Periodic correctors and effective tensor

The library represents ${\mathbb{R}}^d$ as functions on $d=\{0,\dots ,d-1\}$, with standard basis ${\tilde{e}}_j$ for $j<d$ (The standard list $e:n\to F^n$ with $e_i(i)=1_F$ and $e_i(j)=0_F$ for $j\mathrm{\ne }i$ is an ordered basis of $F^n$; hence ${\dim }_F{F}^n=n$, and $F^0$ is the zero space with basis $\mathrm{\varnothing }$ and dimension $0$). For the usual PDE coordinate labels $i=1,\dots ,d$, write $x_i:=x(i-1)$, set $e_i:={\tilde{e}}_{i-1}$, write $p_i:=p(i-1)$, and set ${\mathrm{\partial }}_i:={\mathrm{\partial }}_{i-1}$ in the multi-index convention of $C^k$ maps and multi-index derivative notation in Euclidean space. The same shift applies to vector and tensor components, so component label $i$ means library index $i-1$. By Existence and uniqueness of periodic cell correctors, let ${\chi }_i:={\chi }_{e_i}$ be the unique mean-zero solution of the periodic cell equation. Define ${\chi }_p={\sum }_{i=1}^d{p}_i{\chi }_i$ and define the constant matrix $A^{\ast }$ by its columns: $A^{\ast }{e}_i:={\int }_{Y}a(y)(e_i+\mathrm{\nabla }{\chi }_i(y))dy(1\le i\le d)\mathrm{.}$ Equivalently, $A^{\ast }p={\int }_{Y}a(y)(p+\mathrm{\nabla }{\chi }_p(y))dy$. All integrals are finite because $a\in L^{\mathrm{\infty }}(Y)$ and $\mathrm{\nabla }{\chi }_i\in L^2(Y)$. This definition does not itself assert symmetry or ellipticity.

Symmetry and uniform ellipticity of the effective tensor proves symmetry and uniform ellipticity of the tensor.