Homogenization is a process that assigns to a positive definite-valued map $x\mapsto S(x)$ a non-trivial but physically meaningful average $\bar S$. There are various settings, for instance stochastic homogenization, but I shall restrict to the deterministic one, in which $x\mapsto S(x)$ is ${\mathbb Z}^d$-periodic.

To define $\bar S$, we consider the solutions of the elliptic PDE $${\rm div}(S(x)\nabla\phi)=0,$$ where $\nabla\phi$ is periodic too (but $\phi$ is not !). For every vector $q\in{\mathbb R}^d$, there exists a unique solution (up to an irrelevant additive constant) $\phi_q$ whose gradient averages to $q$ : $\langle\phi_q\rangle=q$. Then $\bar S$ is defined by $$\bar Sq=\langle S\nabla\phi_q\rangle.$$

I am interested in a homogenization process that provides the geometrical mean $A\sharp B$ of two positive definite symmetric matrices, as $\bar S$. I guess that such a process must exist (but I should be grateful to receive a reference). Is there a known explicit process ?

My motivation comes from an observation made in the book Homogenization of differential operators and integral functionals by V. V. Jikov, S. M. Kozlov, O. A. Oleinik: Let $d=2$ and $\sigma$ be the rotation by $\frac\pi2$. If $S(x)\sigma S(\sigma x)\sigma\equiv-k^2I_2$ where $k>0$ is a constant, then $\bar S=kI_2$. When applied to a chess-board with $S(x)=a_\pm I_2$, this provides $\bar S=\sqrt{a_-a_+}I_2$, which answers a particular case of my question.

For the sake of completeness, I recall that $A\sharp B$ is maximal among those positive definite $X$ such that $$\begin{pmatrix} A & X \\ X & B \end{pmatrix}\ge0_n.$$ It is given by the (amazingly non-symmetric) formula $$A\sharp B=A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2}.$$

One argument in favour of the existence of such a process is the fact that $A\sharp B$ satisfies the same necessary conditions (see again the same book) as $\bar S$ : $$\left(\frac{A^{-1}+B^{-1}}2\right)^{-1}\le A\sharp B\le\frac{A+B}2\,.$$