Homogenization of the Eigenvalues of the Neumann–Poincaré Operator

Abstract

In this article, we investigate the spectrum of the Neumann–Poincaré operator ${{\mathcal {K}}}_\varepsilon ^*$ (or equivalently, that of the associated Poincaré variational operator $T_\varepsilon$) associated to a periodic distribution of small inclusions with size $\varepsilon$, and its asymptotic behavior as the parameter $\varepsilon$ vanishes. Combining techniques pertaining to the fields of homogenization and potential theory, we prove that the limit spectrum is composed of the ‘trivial’ eigenvalues 0 and 1, and of a subset which stays bounded away from 0 and 1 uniformly with respect to $\varepsilon$. This non trivial part is the reunion of the Bloch spectrum, accounting for the collective resonances between collections of inclusions, and of the boundary layer spectrum, associated to eigenfunctions which spend a not too small part of their energies near the boundary of the macroscopic device. These results shed new light on the issue of the homogenization of the voltage potential $u_\varepsilon$ caused by a given source in a medium composed of a periodic distribution of small inclusions with an arbitrary (possible negative) conductivity a, surrounded by a dielectric medium, with unit conductivity. In particular, we prove that the limit behavior of $u_\varepsilon$ is strongly related to the (possibly ill-defined) homogenized diffusion matrix predicted by the homogenization theory in the standard elliptic case. Additionally, we prove that the homogenization of $u_\varepsilon$ is always possible when a is either positive, or negative with a ‘small’ or ‘large’ modulus.

A Closer Study of the Particular Case of Rank 1 Laminates

In this appendix, we focus on an interesting particular geometry of microstructures $\omega \subset Y$, that of rank 1 laminates, which is one of the few amenable to explicit calculations. Note that this situation violates some of the prevailing assumptions of this article, notably the fact that $\omega \Subset Y$; it is therefore not surprising that some of the general results established in the previous sections do not hold in the present case.

The ‘macroscopic’ domain ${\varOmega }$ at stake is the two-dimensional square $(0,1)^2$; if is filled with $N^2$ identical cells, homothetic to the unit periodicity cell $Y = (0,1)^2$. The rescaled inclusion pattern $\omega$ in each cell is

$$\begin{aligned}&\omega := \left\{ y=(y_1,y_2) \in Y, \, 0<y_1<\theta \right\} , \\&\quad \text { and so } Y{\setminus }\overline{\omega } = \left\{ y=(y_1,y_2) \in Y, \, \theta<y_1<1 \right\} , \end{aligned}$$

where $\theta \in (0,1)$ is the volume of $\omega$. Accordingly, the total set of inclusions $\omega _N \subset {\varOmega }$ read as

$$\begin{aligned} \omega _N := \bigcup _{\begin{array}{c} j \in {\mathbb {N}}^2 \\ 0 \leqq j \leqq N-1 \end{array}}{\frac{1}{N}(j + \omega )}; \end{aligned}$$

see Fig. 2 for an illustration.

Fig. 2

Situation where the inclusion $\omega \subset Y$ is a rank 1 laminate

To avoid effects caused by the interactions between cells and the boundary $\partial {\varOmega }$, we impose periodic boundary conditions on $\partial {\varOmega }$. In this context, the voltage potential $u_N \in H^1_\#({\varOmega })/{\mathbb {R}}$ associated to a source $f \in (H^1_\#({\varOmega }) / {\mathbb {R}})^*$ and to the distribution of conductivity equal to $a \in {\mathbb {C}}$ in $\omega _N$ and 1 in $Y{\setminus }\overline{\omega _N}$ is solution to

$$\begin{aligned} -\mathrm{div}(A_N \nabla u) = f \text { in } {\varOmega }, \text { where } A_N(x) = \left\{ \begin{array}{ll} a &{}\quad \text {if } x \in \omega _N, \\ 1 &{}\quad \text {otherwise} \end{array} \right. . \end{aligned}$$

(A.1)

We are interested in the spectrum of the Poincaré variational operator $T_N$, which maps an arbitrary function $u \in H_\#^1({\varOmega }) / {\mathbb {R}}$ to the unique element $T_N u \in H_\#^1({\varOmega }) / {\mathbb {R}}$ satisfying

$$\begin{aligned} \forall v \in H^1_\#({\varOmega })/ {\mathbb {R}}, \,\, \int _{{\varOmega }}{\nabla (T_N u)\cdot \nabla v \,\mathrm{d}x} = \int _{\omega _N}{\nabla u \cdot \nabla v\,\mathrm{d}x}, \end{aligned}$$

and, in particular, in the identification of the limit spectrum

$$\begin{aligned} \lim \limits _{N \rightarrow \infty }{\sigma (T_N)} = \left\{ \lambda \in {\mathbb {C}}, \,\, \exists \, N_j \rightarrow \infty , \, \lambda _{N_j} \in \sigma (T_{N_j}),\,\, \lambda _{N_j} \xrightarrow {N_j \rightarrow \infty } \lambda \right\} .\nonumber \\ \end{aligned}$$

(A.2)

1.1 Study of the Homogenization Process for the Operator $T_N$

The main tool in our analysis is the discrete Bloch decomposition [2], which has already been used without proof several times in this article. Although it is quite classical, we sketch the proof for completeness.

Theorem A.1

Let u be a function in $L^2_\#({\varOmega })$. Then, there exists a unique collection $\left\{ u_j \right\}$, indexed by $j \in {\mathbb {N}}^2$, $0 \leqq j \leqq N-1$, composed of $N^2$ complex-valued functions in $L^2_\#(Y)$ such that the following identity holds:

$$\begin{aligned} u(x) = \sum \limits _{0\leqq j \leqq N-1}{u_j(Nx)\, e^{2i\pi j \cdot x}}, \text { almost everywhere } x \in {\varOmega }. \end{aligned}$$

(A.3)

Furthermore, the Parseval identity holds: for $u,v \in L^2_\#({\varOmega })$, with coefficients $\left\{ u_j \right\} _{0\leqq j \leqq N-1}$, $\left\{ v_j \right\} _{0\leqq j \leqq N-1}$, one has

$$\begin{aligned} \int _{\varOmega }{u(x)\overline{v(x)}\,\mathrm{d}x}= \sum \limits _{0\leqq j\leqq N-1}{\int _Y{u_j(y)\overline{v_j(y)}\,\mathrm{d}y}}. \end{aligned}$$

(A.4)

Proof

Let $u \in L^2_\#(Y)$ be given, and assume that there exist $N^2$ functions $u_j \in L^2_\#(Y)$ such that (A.3) holds. Then, for arbitrary $j^\prime \in {\mathbb {N}}^2$, $0 \leqq j^\prime \leqq N-1$,

$$\begin{aligned} u\left( x + \frac{j^\prime }{N}\right) = \sum \limits _{0 \leqq j \leqq N-1}{u_j\left( Nx\right) e^{2i\pi j \cdot \left( x + \frac{j^\prime }{N}\right) }}. \end{aligned}$$

So as to isolate a particular index $0 \leqq j^0 \leqq N-1$, we multiply both sides in the previous identity by $e^{-2i\pi j^0 \cdot (x + \frac{j^\prime }{N})}$, then sum over $j^\prime$ to obtain

$$\begin{aligned} \sum \limits _{\begin{array}{c} 0 \leqq j \leqq N-1 \\ 0 \leqq j^\prime \leqq N-1 \end{array}}{u_j\left( Nx\right) e^{2i\pi \left( j- j^0\right) \cdot \left( x + \frac{j^\prime }{N}\right) }} = \sum \limits _{0 \leqq j \leqq N-1}{u\left( x + \frac{j^\prime }{N}\right) e^{-2i\pi j^0 \cdot \left( x + \frac{j^\prime }{N}\right) }}.\nonumber \\ \end{aligned}$$

(A.5)

Using the fact that, for any $N^{\text {th}}$ root $r = e^{2\pi i j/N}$ of 1,

$$\begin{aligned} \sum \limits _{0\leqq k \leqq N-1}{r^k } = \left\{ \begin{array}{cc} N &{} \text {if } r = 1, \\ 0 &{} \text {otherwise} \end{array} \right. . \end{aligned}$$

(A.6)

The relation (A.5) simplifies into

$$\begin{aligned} u_{j^0}\left( y\right) =\frac{1}{N^2}\sum \limits _{0\leqq j^\prime \leqq N-1}{u\left( \frac{y}{N} + \frac{j^\prime }{N}\right) e^{-2i\pi j^0 \cdot \left( \frac{y}{N} + \frac{j^\prime }{N}\right) } }, \end{aligned}$$

(A.7)

a formula which clearly defines a function in $L^2_\#(Y)$. Conversely, one easily proves that the $u_j$ defined in (A.7) satisfy (A.3), which proves the first statement.

To verify the Parseval identity (A.4), let $u,v \in L^2_\#({\varOmega })$ be decomposed as

$$\begin{aligned} u(x) = \sum \limits _{0\leqq j \leqq N-1}{u_j(Nx)\, e^{2i\pi j \cdot x}}, \text { and } v(x) = \sum \limits _{0\leqq j^\prime \leqq N-1}{ v_{j^\prime } (Nx)\, e^{2i\pi j^\prime \cdot x}}. \end{aligned}$$

A simple calculation yields

$$\begin{aligned} \int _\Omega {u(x)\overline{v(x)}\,\mathrm{d}x}= & {} \sum \limits _{0 \leqq j, j^\prime \leqq N-1}{\int _{\Omega }{u_j(Nx)\overline{v_{j^\prime }(Nx)} \, e^{2i\pi (j-j^\prime )\cdot x}\,\mathrm{d}x}}\\= & {} \sum \limits _{0 \leqq j, j^\prime ,k \leqq N-1}{\int _{\frac{1}{N}(k+Y)}{u_j(Nx)\overline{v_{j^\prime }(Nx)} \, e^{2i\pi (j-j^\prime )\cdot x}\,\mathrm{d}x}}\\= & {} \frac{1}{N^2}\sum \limits _{0 \leqq j, j^\prime ,k \leqq N-1}{\int _{Y}{u_j(y)\overline{v_{j^\prime }(y)} \, e^{2i\pi (j-j^\prime ) \cdot (\frac{k}{N} + \frac{y}{N})}\,\mathrm{d}y}}\\= & {} \sum \limits _{0 \leqq j \leqq N-1}{\int _{Y}{u_j(y)\overline{v_{j}(y)}\,\mathrm{d}y}}, \end{aligned}$$

where we have again made use of (A.6) to pass from the third line to the last. $\quad \square$

Let us now consider the operator B, from $L^2_\#(Y)^{N^2}$ into $L^2_\#({\varOmega })$ which maps a collection $\left\{ u_j \right\} _{0\leqq j \leqq N-1}$ of coefficients to the function $u \in L^2_\#({\varOmega })$ defined by

$$\begin{aligned} u(x) = \sum \limits _{0\leqq j \leqq N-1}{u_j(Nx)\, e^{2i\pi j \cdot x}}, \text { almost everywhere } x \in {\varOmega }. \end{aligned}$$

Equipping both spaces with their natural inner products, Theorem A.1 states that B is a bijective isometry, whose inverse, ${B}^{-1}:L^2_\#({\varOmega }) \rightarrow L^2_\#(Y)^{N^2}$, is also its adjoint operator.

If u belongs to $H^1_\#({\varOmega })$, (A.7) implies that the coefficients $u_j$ of its Bloch decomposition actually belong to $H^1_\#(Y)$, and that the Bloch decomposition of $\nabla u$ read as

$$\begin{aligned} \nabla u(x) = N \sum \limits _{0\leqq j \leqq N-1}{\left( \nabla _y + 2i\pi \frac{j}{N}\right) u_j(Nx)\, e^{2i\pi j \cdot x}}, \text { almost everywhere } x \in {\varOmega }. \end{aligned}$$

Using the fact that the Bloch decomposition of the constant function $u \equiv 1 \in H^1_\#({\varOmega })$ has coefficients

$$\begin{aligned} u_0(y) = 1, \text { and } u_j(y) = 0 \text { if } j \ne 0, \end{aligned}$$

B induces an invertible operator (still denoted by B) $(H^1_\#(Y)/{\mathbb {C}}) \times H^1_\#(Y)^{N^2-1} \rightarrow H^1_\#({\varOmega }) /{\mathbb {C}}$.

The following proposition easily follows from the previous remarks, and in particular from the Parseval identity (A.4):

Proposition A.1

For any $\eta \in \overline{Y}$, $\eta \ne 0$, define $T_{\eta }: H^1_\#(Y) \rightarrow H^1_\#(Y)$ by

$$\begin{aligned}&\forall v \in H^1_\#(Y), \,\, \int _{Y}{(\nabla _y + 2i\pi \eta )(T_{\eta } u) \cdot \overline{(\nabla _y + 2i\pi \eta ) v }\,\mathrm{d}y} \nonumber \\&\quad = \int _{\omega }{(\nabla _y + 2i\pi \eta ) u \cdot \overline{(\nabla _y + 2i\pi \eta ) v }\,\mathrm{d}y }, \end{aligned}$$

(A.8)

and define $T_0 : H^1_\#(Y) / {\mathbb {C}} \rightarrow H^1_\#(Y) / {\mathbb {C}}$ by

$$\begin{aligned} \forall v \in H^1(Y) / {\mathbb {C}}, \,\, \int _Y{\nabla _y (T_0 u) \cdot \overline{\nabla _y v}\,\mathrm{d}y} = \int _\omega {\nabla _y u \cdot \overline{\nabla _y v}\,\mathrm{d}y}. \end{aligned}$$

(A.9)

The operator B diagonalizes $T_N$, that is the self-adjoint operator $B^* T_N B$ maps a collection $\left\{ u_j \right\} _{0 \leqq j \leqq N-1} \in (H^1_\#(Y)/{\mathbb {C}}) \times H^1_\#(Y)^{N^2-1}$ to $\left\{ T_{\frac{j}{N}} u_j \right\} _{0 \leqq j \leqq N-1} \in (H^1_\#(Y)/{\mathbb {C}}) \times H^1_\#(Y)^{N^2-1}$. As a consequence, the spectrum $\sigma (T_N)$ is the union of the spectra $\sigma (T_{\frac{j}{N}})$:

$$\begin{aligned} \sigma (T_N) = \bigcup _{0 \leqq j \leqq N-1}{\sigma \left( T_{\frac{j}{N}}\right) }. \end{aligned}$$

Therefore, the study of the spectrum of $T_N$, boils down to that of the spectra of the operators $T_\eta$ defined by (A.8) and (A.9). Let us now take advantage of the particular geometry of $\omega$ to simplify the problem further. We decompose functions $u \in H^1_\#(Y)$ (or $u \in H^1_\#(Y) / {\mathbb {C}}$) by using partial Fourier series in the variable $y_2$:

$$\begin{aligned} u(y) = \sum \limits _{n=-\infty }^{+\infty }{a_n(y_1) e^{2i\pi ny_2}}, \text { almost everywhere } y \in Y, \end{aligned}$$

where the $a_n \in H^1_\#(0,1)$ (and $a_0 \in H^1_\#(0,1) / {\mathbb {C}}$ if $u \in H^1_\#(Y) / {\mathbb {C}}$).

After some elementary calculations, the operators $T_\eta$ are diagonalized by this Fourier decomposition, that is the spectrum of $T_N$ read as

$$\begin{aligned} \sigma (T_N) = \bigcup _{\begin{array}{c} 0 \leqq j \leqq N-1 \\ n \in {\mathbb {N}} \end{array}}{\sigma \left( T_{\frac{j}{N}}^n\right) }, \end{aligned}$$

where, for any $\eta = (\eta _1,\eta _2) \in \overline{Y}$, $T_\eta ^n : H^1_\#(0,1) \rightarrow H^1_\#(0,1)$ is defined by

$$\begin{aligned}&\forall v \in H^1_\#(0,1) \in , \,\, \int _{0}^1\left( ((T_\eta ^n u)^\prime + 2i\pi \eta _1 (T_\eta ^n u))\overline{(v^\prime + 2i\pi \eta _1 v)}\right. \nonumber \\&\qquad \left. + 4\pi ^2(n+\eta _2)^2 u\overline{v}\right) \,\mathrm{d}y \nonumber \\&\quad = \int _{0}^\theta {\left( (u^\prime + 2i\pi \eta _1 u)\overline{(v^\prime + 2i\pi \eta _1 v)} + 4\pi ^2(n+\eta _2)^2 u\overline{v}\right) \,\mathrm{d}y}, \end{aligned}$$

(A.10)

and $T_0^0 : H^1_\#(0,1)/ {\mathbb {C}} \rightarrow H^1_\#(0,1) / {\mathbb {C}}$ is given by

$$\begin{aligned} \forall v \in H^1_\#(0,1)/{\mathbb {C}}, \,\, \int _{0}^1{((T_0^0 u)^\prime )\overline{v^\prime }\,\mathrm{d}y} = \int _{0}^\theta {u^\prime \overline{v^\prime }\,\mathrm{d}y}. \end{aligned}$$

(A.11)

It is proved in the same way as in Section 3.3 that the spectrum of each operator $T_\eta ^n$ consists of a sequence of eigenvalues in [0, 1] with $\frac{1}{2}$ as unique accumulation point, and we now proceed to identify these eigenvalues.

Let us first study the eigenvalues of the operator $T_\eta ^n$ in the case where either $\eta _2 \ne 0$ or $n \ne 0$. A value $\beta \in {\mathbb {C}}$ is an eigenvalue for $T_\eta ^n$ as defined in (A.10) if there exists $u \in H^1_\#(0,1)$, $u\ne 0$ such that

$$\begin{aligned} -\left( \frac{\partial }{\partial y_1} + 2i\pi \eta _1 \right) \left( A_\beta (y_1)\left( \frac{\partial u}{\partial y_1} + 2i\pi \eta _1 u \right) \right) + 4\pi ^2 A_\beta (y_1)(n + \eta _2)^2 u =0,\nonumber \\ \end{aligned}$$

(A.12)

where

$$\begin{aligned} A_\beta (y) = \left\{ \begin{array}{ll} \beta -1 &{}\quad \text {if } y_1< \theta , \\ \beta &{}\quad \text {if } \theta< y_1 < 1. \\ \end{array} \right. \end{aligned}$$

Assuming $\beta \notin \left\{ 0,1\right\}$, (A.12) is equivalent to

$$\begin{aligned} u^{\prime \prime }(z) + 4i \pi \eta _1 u^\prime (z) - 4\pi ^2(\eta _1^2 + (n+\eta _2)^2) u(z) = 0 \text { almost everywhere } z\in (0,\theta ) \text { and } z \in (\theta ,1),\nonumber \\ \end{aligned}$$

(A.13)

complemented with the transmission conditions at $z=0$ and $z = \theta$:

$$\begin{aligned} u(0^+) = u(1^-), \,\,\,\, \beta (u^\prime + 2i\pi \eta _1u)(1^-) = (\beta -1) (u^\prime + 2i\pi \eta _1u)(0^+), \end{aligned}$$

(A.14)

$$\begin{aligned} u(\theta ^-) = u(\theta ^+), \,\,\,\, (\beta -1) (u^\prime + 2i\pi \eta _1u)(\theta ^-) = \beta (u^\prime + 2i\pi \eta _1u)(\theta ^+).\nonumber \\ \end{aligned}$$

(A.15)

The ordinary differential equation (A.13) has discriminant ${\varDelta } = 16\pi ^2(n + \eta _2)^2$, and the associated characteristic equation has two solutions:

$$\begin{aligned} r_1 = -2i \pi \eta _1 - 2\pi (n + \eta _2), \,\, r_2 = -2i\pi \eta _1 + 2\pi (n + \eta _2), \end{aligned}$$

which are distinct, since $n+\eta _2 \ne 0$. Therefore, there exist 4 coefficients $A,B,C,D \in {\mathbb {C}}$ such that

$$\begin{aligned} u(z) = Ae^{r_1z} + Be^{r_2z} \text { for } z \in (0,\theta ), \,\, u(z) = Ce^{r_1z} + De^{r_2z} \text { for } z \in (\theta ,1). \end{aligned}$$

The fact that $u \in H^1_\#(0,1)$ imposes on us that

$$\begin{aligned} A + B = Ce^{r_1} + De^{r_2}, \text { and } Ae^{r_1\theta } + Be^{r_2\theta } = Ce^{r_1\theta } + De^{r_2\theta }, \end{aligned}$$

to be complemented with the transmission conditions

$$\begin{aligned}&(\beta -1) (r_1+2i\pi \eta _1) A+ (\beta -1) (r_2+2i\pi \eta _1) B \\&\quad = \beta (r_1+2i\pi \eta _1)Ce^{r_1} + \beta (r_2+2i\pi \eta _1)De^{r_2} \\&(\beta -1) (r_1+2i\pi \eta _1) Ae^{r_1\theta } + (\beta -1) (r_2+2i\pi \eta _1) Be^{r_2\theta } \\&\quad = \beta (r_1+2i\pi \eta _1)Ce^{r_1\theta } + \beta (r_2+2i\pi \eta _1)De^{r_2\theta } \end{aligned}$$

As a consequence, (A.13,A.14,A.15) has a non-trivial solution provided the following determinant vanishes:

$$\begin{aligned} \left|\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 1 &{} 1 &{} -e^{r_1} &{} -e^{r_2} \\ e^{r_1\theta } &{} e^{r_2\theta } &{} -e^{r_1\theta } &{} -e^{r_2\theta } \\ (\beta -1) (r_1+2i\pi \eta _1) &{} (\beta -1) (r_2+2i\pi \eta _1) &{} -\beta (r_1+2i\pi \eta _1) e^{r_1} &{} -\beta (r_2+2i\pi \eta _1)e^{r_2}\\ (\beta -1) (r_1+2i\pi \eta _1) e^{r_1\theta } &{} (\beta -1) (r_2+2i\pi \eta _1) e^{r_2\theta } &{} -\beta (r_1+2i\pi \eta _1) e^{r_1\theta } &{} -\beta (r_2+2i\pi \eta _1)e^{r_2\theta } \end{array} \right|= 0,\nonumber \\ \end{aligned}$$

(A.16)

which is a quadratic equation for $\beta$. After tedious calculations, (A.16) simplifies into

$$\begin{aligned} \beta ^2 - \beta + \gamma = 0, \text { where } \gamma := \frac{1}{4} \frac{\cosh (2\pi (n + \eta _2)) - \cosh ((2\pi (n + \eta _2))(2\theta -1))}{\cosh (2\pi (n + \eta _2)) - \cos (2\pi \eta _1)}. \end{aligned}$$

The discriminant of this second order equation read as

$$\begin{aligned} {\varDelta }_\eta ^n = \frac{\cosh \left( (2\pi (n + \eta _2))(2\theta -1)\right) -\cos (2\pi \eta _1)}{\cosh (2\pi (n + \eta _2)) - \cos (2\pi \eta _1)}. \end{aligned}$$

(A.17)

We observe that ${\varDelta }_\eta ^n \in (0,1)$, leading to two distinct eigenvalues $\beta _\eta ^{n\pm } = (1 \pm \sqrt{{\varDelta }_\eta ^n})/2$, which are symmetric with respect to $\frac{1}{2}$.

As for the eigenvalues of $T_\eta ^n$ in the case where $\eta _2 = 0$ and $n=0$, simple calculations show that $\sigma (T_\eta ^n) = \left\{ 0, 1\right\}$ if $\eta _1 \ne 0$ and that $\sigma (T_0^0) = \left\{ 0,1-\theta ,1\right\}$. All in all, we have proved that the spectrum of $T_N$ is

$$\begin{aligned} \sigma (T_N) = \left\{ 0,1-\theta ,1\right\} \cup \left\{ \frac{1}{2}\left( 1\pm \sqrt{{\varDelta }_{\frac{j}{N}}^n}\right) \right\} _{\begin{array}{c} 0 < j \leqq N-1 \\ n \in {\mathbb {N}} \end{array}}, \end{aligned}$$

where ${\varDelta }_\eta ^n$ is defined by (A.17). This allows for the identification of the limit spectrum (A.2) as

$$\begin{aligned} \lim \limits _{N\rightarrow \infty }{\sigma (T_N)} = \left\{ 0,1-\theta ,1\right\} \cup \left\{ \frac{1}{2}\left( 1\pm \sqrt{{\varDelta }_{\eta }^n}\right) \right\} _{\eta \in \overline{Y}, n \in {\mathbb {N}}} = [0,1]. \end{aligned}$$

Hence, in the present situation of rank 1 laminates, the limit spectrum of $T_N$ is the whole interval [0, 1], in sharp contrast with the situation where $\omega \Subset Y$, as studied in Section 4.

1.2 Analysis of the Homogenized Tensor $A^*$

As we have seen in Section 5.2 (see in particular Propositions 5.5 and 5.6, which are straightforwardly adapted to this context), the asymptotic behavior of the voltage potential $u_N$ associated to a conductivity $a \in {\mathbb {C}}$ inside the set of inclusions $\omega _N$, solution to (A.1), is partly driven by the formally homogenized tensor $A^*$ whose components $A^*_{ij}$ are given by ($i,j=1,2$):

$$\begin{aligned} A^*_{ij} = \int _Y{A(y) (e_i + \nabla _y \chi _i) \cdot (e_j + \nabla _y \chi _j) \,\mathrm{d}y}, \text { where } A(y) = \left\{ \begin{array}{ll} a &{}\quad \text {if } y \in \omega , \\ 1 &{}\quad \text {otherwise}, \end{array} \right. \nonumber \\ \end{aligned}$$

(A.18)

and the cell functions $\chi _i \in H^1_\#(Y) / {\mathbb {R}}$ ($i=1,2$) are solutions to

$$\begin{aligned} -\mathrm{div}_y(A(y)(e_i + \nabla _y \chi _i)) = 0. \end{aligned}$$

(A.19)

As we have seen in Section 5.1, both cell problems (A.19) have a unique solution, provided the conductivity a lies outside the exceptional set ${\varSigma }_\omega$ given by

$$\begin{aligned} {\varSigma }_\omega = \left\{ a \in {\mathbb {C}}, \,\, \frac{1}{1-a} \in \sigma (T_0)\right\} , \end{aligned}$$

where $T_0 : H^1_\#(Y) / {\mathbb {R}} \rightarrow H^1_\#(Y) / {\mathbb {R}}$ is (the real version of that ) defined by (A.9) as follows:

$$\begin{aligned} \forall v \in H^1_\#(Y) / {\mathbb {R}}, \,\, \int _Y{\nabla _y (T_0 u) \cdot \nabla _y v\,\mathrm{d}y} = \int _\omega {\nabla _y u \cdot \nabla _y v\,\mathrm{d}y}. \end{aligned}$$

(A.20)

On the contrary, in the case $a \in {\varSigma }_\omega$, there may be multiple solutions to (A.19) (or none), but when this happens, it is easily seen that (A.18) is independent of which of these solutions is used.

Calculations similar to those performed in the previous section (based on a partial Fourier decomposition) lead to an explicit characterization of the set ${\varSigma }_\omega$ in the present context of rank 1 laminates:

$$\begin{aligned} {\varSigma }_\omega := \left\{ -\frac{\theta }{1-\theta }, 0\right\} \cup \left\{ a_n^\pm \right\} _{n \in {\mathbb {N}}^*}, \end{aligned}$$

where the $a_n^\pm$ read as

$$\begin{aligned} a^n_{\pm } = \frac{2(1 \pm 2\sinh (\pi n)\sinh (\pi n (2\theta -1)) ) - \cosh (2\pi n (2\theta -1)) - \cosh (2\pi n)}{\cosh (2\pi n) - \cosh (2\pi n (2\theta -1))}. \end{aligned}$$

Let us now turn to the cell problems (A.19) in this setting for an arbitrary conductivity $a \in {\mathbb {C}}$. Relying again on a partial Fourier decomposition in the $y_2$ variable, it is easy to see that, if (A.19) has solutions, one of them is necessarily of the form

$$\begin{aligned} \chi _i(y) = u_i(y_1), \text { for some function } u_i \in H^1_\#(0,1) / {\mathbb {R}}, \, i = 1,2. \end{aligned}$$

(A.21)

Now, easy calculations reveal that if $a \ne -\frac{\theta }{1-\theta }$, (A.19) has (possibly non-unique) solutions $\chi _i$ for $i=1,2$:

$$\begin{aligned} \chi _1(y) = \left\{ \begin{array}{ll} Ay_1 + B &{}\quad \text {if } y_1 < \theta , \\ Cy_1 + D &{}\quad \text {if } y_1 > \theta , \\ \end{array} \right. ,\text { and } \chi _2(y) = 0, \end{aligned}$$

where the coefficients A and C read as

$$\begin{aligned} A = \frac{1-a}{a - \frac{\theta }{1-\theta }}, \, C = A \frac{\theta }{\theta -1}. \end{aligned}$$

Then, the homogenized tensor (A.18) equals

$$\begin{aligned} A^* = \left( \begin{array}{cc} \lambda _{\theta ,a}^- &{} 0, \\ 0 &{} \lambda _{\theta ,a}^+ \end{array} \right) , \text { where } \lambda _{\theta ,a}^- = \left( \frac{\theta }{a} + 1-\theta \right) ^{-1}, \text { and }\lambda _{\theta ,a}^+ = a\theta + (1-\theta ). \end{aligned}$$

Note that $A^*$ is invertible, and becomes degenerate when a gets close to $-\frac{\theta }{1-\theta }$. The behaviors of the mappings $a \mapsto \lambda _{\theta ,a}^\pm$ change depending on whether the volume fraction $\theta$ is larger or smaller than $\frac{1}{2}$, as can be seen on Fig. 3. In the case where $\theta < \frac{1}{2}$, three regimes are to be distinguished:

• When $-\frac{\theta }{1-\theta }< a < 0$, $A^*$ has eigenvalues with opposite signs.

• When $-\frac{1-\theta }{\theta }<a<-\frac{\theta }{1-\theta }$, $A^*$ is positive definite.

• When $a < -\frac{1-\theta }{\theta }$, $A^*$ has again eigenvalues with opposite signs.

This behavior differs in the case where $\theta > \frac{1}{2}$:

• When $-\frac{1-\theta }{\theta }< a < 0$, $A^*$ has eigenvalues with opposite signs.

• When $-\frac{\theta }{1-\theta }<a<-\frac{1-\theta }{\theta }$, $A^*$ is negative definite.

• When $a < -\frac{\theta }{1-\theta }$, $A^*$ has again eigenvalues with opposite signs. These results are in sharp contrast with the case of inclusions $\omega \Subset Y$, dealt with in Section 5. In the case $a = -\frac{\theta }{1-\theta }$, the cell problem (A.19) has infinitely many solutions of the form (A.21) for $i=2$, and none for $i=1$.

It is remarkable that the only value of a for which the cell problems do not have solutions is $a = -\frac{\theta }{1-\theta }$, corresponding to the essential spectrum of the operator defined in (A.20). We do not know whether this fact holds for general inclusion patterns $\omega \subset Y$ or if it is particular to rank 1 laminates.

Fig. 3

Behavior of the eigenvalues $a \mapsto \lambda _{\theta ,a}^\pm$ of the homogenized matrix $A^*$ for the volume fractions (left) $\theta = \frac{1}{3}$, and (right) $\theta = \frac{2}{3}$