)$, there exists a unique weak solution $\mathbf {u}^{out}(\mathbf {y};\mathbf {f}^{out})\in H(Q_1)$ that satisfies the following system

(2.30)\begin{equation} \left\{\begin{array}{@{}ll} \text{ div}_{\mathbf{y}} \left( 2\mu_1 e(\mathbf{u}^{out}) - p^{out}\mathbf{I} \right) = \mathbf{f}^{out} & \text{ in } Q_1\\ \mathbf{u}^{out} = \mathbf{u}^{in} & \text{ on }\Gamma \end{array}\right. \end{equation}

where in our context, $\mathbf {f}^{out} = {{\mathbf 0}} \text { or } \mathbf {f}^{out} = -\mathbf {e}$, a constant unit vector. Moreover,

(2.31)\begin{equation} \left\lVert{\mathbf{u}^{out}}\right\rVert_{Q_1} \leq \frac{1}{B_1(Q)} \left\lVert{\mathbf{f}^{out}}\right\rVert_{L^{2}(Q_1)} + 2E_1 \left\lVert{\mathbf{u}^{in}}\right\rVert_{Q_2}. \end{equation}

where the positive constants $B_1(Q_{{1}})$ is defined in (2.10) and $E_1\ge 1$ depends only on $Q_1$ and $Q_2$.

Proof. To handle the inhomogeneous boundary condition, we proceed as follows. By [Reference Kato, Mitrea, Ponce and Taylor24, corollary 3.2], there exists a bounded, divergence free extension $T(\mathbf {u}^{in})$ of $\mathbf {u}^{in}$ to a small neighbourhood $O$ of $Q_2$ and vanishes at $\partial O\subset Q_1$ such that

(2.32)\begin{equation} \left\lVert{T\left(\mathbf{u}^{in} \right)}\right\rVert_{Q}\leq E_1 \left\lVert{\mathbf{u}^{in}}\right\rVert_{Q_2}\end{equation}

where $E_1\ge 1$ depends only on $Q_1$ and $Q_2$. Furthermore, the extension $T(\mathbf {u}^{in})$ on $O$ can be extended periodically to $\mathbb {R}^{n}$ [Reference Conca18] since $T(\mathbf {u}^{in})$ vanishes on $\partial O$ and hence on $\partial Q$. We denote the restriction $T(\mathbf {u}^{in})|_{Q_1}$ as $\tilde {\mathbf {u}}^{out}\in H(Q_1)$ and $\mathring {\mathbf {u}}^{out} := \mathbf {u}^{out} - \tilde {\mathbf {u}}^{out} \in \mathring {H}(Q_1)$ and (2.30) becomes

(2.33)\begin{equation} \text{div}_{\mathbf{y}} \left( 2\mu_1 e(\mathring{\mathbf{u}}^{out}) - p^{out}\mathbf{I}\right) = \mathbf{f}^{out} - \mu_1\Delta \tilde{\mathbf{u}}^{out} \quad \text{ in } Q_1 \end{equation}

Consider the variational formulation: find $\mathring {\mathbf {u}}^{out} \in \mathring {H}(Q_1)$ such that $\forall \Phi \in \mathring {H}(Q_1)$,

(2.34)\begin{equation} \int_{Q_1} 2\mu_1 e(\mathring{\mathbf{u}}^{out}) :\overline{e(\Phi)}\,{\rm d}\mathbf{y} ={-}\int_{Q_1} \mathbf{f}^{out} \cdot \overline{\Phi}\,{\rm d}\mathbf{y} - \int_{Q_1} 2\mu_1 e(\tilde{\mathbf{u}}^{out}) : \overline{e(\Phi)}\,{\rm d}\mathbf{y}, \end{equation}

The right-hand side of (2.34) can be bounded as follows

\[ \left| \int_{Q_1} \mathbf{f}^{out} \cdot \overline{\Phi}\,{\rm d}\mathbf{y} + \int_{Q_1} 2\mu_1 e( \tilde{\mathbf{u}}^{out}) : \overline{e(\Phi)}\,{\rm d}\mathbf{y}\right|\leq \left(\frac{\left\lVert{\mathbf{f}^{out}}\right\rVert_{L^{2}(Q_1)}}{B_1(Q_{{1}})} + \left\lVert{\tilde{\mathbf{u}}^{out}}\right\rVert_{Q_1} \right) \left\lVert{\Phi}\right\rVert_{Q_1} \]

The sesquilinear form $\int _{Q_1} 2\mu _1 e( \mathring {\mathbf {u}}^{out}) : \overline {e(\Phi )}\,{\rm d}\mathbf {y}$ is clearly bounded and coercive with constant 1. Hence by the Lax–Milgram lemma there exists a unique weak solution $\mathring {\mathbf {u}}^{out}\in \mathring {H}(Q_1)$ to (2.33) such that $\left \lVert {\mathring {\mathbf {u}}^{out}}\right \rVert _{Q_1}\le (({1}/{B_1(Q_{{1}})})\left \lVert {\mathbf {f}^{out}}\right \rVert _{L^{2}(Q_1)} + \left \lVert {\tilde {\mathbf {u}}^{out}}\right \rVert _{Q_1})$. In terms of $\mathring {\mathbf {u}}^{out}$, $\mathbf {u}^{out}$ can be expressed as $\mathbf {u}^{out} = \tilde {\mathbf {u}}^{out} + \mathring {\mathbf {u}}^{out}$ and satisfies the estimate

(2.35)\begin{equation} \left\lVert{\mathbf{u}^{out}}\right\rVert_{Q_1} \leq \left\lVert{\mathring{\mathbf{u}}^{out}}\right\rVert_{Q_1} +\left\lVert{\tilde{\mathbf{u}}^{out}}\right\rVert_{Q_1}\leq \frac{1}{B_1(Q_{{1}})} \left\lVert{\mathbf{f}^{out}}\right\rVert_{L^{2}(Q_1)} + 2E_1 \left\lVert{\mathbf{u}^{in}}\right\rVert_{Q_2}\end{equation}

To show the solution $\mathbf {u}^{out}$ is unique, suppose $\mathbf {u}^{out}_1$ and $\mathbf {u}^{out}_2$ both solve (2.30) then the difference ${{\mathbf {w}}}^{\text {diff}} = \mathbf {u}^{out}_1 - \mathbf {u}^{out}_2 \in \mathring {H}(Q_1)$ must satisfy

\[ \int_{Q_1} 2\mu_1 e( {{\mathbf{w}}}^{\text{diff}}) : \overline{e(\Phi)}\,{\rm d}\mathbf{y} = 0, \quad \forall \Phi \in \mathring{H}(Q_1) \]

Hence ${{\mathbf {w}}}^{\text {diff}}=0$ in $Q_1$ because ${{\mathbf {w}}}\in \mathring {H}(Q_1)$. We note the two special cases:

(2.36)\begin{align} \left\lVert{\mathbf{u}^{out}}\right\rVert_{Q_1} & \leq 2E_1 \left\lVert{\mathbf{u}^{in}}\right\rVert_{Q_2} \text{ for } \mathbf{f}^{out} = {{\mathbf 0}} \end{align}

(2.37)\begin{align} \left\lVert{\mathbf{u}^{out}}\right\rVert_{Q_1} & \leq \frac{1}{B_1(Q_{{1}})} \sqrt{|Q_1|} + 2E_1 \left\lVert{\mathbf{u}^{in}}\right\rVert_{Q_2} \text{ for } \mathbf{f}^{out} ={-}\mathbf{e}_{{j}},\quad j=1,\ldots,n, \end{align}

where $|Q_1|$ is the volume of $Q_1$.

Lemma 2.2 Let $Q_2$ satisfy the same assumptions as those in lemma 2.1. For any pair of $(\mathbf {u}^{out}, p^{out}) \in H(Q_1) \times L^{2}(Q_1)/ \mathbb {C}$ that satisfies (2.30), there exists a unique vector $\mathbf {u}^{in}(\mathbf {y};\mathbf {f}^{in})\in H(Q_2)$ that satisfies the Stokes equation with continuity of tangential traction on $\Gamma$

(2.38)\begin{equation} \left\{\begin{array}{@{}ll} {\rm div}_{\mathbf{y}} \left( 2\mu_1 e(\mathbf{u}^{in}) - p^{in}\mathbf{I} \right) = \mathbf{f}^{in} & \text{in }Q_2,\\ \mathbf{n}\times\mathbf{n}\times\left[\left(\left(2\mu_1 e(\mathbf{u}^{out}) - p^{out}\mathbf{I} \right) - \left( 2\mu_1 e(\mathbf{u}^{in}) - p^{in}\mathbf{I} \right) \right)\cdot\mathbf{n}\right] = {{\mathbf 0}} & \text{on }\Gamma, \end{array}\right. \end{equation}

where in our context, $\mathbf {f}^{in} = {{\mathbf 0}} \text { or } \mathbf {f}^{in} = -\mathbf {e}$. Moreover,

\[ \left\lVert{\mathbf{u}^{in}}\right\rVert_{Q_2} \leq \frac{1}{B_1(Q_2)}\left\lVert{\mathbf{f}^{in}}\right\rVert_{L^{2}(Q_2)} + \frac{E_1(Q_1)}{B_1(Q_1)}\left\lVert{\mathbf{f}^{out}}\right\rVert_{L^{2}(Q_1)} + E_1(Q_1)\left\lVert{\mathbf{u}^{out}}\right\rVert_{Q_1} \]

and $E_1(Q_1)$, $B_1(Q_1)$ and $B_1(Q_2)$ depend only on $Q$ and $\Gamma$.

Proof. Take ${\Phi } \in H(Q_2)$, the variation formulation for the PDE is

\[ \int_{Q_2} \left({\rm div}_{\mathbf{y}} \left( 2\mu_1 e(\mathbf{u}^{in}) - p^{in}\mathbf{I} \right) \right) \cdot \bar{\Phi}\,{\rm d}\mathbf{y} = \int_{Q_2} \mathbf{f}^{in} \cdot \bar{\Phi}\,{\rm d}\mathbf{y}. \]

For the ease of notation, let $\boldsymbol \pi ^{in}:=\boldsymbol \tau (\mathbf {u}^{in},\mu _1,p^{in})$ and $\boldsymbol \pi ^{out}:=\boldsymbol \tau (\mathbf {u}^{out},\mu _1,p^{out})$ where the stress function $\boldsymbol \tau$ is defined in (2.1). Applying integration by parts on the left-hand side, followed by an application of the divergence theorem leads to

(2.39)\begin{equation} - \int_{\Gamma} ({ \boldsymbol\pi^{in} \cdot \bar{\Phi}}) \cdot \mathbf{n}\,{\rm d}S - \int_{Q_2} 2\mu_1 e(\mathbf{u}^{in}) : \overline{e(\Phi)}\,{\rm d}\mathbf{y} = \int_{Q_2} \mathbf{f}^{in} \cdot \bar{\Phi}\,{\rm d}\mathbf{y} \end{equation}

Let $\mathbf {t}$ denote the unit vector in the tangent plane such that $\mathbf {t}\cdot \mathbf{n} = 0$. The conditions on $\Gamma$ in (2.38) imply

\begin{align*} \begin{cases} \Phi\cdot\mathbf{n} =0\Rightarrow \Phi = {\rm d}(\mathbf{y}) \mathbf{t} \text{ for some function }{\rm d}(\mathbf{y}),\\ \mathbf{n}\times\mathbf{n}\times\left[\left(\boldsymbol{\pi}^{out}- \boldsymbol{\pi}^{in}\right)\cdot\mathbf{n}\right] = {{\mathbf 0}} \Rightarrow \left( \boldsymbol{\pi}^{out} -\boldsymbol{\pi}^{in} \right)\cdot\mathbf{n}\\ \quad =C(\mathbf{y})\mathbf{n} \text{ for some function }C(\mathbf{y}) \end{cases} \end{align*}

With these observations and the fact that $\boldsymbol \pi ^{in}$ is symmetric, the first term in (2.39) can be expressed as

\begin{align*} - \int_{\Gamma} { \bar{\Phi} \cdot \boldsymbol\pi^{in} }\cdot \mathbf{n} \,{\rm d}S & =\int_{\Gamma} \overline{{\rm d}(\mathbf{y})}\mathbf{t} \cdot \left[\left( \boldsymbol\pi^{out} - \boldsymbol\pi^{in} \right)\cdot\mathbf{n}\right] - \int_{\Gamma} \overline{{\rm d}(\mathbf{y})}\mathbf{t} \cdot (\boldsymbol\pi^{out}\cdot\mathbf{n})\,{\rm d}S\\ & ={-} \int_{\Gamma} \bar{\Phi} \cdot \boldsymbol\pi^{out} \cdot\mathbf{n} \,{\rm d}S \end{align*}

and hence the variational form (2.39) becomes for all $\Phi \in H(Q_2)$

(2.40)\begin{equation} - \int_{Q_2}2\mu_1 e(\mathbf{u}^{in}) : \overline{e(\Phi)}\,{\rm d}\mathbf{y} = \int_{Q_2} \mathbf{f}^{in}\cdot \bar{\Phi}\,{\rm d}\mathbf{y} + \int_{\Gamma} \bar{\Phi} \cdot \boldsymbol\pi^{out} \cdot\mathbf{n}\,{\rm d}S \end{equation}

To bound the right-hand side of (2.40), we first extend $\Phi \in H(Q_2)$ by the operator $T$ described in (2.32)

(2.41)\begin{equation} \left\lVert{T(\Phi)}\right\rVert_{Q}\leq E_1\left\lVert{\Phi}\right\rVert_{Q_2}\end{equation}

$T(\Phi )$ rapidly decays to zero in a small neighbourhood of $Q_2$ and stays $0$ for the rest of $Q_1$. The restriction of $T(\Phi )$ in $Q_1$, denoted by $\Phi ^{out}$, has the following estimate

(2.42)\begin{equation} \left\lVert{\Phi^{out}}\right\rVert_{Q_1} \leq \left\lVert{T(\Phi)}\right\rVert_{Q}\leq E_1\left\lVert{\Phi}\right\rVert_{Q_2}\end{equation}

Hence

\begin{align*} \left| \int_{\Gamma} \bar{\Phi} \cdot \boldsymbol\pi^{out} \cdot\mathbf{n}\,{\rm d}S\right|& = \left| \int_{\Gamma} \bar{\Phi}^{out} \cdot \boldsymbol\pi^{out} \cdot\mathbf{n}\,{\rm d}S \right|\\ & = \left| \int_{Q_1} \left[ {\rm div}_{\mathbf{y}} \left( 2\mu_1 e(\mathbf{u}^{out}) - p^{out}\mathbf{I} \right) \right] \cdot \overline{\Phi^{out}}\,{\rm d}\mathbf{y}\right.\\ & \quad + \left.\int_{Q_1} 2\mu_1 e(\mathbf{u}^{out}): \overline{e(\Phi^{out})}\,{\rm d}\mathbf{y} \right|\\ & \leq \left| \int_{Q_1} \mathbf{f}^{out} \cdot \overline{\Phi^{out}}\,{\rm d}\mathbf{y}\right| + \left|\int_{Q_1} 2\mu_1 e(\mathbf{u}^{out}): \overline{e(\Phi^{out})}\,{\rm d}\mathbf{y} \right|\\ & \leq \frac{E_1}{B_1(Q_{{1}})}\left\lVert{\mathbf{f}^{out}}\right\rVert_{L^{2}(Q_1)}\left\lVert{\Phi}\right\rVert_{Q_2} + E_1\left\lVert{\mathbf{u}^{out}}\right\rVert_{Q_1}\left\lVert{\Phi}\right\rVert_{Q_2} \end{align*}

Therefore the right-hand side of (2.40) is bounded by

\[ \left\lVert{\Phi}\right\rVert_{Q_2} \left(\frac{E_1 \left\lVert{\mathbf{f}^{in}}\right\rVert_{L^{2}(Q_2)}}{B_1(Q_{{2}})} + \frac{E_1 \left\lVert{\mathbf{f}^{out}}\right\rVert_{L^{2}(Q_1)}}{B_1(Q_{{1}})} + E_1\left\lVert{\mathbf{u}^{out}}\right\rVert_{Q_1} \right) \]

Finally, by the Lax–Milgram lemma a unique solution $\mathbf {u}^{in}$ exists such that

(2.43)\begin{equation} \left\lVert{\mathbf{u}^{in}}\right\rVert_{Q_2} \leq\frac{E_1}{B_1(Q_{{2}})}\left\lVert{\mathbf{f}^{in}}\right\rVert_{L^{2}(Q_2)} + \frac{E_1}{B_1(Q_{{1}})}\left\lVert{\mathbf{f}^{out}}\right\rVert_{L^{2}(Q_1)} + E_1\left\lVert{\mathbf{u}^{out}}\right\rVert_{Q_1} \end{equation}

The construction of the solution for $z$ with large magnitude will be carried out using the following steps.

1. (i) $O(w^{-1})$: Consider the system of (2.24) and (2.27) for $\mathbf{u}_0^{in}(\mathbf {y};\mathbf {e})\in H(Q_2)$.

   (2.44)\begin{equation} \left \{\begin{array}{@{}ll} {\rm div}_{\mathbf{y}} \left( 2\mu_1 e(\mathbf{u}_0^{in}) \right) = {{\mathbf 0}} & \text{in } Q_2\\ 2\mu_1e(\mathbf{u}_0^{in}) \cdot \mathbf{n} = C(\mathbf{y}) \mathbf{n} & \text{on } \Gamma \end{array} \right. \end{equation}
   The variational formulation is
   (2.45)\begin{equation} -\int_{\Gamma} \bar{\mathbf{v}}\cdot 2\mu_1e(\mathbf{u}_0^{in}) \cdot\mathbf{n} - \int_{Q_2}2\mu_1 e(\mathbf{u}_0^{in}) : \overline{e(\mathbf{v})} = 0 \quad \forall \mathbf{v}\in H(Q_2) \end{equation}
   The first term vanishes because of the boundary conditions. Hence
   \[ \mathbf{u}_0^{in}(\mathbf{y};\mathbf{e}) = {{\mathbf 0}} \text{ in } Q_2 \text{ because}\ H(Q_2)\perp \mathcal{R}(Q_2). \]

2. (ii) $O(w^{0})$ in $Q_1$: Solve the system of (2.29) and (2.22) for $\mathbf{u}_0^{out}(\mathbf {y};\mathbf {e}) \in H(Q_1)$.

   (2.46)\begin{equation} \left \{\begin{array}{@{}ll} {\rm div}_{\mathbf{y}} \left( 2\mu_1 e(\mathbf{u}_0^{out}) - p_0^{out}\mathbf{I} \right)={-}\mathbf{e} & \text{in } Q_1\\ \mathbf{u}_0^{out} = \mathbf{u}_0^{in} = {{\mathbf 0}} & \text{on } \Gamma \end{array}\right. \end{equation}
   An application of lemma 2.1 and (2.37) leads to the following result
   (2.47)\begin{equation} \left\lVert{\mathbf{u}_0^{out}}\right\rVert_{Q_1} \leq \frac{\sqrt{|Q_1|}}{B_1(Q_{{1}})} + 2E_1 \left\lVert{\mathbf{u}^{in}_0}\right\rVert_{Q_2}= \frac{\sqrt{|Q_1|}}{B_1(Q_{{1}})} \end{equation}

3. (iii) $O(w^{0})$ in $Q_2$: Consider the system of (2.25) and (2.28) for $\mathbf{u}_1^{in}\in H(Q_2)$:

   \[ \left\{\begin{array}{@{}ll} {\rm div}_{\mathbf{y}}\left( 2\mu_1 e(\mathbf{u}_1^{in}) - p_0^{in}\mathbf{I} \right) ={-}\mathbf{e} & \text{in } Q_2\\ \mathbf{n}\times\mathbf{n}\times\left[\left( \left( 2\mu_1 e(\mathbf{u}_0^{out}) - p_0^{out}\mathbf{I} \right) - \left( 2\mu_1 e(\mathbf{u}_{1}^{in}) - p_0^{in}\mathbf{I} \right) \right)\cdot \mathbf{n}\right] = {{\mathbf 0}} & \text{on } \Gamma \end{array}\right. \]
   By applying lemma 2.2 and (2.43) with $\mathbf {f}^{out} = -\mathbf {e}$ and $\mathbf {f}^{in} = -\mathbf {e}$, we obtain
   (2.48)\begin{equation} \left\lVert{\mathbf{u}_{1}^{in}}\right\rVert_{Q_2} \leq {C_1E_1},\quad {C_1}:=\frac{\sqrt{|Q_2|}}{B_1(Q_{{2}})} + 2\frac{\sqrt{|Q_1|}}{B_1(Q_{{1}})} \end{equation}

4. (iv) Induction step, $k\geq 1$: Given $\mathbf{u}_{k}^{in} \in H(Q_2)$ and $\mathbf{u}_{k-1}^{out}\in H(Q_1)$, find $\mathbf{u}_{k+1}^{in}(\mathbf {y};{{\mathbf 0}}) \in H(Q_2)$ and $\mathbf{u}_{k}^{out}(\mathbf {y};{{\mathbf 0}})\in H(Q_1)$.

   1. (a) Applying lemma 2.1 with $\mathbf {f} = {{\mathbf 0}}$, we conclude that for a given $\mathbf{u}_{k}^{in} \in H(Q_2)$, $k\ge 1$, there exists a unique $\mathbf{u}_{k}^{out}\in H(Q_1)$ that solves the system of (2.23) and (2.29) and assumes the estimate

      (2.49)\begin{gather} \left\{\begin{array}{@{}l} {\rm div}_{\mathbf{y}} \left( 2\mu_1 e(\mathbf{u}_{k}^{out}) - p_{k}^{out}\mathbf{I} \right) = {{\mathbf 0}}\quad \text{in } Q_1\\ \mathbf{u}_{k}^{out} = \mathbf{u}_{k}^{in} \quad \text{on } \Gamma, \end{array}\right. \end{gather}
      (2.50)\begin{gather} \left\lVert{\mathbf{u}_{k}^{out}}\right\rVert_{Q_1} \leq 2E_1\left\lVert{\mathbf{u}_{k}^{in}}\right\rVert_{Q_2} \end{gather}

   2. (b) By applying lemma 2.2 with $\mathbf {f}^{in} = {{\mathbf 0}}=\mathbf {f}^{out}$, we see that for any given $\mathbf{u}_{k}^{out}\in H(Q_1)$, $k\ge 1$ that satisfies (2.49), there exists a unique solution $\mathbf{u}_{k+1}^{in} \in H(Q_2)$ to the system of equations (2.26) and (2.28)

      (2.51)\begin{equation} \left\{\begin{array}{@{}ll} {\rm div}_{\mathbf{y}} \left( 2\mu_1 e(\mathbf{u}_{k+1}^{in}) - p_{k}^{in}\mathbf{I} \right) = {{\mathbf 0}} & \text{in } Q_2\\ \mathbf{n}\times\mathbf{n}\times\left[\left( \left( 2\mu_1e(\mathbf{u}_k^{out}) - p_k^{out}\mathbf{I} \right) - \left( 2\mu_1 e(\mathbf{u}_{k+1}^{in}) - p_k^{in}\mathbf{I} \right) \right)\mathbf{n}\right] = {{\mathbf 0}} & \text{on } \Gamma \end{array}\right. \end{equation}
      Moreover,
      (2.52)\begin{equation} \left\lVert{\mathbf{u}_{k+1}^{in}}\right\rVert_{Q_2} \leq E_1\left\lVert{\mathbf{u}_{k}^{out}}\right\rVert_{Q_1} \end{equation}

Now we have found the coefficients $\mathbf {u}^{in}_n(\mathbf {y};\mathbf {e})$ and $\mathbf {u}^{out}_n(\mathbf {y};\mathbf {e})$ in (2.21) iteratively. We prove the convergence of the series in the following theorem by taking into account the fact that $\mathbf{u}_0^{in} = {{\mathbf 0}}$.

Theorem 2.3 Define the partial sums

\[ \mathbf{S}_q^{in}(\mathbf{y};\mathbf{e},w) := \sum_{k = 0}^{q} \mathbf{u}^{in}_{k+1}(\mathbf{y},\mathbf{e})w^{k+1},\,\mathbf{S}_q^{out}(\mathbf{y};\mathbf{e},w) := \sum_{k = 0}^{q} \mathbf{u}^{out}_{k}(\mathbf{y};\mathbf{e}) w^{k}. \]

Let $R\in (0,1)$, in the disc $|w|\le {R}/{2E_1^{2}}$, the series $\mathbf {S}_q^{in}(\mathbf {y};\mathbf {e},w)$ and $\mathbf {S}_q^{out}(\mathbf {y};\mathbf {e},w)$ converge uniformly to $\mathbf {u}^{in}_{\infty }(\mathbf {y};\mathbf {e},w)\in H(Q_2)$ and $\mathbf {u}^{out}_{\infty }(\mathbf {y};\mathbf {e},w)\in H(Q_1)$, respectively. Therefore, $\mathbf{u}_{\infty }(\mathbf {y};\mathbf {e},w) := \mathbf {u}^{in}_{\infty }(\mathbf {y};\mathbf {e},w)\chi _{2} + \mathbf {u}^{out}_{\infty }(\mathbf {y};\mathbf {e},w)\chi _{1} \in H(Q)$ solves the cell problem (2.6) and is analytic for $|w|< {1}/{2E_1^{2}}$.

Proof. For each $q\in \mathbb {N}$, $\mathbf {S}_{q}^{in}(\mathbf {y};\mathbf {e},w)$ is a polynomial function of $w$ and maps from $\mathbb {C}$ to the Hilbert space $H(Q_2)$. Similarly, $\mathbf {S}_q^{out}(\mathbf {y};\mathbf {e},w)$ maps from $\mathbb {C}$ to $H(Q_1)$. To show uniform convergence, we note that (2.50) and (2.52) imply there exists a positive constant $E_1$ that depends only on $Q_1$ and $Q_2$ such that $\left \lVert {\mathbf {u}^{in}_{k+1}}\right \rVert _{Q_2} \leq E_1\left \lVert {\mathbf {u}^{out}_{k}}\right \rVert _{Q_1} \leq 2E_1^{2} \left \lVert {\mathbf {u}^{in}_{k}}\right \rVert _{Q_2}$. Therefore,

(2.53)\begin{equation} \left\lVert{\mathbf{u}_k^{in}}\right\rVert_{Q_2} \leq\left(2E_1^{2} \right)^{k-1}\left\lVert{\mathbf{u}_1^{in}}\right\rVert_{Q_2},\quad k\ge 1 \end{equation}

Let $m>q>N$, and define $r := 2E_1^{2}\left | w \right |$. Then by (2.48) implies

\begin{align*} \left\lVert{\mathbf{S}_m^{in}(w) - \mathbf{S}_q^{in}(w)}\right\rVert_{Q_2}& \leq \left\lVert{\mathbf{u}_1^{in}}\right\rVert_{Q_2} \left( (2E_1^{2})^{q} |w|^{q+1} + \cdots + (2E_1^{2})^{m-1}\left| w\right|^{m} \right)\\ & \leq \frac{r^{q+1} - r^{m+1}}{1-r} \frac{\left\lVert{\mathbf{u}_1^{in}}\right\rVert_{Q_2}}{2E_1^{2}}\\ & \leq \frac{r^{q+1} - r^{m+1}}{1-r}\left( {\frac{C_1}{2E_1^{2}}} \right), C_1\ \text{is defined in (2.48)}. \end{align*}

Therefore, for $r\le R<1$, i.e. $|w|\le {R}/{2E_1^{2}}$, where $R$ is any fixed number in $(0,1)$,

(2.54)\begin{equation} \left\lVert{\mathbf{S}_m^{in}(w) - \mathbf{S}_q^{in}(w)}\right\rVert_{Q_2} \le \left({\frac{C_1}{2E_1^{2}}} \right)\left(\frac{R^{N+1}}{1-R}\right),\quad\forall m>q>N. \end{equation}

For $\mathbf {S}_q^{out}(\mathbf {y};\mathbf {e},w)$ we have $\left \lVert {\mathbf{u}_k^{out}}\right \rVert _{Q_1} \leq (2E_1^{2})^{k-1}\left \lVert {\mathbf{u}_1^{out}}\right \rVert _{Q_1}$. By a similar procedure, for $m>q>N$ and $|w|\le {R}/{2E_1^{2}}$ the following estimate is valid

\[ \left\lVert{\mathbf{S}_m^{out}(w) -\mathbf{S}_q^{out}(w)}\right\rVert_{Q_1} \leq {C_1}\left(\frac{R^{N+1}}{1-R} \right) \]

Therefore, for every fixed $w$ satisfying $|w|\le {R}/{2E_1^{2}}$ for any $0< R<1$, $\mathbf {S}_q^{in}(\mathbf {y};w)$ and $\mathbf {S}_q^{out}(\mathbf {y};w)$ converge uniformly to $\mathbf {u}^{in}_{\infty }(\mathbf {y};w)\in H(Q_2)$ and $\mathbf {u}^{out}_{\infty }(\mathbf {y};w)\in H(Q_1)$, respectively. Since for each $q$, $\mathbf {S}_q^{in}(\mathbf {y};w)$ and $\mathbf {S}_q^{out}(\mathbf {y};w)$ are polynomials of $w$, hence analytic, the uniform convergence implies that the limit functions $\mathbf {u}^{in}_{\infty }(\mathbf {y};w)$ and $\mathbf {u}^{out}_{\infty }(\mathbf {y};w)$ are also analytic in $|w|< {1}/{2E_1^{2}}$ with values in $H(Q_1)$ and $H(Q_2)$, respectively, by applying Morera's theorem for Banach space valued analytic functions [Reference Limaye27] to the uniformly converging sequences. By construction, the function $\mathbf{u}_{\infty }(\mathbf {y};\mathbf {e},w)$ defined in (2.20) solves the cell problem (2.6) for all $w$ in the disc $\left \{w: |w|<{1}/{2E_1^{2}}\right \}:=B_0\left ({1}/{2E_1^{2}}\right )$. Moreover, the uniqueness of the solution implies that $\mathbf{u}_{\infty }(\mathbf {y};\mathbf{e} _k,w) = \mathbf {u}^{k}\left (\mathbf {y};\frac {1}{w}\right ) \text { in } H(Q)$ for $w\in B_0\left ({1}/{2E_1^{2}}\right )\cap \{w\in \mathbb {C}\setminus (-\infty,0]\}$.

The following theorem shows the relation between the two-fluid self-permeability $\boldsymbol K$ in (2.5) and the Darcy permeability $\boldsymbol K^{(D)}$ in (1.5)

Proof. The uniform convergence allows passing the limit $w \rightarrow 0$ inside the summation of (2.21) to obtain $\mathbf{u}_{\infty }^{in}(\mathbf {y};\mathbf{e} _i,0) = {{\mathbf 0}}$. Similarly, the uniform convergence allows passing the limit $w \rightarrow 0$ inside the summation of (2.21) to obtain

\[ \mathbf{u}^{out}_{\infty}(\mathbf{y};\mathbf{e}_{{i}},0) =\mathbf{u}_0^{out}(\mathbf{y};\mathbf{e}_i) \]

Furthermore, $\mathbf{u}_0^{out}(\mathbf {y};\mathbf{e} _i) \in H(Q_1)$ satisfies (2.46) and in fact $\mathbf{u}_0^{out}(\mathbf {y};\mathbf{e} _i) \in \mathring {H}(Q_1)$ since $\mathbf{u}_0^{out}(\mathbf {y};\mathbf{e} _i)\vert _{\Gamma } = {{\mathbf 0}}$, which is identical to the equation for $\mathbf{u}_D$ (1.4). The uniqueness of the solution then ensures that $\mathbf{u}_0^{out}(\mathbf {y};\mathbf{e} _i) = \mathbf{u}_D^{i}(\mathbf {y})$. Therefore the series $\mathbf{u}_{\infty }^{out}(\mathbf {y};\mathbf{e} _i,w) \to \mathbf{u}_D^{i}(\mathbf {y})$ uniformly as $|w|\to 0$ in $\mathring {H}(Q_1)$. For (iii), we note that

\begin{align*} \left| K_{ij} (w)- K^{(D)}_{ij} \right| & = \left| \int_{Q} \left( \mathbf{u}^{i} - \chi_1 \mathbf{u}_D^{i}\right) \cdot \mathbf{e}_j\,{\rm d}\mathbf{y}\right| \leq \left\lVert{\mathbf{u}^{i} - \chi_1\mathbf{u}^{i}_D}\right\rVert_{L^{2}(Q)}\\ & \le \frac{1}{B_1(Q)} \left\lVert{\sum_{k=1}^{\infty} \left( \mathbf{u}_{k}^{in}(\mathbf{y};\mathbf{e}_i)\chi_{2} + \mathbf{u}_{k}^{out}(\mathbf{y};\mathbf{e}_i)\chi_{1} \right)w^{k} }\right\rVert_Q \end{align*}

From (2.50), (2.53) and (2.48), we have for $|w|<{1}/{2E_1^{2}}$, or equivalently $|z| > 2E_1^{2}$,

(2.55)\begin{equation} \left| K_{ij} (w)- K^{(D)}_{ij} \right| \leq {C_1\left(\frac{E_1+1}{2E_1 B_1(Q)}\right)}\frac{2E_1^{2} |w|}{1-2E_1^{2} |w|} \end{equation}

In the following section, we study the behaviour of $\boldsymbol K(z)$ near $z=0$, i.e. the inclusion is an air bubble.

2.4 Analyticity of the solution for small $|z|$

Let $\mathbf {e}$ be a constant unit vector in $\mathbb {R}^{n}$. We seek solutions of the following form

(2.56)\begin{align} \mathbf{u}_{null}^{in}(\mathbf{y};\mathbf{e},z) & = \sum_{k = 0}^{\infty} \mathbf{u}^{in}_k(\mathbf{y};\mathbf{e}) z^{k} ,\quad p^{in}(\mathbf{y};\mathbf{e},z) = \sum_{k = 0}^{\infty} p^{in}_k(\mathbf{y};\mathbf{e}) z^{k} \text{ in } Q_2, \end{align}

(2.57)\begin{align} \mathbf{u}_{null}^{out}(\mathbf{y};\mathbf{e},z) & = \sum_{k = 0}^{\infty} \mathbf{u}^{out}_k(\mathbf{y};\mathbf{e}) z^{k},\quad p^{out}(\mathbf{y};\mathbf{e},z) = \sum_{k = 0}^{\infty} p^{out}_k(\mathbf{y};\mathbf{e}) z^{k} \text{ in } Q_1 \end{align}

By a procedure similar to that in § 2.3, the following equations are obtained via collecting terms with respect to the order of $z$. The PDEs for $Q_1$ are as follows:

(2.58)\begin{align} & O(1):\quad {\rm div}_{\mathbf{y}} \left( 2\mu_1 e(\mathbf{u}_0^{out}) - p_0^{out}\mathbf{I} \right) ={-}\mathbf{e} \end{align}

(2.59)\begin{align} & O(z^{k}), \,k\geq 1:\quad {\rm div}_{\mathbf{y}} \left( 2\mu_1 e(\mathbf{u}_k^{out}) - p_k^{out}\mathbf{I} \right) = {{\mathbf 0}} \end{align}

Similarly, the PDEs for $Q_2$ are

(2.60)\begin{align} & O(1):\quad -\nabla p_0^{in}={-}\mathbf{e} \end{align}

(2.61)\begin{align} & O(z^{k}),\,k\geq 1 :{\rm div}_{\mathbf{y}} \left( 2\mu_1 e(\mathbf{u}_{k-1}^{in}) - p_k^{in}\mathbf{I} \right) = {{\mathbf 0}} \end{align}

The interface condition (2.29) remains the same for the small $|z|$ case while (2.27) and (2.28) now read

(2.62)\begin{align} & \mathbf{n} \times \mathbf{n} \times \left[ \left( \left( 2\mu_1 e(\mathbf{u}_0^{out}) - p_0^{out}\mathbf{I} \right)\cdot\mathbf{n} - ( - p_0^{in}\mathbf{I} \right) \cdot\mathbf{n} \right] = {{\mathbf 0}} \end{align}

(2.63)\begin{align} & \mathbf{n} \times \mathbf{n} \times \left[ \left(\left(2\mu_1 e(\mathbf{u}_k^{out}) - p_k^{out}\mathbf{I} \right)- \left( 2\mu_1 e(\mathbf{u}_{k-1}^{in})- p_k^{in}\mathbf{I} \right)\right) \cdot\mathbf{n} \right]= {{\mathbf 0}},\quad k\ge 1 \end{align}

The first equation to be solved is (2.60), whose solution is simply

(2.64)\begin{equation} p_0^{in}(\mathbf{y})=\mathbf{e}\cdot \mathbf{y}+{c}-\int_{Q_2} (\mathbf{e}\cdot \mathbf{y}+{c})\,{\rm d}\mathbf{y} \text{ in } Q_2 \end{equation}

where ${c}$ is a constant. The next problem is the system of (2.58) and (2.62). Similar to the calculation in lemma 2.2, the weak formulation of this system is: find $u_0^{out} \in H(Q_1)$ such that for all $\boldsymbol \Phi \in H(Q_1)$ and $\pi _0^{out}:=2\mu _1 e(\mathbf{u}_0^{out}) - p_0^{out}\mathbf {I}$

\[ - \int_{\Gamma} (\bar{\Phi} \cdot(\pi_0^{out}+p_0^{in}\mathbf{I})-p_0^{in}\mathbf{I})\cdot \mathbf{n}\,{\rm d}S - \int_{Q_1} 2\mu_1 e(\mathbf{u}_0^{out}) : \overline{e(\Phi)}\,{\rm d}\mathbf{y} = \int_{Q_1} -\mathbf{e} \cdot \bar{\Phi}\,{\rm d}\mathbf{y} \]

Since $\boldsymbol \Phi \cdot \mathbf {n}=0$ and $p_0^{in} \mathbf {I} \cdot \mathbf {n}$ is parallel to $\mathbf {n}$, (2.62) implies the integral on $\Gamma$ vanishes. Hence by the Lax–Milgram lemma, we have

(2.65)\begin{equation} \|\mathbf{u}_0^{out}\|_{Q_1}\le \frac{\sqrt{|Q_1|}}{B_1(Q_{{1}})} \end{equation}

The system for $\mathbf{u}_{k-1}^{in}$, $k\ge 1$ (inner problem) is to find $\mathbf{u}_{k-1}^{in}\in H(Q_2)$ with given $\mathbf{u}_0^{out}\in H(Q_1)$ such that

(2.66)\begin{equation} \left\{\begin{array}{@{}l} {\rm div} \left( 2\mu_1 e(\mathbf{u}_{k-1}^{in}) - p_k^{in}\mathbf{I} \right) = {{\mathbf 0}} \text{ in } Q_2\\ \mathbf{u}_{k-1}^{in}|_\Gamma= \mathbf{u}_{k-1}^{out}|_\Gamma \end{array}\right. \end{equation}

With an argument similar to the derivation of lemma 2.1, the following estimate can be derived for system (2.66)

Lemma 2.5 Let $Q_2$ satisfy the same assumption in lemma 2.1. For any given vector field $\mathbf {u}^{out} \in H(Q_1)$, there exists a unique weak solution $\mathbf {u}^{in}(\mathbf {y})\in H(Q_2)$ s.t.

(2.67)\begin{align} & \left\{\begin{array}{@{}ll}{\rm div}_{\mathbf{y}} \left( 2\mu_1 e(\mathbf{u}^{in}) - p^{in}\mathbf{I} \right) = \mathbf{f}^{in} & \text{ in } Q_2\\ \mathbf{u}^{in} = \mathbf{u}^{out} & \text{ on }\Gamma \end{array}\right. \end{align}

(2.68)\begin{align} & \left\lVert{\mathbf{u}^{in}}\right\rVert_{Q_2} \leq \frac{1}{B_1(Q_{{2}})} \left\lVert{\mathbf{f}^{in}}\right\rVert_{L^{2}(Q_2)} + 2E_2\left\lVert{\mathbf{u}^{out}}\right\rVert_{Q_1}. \end{align}

where $E_2>1$ is the constant associated with the extension operator $T$, $\|T(\boldsymbol \Phi )\|_Q\le E_2 \| \boldsymbol \Phi \|_{Q_{{1}}}$ for all $\boldsymbol \Phi \in H(Q_{{1}})$ and $T(\boldsymbol \Phi )$ decays rapidly to 0 inside $Q_2$. Note that the periodic condition of space $H(Q_1)$ implies $\int _\Gamma \mathbf {u}^{out} \cdot \mathbf {n} \,{\rm d}S=0$.

The system for $\mathbf{u}_k^{out}$ and $p_k^{out}$ with given $\mathbf{u}_{k-1}^{in}\in H(Q_2)$ and $p_k^{in}$, $k\ge 1$ is

(2.69)\begin{equation} \left\{\begin{array}{@{}l} {\rm div}\left( 2\mu_1 e(\mathbf{u}_k^{out}) - p_k^{out}\mathbf{I} \right) = {{\mathbf 0}}\\ \mathbf{n} \times \mathbf{n} \times \left[ \left( \left( 2\mu_1 e(\mathbf{u}_k^{out}) - p_k^{out}\mathbf{I} \right)- \left( 2\mu_1 e(\mathbf{u}_{k-1}^{in})- p_k^{in}\mathbf{I} \right)\right) \cdot\mathbf{n} \right]= {{\mathbf 0}} \end{array}\right. \end{equation}

By an argument similar to the one for lemma 2.2, the system above can be shown to satisfy the following estimate.

Lemma 2.6 Let $Q_2$ satisfy the same assumption in lemma 2.1. For any given pair of $( \mathbf {u}^{in}, p^{in}) \in H(Q_2) \times L^{2}(Q_1)/ \mathbb {C}$ that satisfies (2.67), there exists a unique vector $\mathbf {u}^{out}(\mathbf {y};\mathbf {f}^{out})\in H(Q_1)$ solving the following system

(2.70)\begin{align} & \left\{\begin{array}{@{}l} {\rm div} \left( 2\mu_1 e(\mathbf{u}^{out}) - p^{out}\mathbf{I} \right) = \mathbf{f}^{out} \text{ in }Q_1\\ \mathbf{n}\times\mathbf{n}\times\left[\left(\left( 2\mu_1 e(\mathbf{u}^{in}) - p^{in}\mathbf{I} \right) - \left( 2\mu_1 e(\mathbf{u}^{out}) - p^{out}\mathbf{I} \right) \right)\cdot\mathbf{n}\right] = {{\mathbf 0}} \text{ on }\Gamma\end{array}\right. \end{align}

(2.71)\begin{align} & \left\lVert{\mathbf{u}^{out}}\right\rVert_{Q_1} \leq \frac{E_2}{B_1(Q_{{1}})}\left\lVert{\mathbf{f}^{out}}\right\rVert_{L^{2}(Q_1)} + \frac{E_2}{B_1(Q_{{2}})}\left\lVert{\mathbf{f}^{in}}\right\rVert_{L^{2}(Q_2)} + E_2\left\lVert{\mathbf{u}^{in}}\right\rVert_{Q_2} \end{align}

where $E_2$, $B_1$ depend only on $Q$ and $\Gamma$.

Equation (2.69), lemma 2.5, equation (2.66) and lemma 2.6 imply that for all $k\ge 0$, we have $\|\mathbf{u}_k^{in}\|_{Q_2}\le 2E_2 \|\mathbf{u}_k^{out}\|_{Q_1}$ and $\|\mathbf{u}_{k+1}^{out}\|_{Q_1}\le E_2 \|\mathbf{u}_k^{in}\|_{Q_2}$. Therefore,

(2.72)\begin{align} & \| \mathbf{u}^{in}_k\|_{Q_2} \le\frac{(2E_2^{2})^{k+1}}{E_2} \|\mathbf{u}_0^{out}\|_{Q_1}\le {(2E_2^{2})^{k+1}}\left( \frac{|Q_1|}{E_2B_1(Q_{{1}})}\right) \end{align}

(2.73)\begin{align} & \| \mathbf{u}^{out}_k\|_{Q_1} \le (2E_2^{2})^{k} \|\mathbf{u}_0^{out}\|_{Q_1}\le (2E_2^{2})^{k} \left( \frac{|Q_1|}{B_1(Q_1)}\right) \end{align}

Therefore, the series in (2.56) and (2.57) converge uniformly in the disk $|z|<{1}/{2(E_2)^{2}}$ to an analytic function in $Q_2$ and $Q_1$, respectively. The limit functions $\mathbf{u}_{null}^{in}(\mathbf {y},\mathbf {e},z)$, $\mathbf{u}_{null}^{out}(\mathbf {y},\mathbf {e},z)$ and the corresponding permeability $K_{ij}(z)$ in (2.12) are analytic at $z=0$. Define the permeability (‘B’ for ‘bubbles’)

(2.74)\begin{equation} K_{ij}^{(B)}:=\int_Q [\chi_1 \mathbf{u}_0^{out}(\mathbf{y};\mathbf{e}_i) + \chi_2 \mathbf{u}_0^{in}(\mathbf{y};\mathbf{e}_i)]\cdot \mathbf{e}_j\,{\rm d}\mathbf{y} \end{equation}

then the following estimate, valid for $|z|<{1}/{2E_2^{2}}$, holds

(2.75)\begin{equation} |K_{ij}(z)-K^{B}_{ij}|\le \frac{|Q_1|(1+2E_2)}{{B_1(Q) B_1(Q_1)}}\left(\frac{2E_2^{2}|z|}{1-2E_2^{2}|z|} \right) =O(|z|). \end{equation}

In conclusion, $\boldsymbol K(z)$ in (2.5) is analytic for $z\in \mathbb {C}\setminus [-2E_1^{2},-({1}/{2E_2^{2}})]$, $E_1, E_2\ge 1$. In the next section, and IRF for $\boldsymbol K(z)$ will be derived in two different ways.

3.1 Spectral representation of $\boldsymbol K(z)$

Adding $\int _{Q_2}2\mu _1 e( \mathbf {u}^{k}) : \overline {e(\mathbf {v})}\,{\rm d}\mathbf {y}$ to both sides of (2.14), we have

(3.3)\begin{equation} \int_{Q}2\mu_1 e( \mathbf{u}^{k}) : \overline{e(\mathbf{v})}\,{\rm d}\mathbf{y} ={-}\frac{1}{s}\int_{Q}2\mu_1 \chi_2 e(\mathbf{u}^{k}) : \overline{e(\mathbf{v})}\,{\rm d}\mathbf{y} + \int_{Q} \mathbf{e}_k\cdot \bar{\mathbf{v}}\,{\rm d}\mathbf{y} \end{equation}

where the new variable $s$ is defined as

\[ s:=\frac{1}{z-1} \]

Let $\Delta _{\#}^{-1}$ be the operator that solves for $\mathbf {w}(\mathbf {y};\mathbf {f}) \in H(Q)$ in the following variational formulation

(3.4)\begin{equation} \int_{Q} 2\mu_1 e({{\mathbf{w}}}):\overline{e(\mathbf{v})}\,{\rm d}\mathbf{y} = \int_{Q}\mathbf{f}\cdot \bar{\mathbf{v}}\,{\rm d}\mathbf{y} \end{equation}

where $\mathbf {f} \in L^{2}(Q)$ and $Q$-periodic. In other words, solution ${{\mathbf {w}}}(\mathbf {y}) = \Delta _{\#}^{-1} \mathbf {f} \in H(Q)$ is a weak solution to the cell problem

(3.5)\begin{equation} \left\{\begin{array}{@{}l} -\mu_1\Delta {{\mathbf{w}}} = \mathbf{f} \quad \text{in } Q_1 \cup Q_2\\ [\kern-1pt[ \boldsymbol\pi ]\kern-1pt]\mathbf{n} = \left([\kern-1pt[ \boldsymbol\pi \mathbf{n}]\kern-1pt] \cdot \mathbf{n} \right) \mathbf{n} \text{ on } \Gamma \end{array}\right. \end{equation}

In order to get the spectral representation, we apply $\Delta _{\#}^{-1}$ on both sides of (3.3) and symbolically represent the resulted equations as

\[ {{\mathbf{w}}}_1 = {-\frac{1}{s}}{{\mathbf{w}}}_2 + {{\mathbf{w}}}_3 \]

Then clearly, we have ${{\mathbf {w}}}_1 = \mathbf {u}^{k}$ and ${{\mathbf {w}}}_3 =\Delta _{\#}^{-1}\mathbf{e} _k$. Observe that ${{\mathbf {w}}}_2$ solves

(3.6)\begin{equation} \int_{Q} 2\mu_1e({{\mathbf{w}}}_2):\overline{e(\mathbf{v})}\,{\rm d}\mathbf{y} = \int_{Q} 2\mu_1 \chi_2 e( \mathbf{u}^{k}) : \overline{e(\mathbf{v})}\,{\rm d}\mathbf{y} \text{ for all } \mathbf{v}\in H(Q) \end{equation}

Define the operator $\Gamma _{\chi }$ such that ${{\mathbf {w}}}_2=\Gamma _{\chi } \mathbf {u}^{k}$ and (3.6) can be expressed as

(3.7)\begin{equation} (\Gamma_\chi \mathbf{u}^{{k}},\mathbf{v})_Q = \int_{Q} 2\mu_1 \chi_2 e( \mathbf{u}^{{k}}) : \overline{e(\mathbf{v})}\,{\rm d}\mathbf{y} \text{ for all } \mathbf{v}\in H(Q). \end{equation}

The subscript $\chi$ is used to signify the dependence of $\Gamma _\chi$ on $\chi _2$, the characteristic function of $Q_2$. Clearly, $\Gamma _\chi$ is self-adjoint with respect to the inner product $(\cdot,\cdot )_Q$ because

\[ (\Gamma_\chi \mathbf{u},\mathbf{v})_Q=\overline{\int_{Q} 2\mu_1\chi_2 e(\mathbf{v}) : \overline{e(\mathbf{u})}\,{\rm d}\mathbf{y}}= \overline{(\Gamma_\chi \mathbf{v},\mathbf{u})_Q } = {(\mathbf{u},\Gamma_\chi \mathbf{v})_Q.} \]

Formally, we have $\Gamma _\chi \mathbf {u}=\triangle _\#^{-1}(\nabla \cdot \chi _2 e(\mathbf {u}))$. Now (3.3) becomes

(3.8)\begin{equation} \mathbf{u}^{k} ={-}\frac{1}{s} \Gamma_{\chi} \mathbf{u}^{k} + \Delta_{\#}^{{-}1}\mathbf{e}_k \Leftrightarrow \left( I +\frac{\Gamma_{\chi}}{s} \right)\mathbf{u}^{k} = \Delta_{\#}^{{-}1}\mathbf{e}_k \end{equation}

Theorem 3.3 For $|s|>1$, the solution $\mathbf {u}^{k}\in H(Q)$ admits a series representation

(3.9)\begin{equation} \mathbf{u}^{k}(\mathbf{y};s) = \sum_{m = 0}^{\infty}\left( -\frac{1}{s} \right)^{m}\left(\Gamma_{\chi} \right)^{m} \Delta_{\#}^{{-}1}\mathbf{e}_k\end{equation}

and the components of $\boldsymbol K$ can be represented by the following IRF

(3.10)\begin{equation} K_{kl}(s) = s\int_0^{1}\int_{Q}\frac{(\tilde{M}(d\lambda)\Delta_{\#}^{{-}1}\mathbf{e}_k)_l}{s+\lambda}\,{\rm d}\mathbf{y},\quad k,l=1,\ldots,n,\end{equation}

for some projection-valued measures $\tilde {M}(d\lambda )$ and a series representation

\[ K_{kl}(s) = \int_{Q} \left( \Delta_{\#}^{{-}1}\mathbf{e}_k\right)_l\,{\rm d}\mathbf{y} + \sum_{m = 1}^{\infty} \frac{\tilde{\lambda}_{kl}^{m}}{({-}s)^{m}}\quad \text{with }\tilde{\lambda}_{kl}^{m} := \int_{Q} \left(\left(\Gamma_{\chi} \right)^{m} \Delta_{\#}^{{-}1}\mathbf{e}_k \right)_l\,{\rm d}\mathbf{y}. \]

Proof. From (3.8), since $\Gamma _{\chi }$ is self-adjoint with norm bounded by 1, for $|s|>1$, the spectral theory for self-adjoint operator implies the existence of a projection-valued measure $\tilde {M}$ such that

(3.11)\begin{equation} \mathbf{u}^{k}(\mathbf{y};s) = \left(I + \frac{\Gamma_{\chi}}{s} \right)^{{-}1} \Delta_{\#}^{{-}1}\mathbf{e}_k = {s}\int_0^{1} \frac{\tilde{M}({\rm d}\lambda)(\Delta_{\#}^{{-}1}\mathbf{e}_k)}{s+\lambda}\end{equation}

Hence the $kl$-the element of permeability $\boldsymbol K$ has the following IRF

(3.12)\begin{equation} K_{kl}(s) = \int_{Q} (\mathbf{u}^{k})_l\,{\rm d}\mathbf{y}=s\int_0^{1}\int_{Q}\frac{(\tilde{M}({\rm d}\lambda)\Delta_{\#}^{{-}1}\mathbf{e}_k)_l}{s+\lambda}\,{\rm d}\mathbf{y} \end{equation}

On the other hand, for $|s|>1$, the geometric expansion of the middle term near $s=\infty$ in (3.11) results in the following expression

(3.13)\begin{equation} K_{kl}(s) = \int_{Q} \left[ \sum_{m = 0}^{\infty}\left(-\frac{1}{s} \right)^{m}\left( \Gamma_{\chi} \right)^{m} \Delta_{\#}^{{-}1}\mathbf{e}_k \right]\cdot \mathbf{e}_l\,{\rm d}\mathbf{y}= \sum_{m = 0}^{\infty} \frac{\tilde{\lambda}_{kl}^{m}}{({-}s)^{m}}\end{equation}

where $\tilde {\lambda }_{kl}^{m}$ is defined as $\tilde {\lambda }_{kl}^{m} := \int _{Q}((\Gamma _{\chi })^{m} \Delta _{\#}^{-1}\mathbf{e} _k )_l\,{\rm d}\mathbf {y}.$

For the three-dimensional space $n=3$, the expansion (3.12) can be cast in the matrix form

(3.14)\begin{equation} \boldsymbol K(s)=\sum_{m = 0}^{\infty} \frac{\tilde{\boldsymbol\Lambda}_{m}}{({-}s)^{m}} \end{equation}

with the matrix-valued moments defined as

(3.15)\begin{equation} \tilde{\boldsymbol\Lambda}_m:=\begin{pmatrix} \int_{Q} \left(\Gamma_{\chi} \right)^{m} \Delta_{\#}^{{-}1}\mathbf{e}_1\,{\rm d}\mathbf{y} & \int_{Q} \left(\Gamma_{\chi} \right)^{m} \Delta_{\#}^{{-}1}\mathbf{e}_2\,{\rm d}\mathbf{y} & \int_{Q} \left(\Gamma_{\chi} \right)^{m} \Delta_{\#}^{{-}1}\mathbf{e}_3\,{\rm d}\mathbf{y} \end{pmatrix} \end{equation}

3.2 Relationships between two representations and characterization of the microstructural information on permeability

The calculations in the previous section reveal that the variable $s:={1}/{(z-1)}$ is the natural one to use. Because of this, we will consider $\boldsymbol K$ as a function of $s$. Note that $s$ maps $(-\infty,0]$ on the $z$-plane to $[-1,0]$ on the $s$-plane. The following properties of $\boldsymbol K(s)$ can be easily deduced from the results in proposition 3.1.

1. (i) $\boldsymbol K(s)$ is holomorphic in $\mathbb {C}\setminus {\left [-\frac {2E_2^{2}}{1+2E_2^{2}},-\frac {1}{1+2E_1^{2}}\right ]}.$

2. (ii) $\frac {\boldsymbol K(s)-(\boldsymbol K(s))^{*}}{s-\overline {s}}\ge 0$ for all $Im{(s)}\ne 0$

3. (iii) $\boldsymbol K(s)\ge 0 \text { for } \mathbb {R}\ni s>0$ because $s>0$ iff $\mathbb {R}\ni z>1$.

Then by the representation theorem in [Reference Dyukarev and Katsnelson20, theorem 3.1], there exists a monotonically increasing matrix-valued function $\pmb {\sigma }(t)$, matrices $\pmb {A}\ge 0$ and $\pmb {C}\ge 0$ such that $\int _{+0}^{\infty } \frac {{\rm d}\pmb {\sigma }}{1+t}<\infty$, $\pmb {A}+\pmb {C}+\int _{+0}^{\infty } \frac {{\rm d}\pmb {\sigma }}{1+t}>0$ and

(3.16)\begin{equation} \boldsymbol K(s)=\pmb{A}+{\pmb{C}}{s}+\int_{{+}0}^{\infty} \frac{s}{s+t}\,{\rm d}\pmb{\sigma}(t), \end{equation}

As $s\rightarrow \infty$, $z\rightarrow 1$ and hence $\boldsymbol K\rightarrow \boldsymbol K(z=1)$. Therefore, we must have ${\pmb {C}=\pmb {0}}$. Moreover, $\pmb {A}=\boldsymbol K(s=0)=\boldsymbol K^{(D)}$. Also, since $\boldsymbol K(s)$ is holomorphic in $\mathbb {C}\setminus {\left [-\frac {2E_2^{2}}{1+2E_2^{2}},-\frac {1}{1+2E_1^{2}}\right ]}$, we have

(3.17)\begin{equation} \boldsymbol K(s)=\pmb{K}^{(D)}+\int_{{1}/({1+2E_1^{2}})}^{{2E_2^{2}}/({1+2E_2^{2}})} \frac{s}{s+t}\,{\rm d}\pmb{\sigma}(t), \end{equation}

which is valid for all $s\in \mathbb {C}\setminus {\left [-\frac {2E_2^{2}}{1+2E_2^{2}},-\frac {1}{1+2E_1^{2}}\right ]}\subset (-1,0)$. To compare with (3.14), which is valid only for $|s|>1$, we expand (3.17) near $s=\infty$ to obtain the following series expansion

(3.18)\begin{equation} \boldsymbol K(s)=\boldsymbol K^{(D)}+\sum_{m=0}^{\infty} ({-}1)^{m} \left(\frac{1}{s} \right)^{m+1} \boldsymbol{\mu}^{\sigma}_m \end{equation}

where $\boldsymbol {\mu }^{\sigma }_m$ is the $m$-th moment of the measure ${\rm d}\pmb {\sigma }$. Equating the coefficients term by term with (3.14) leads to the following relation between $\boldsymbol {\mu }^{\sigma }_m$ and the ‘geometrical information’ coefficients in (3.15)

(3.19)\begin{align} & \boldsymbol K^{(D)}+\boldsymbol{\mu}^{\sigma}_0=\tilde{\boldsymbol\Lambda}_0={\boldsymbol K(s=\infty)},\text{ i.e. }\ \boldsymbol{\mu}^{\sigma}_0=\boldsymbol K(z=1)-\boldsymbol K^{(D)} \end{align}

(3.20)\begin{align} & \boldsymbol{\mu}^{\sigma}_{m}=\tilde{\boldsymbol\Lambda}_m, \ m\ge 1 \end{align}

Recall that $\boldsymbol K$ can be regarded as a function of $s$ as well as a function of $z$, $s:={1}/{(z-1)}$. In particular, the first moment $\boldsymbol {\mu }^{\sigma }_1$ can be calculated explicitly as follows

(3.21)\begin{equation} \tilde{\lambda}_{kl}^{1}=(\Gamma_{\chi}\mathbf{u}^{k}(\mathbf{y};1),\mathbf{u}^{l}(\mathbf{y};1))_Q= 2\mu_1\int_Q \chi_2 e(\mathbf{u}^{k}(\mathbf{y};1)):\overline{e(\mathbf{u}^{l}(\mathbf{y};1))}\,{\rm d}\mathbf{y} \end{equation}