In the standard homogenization problem

$$-\nabla.\left(A\left(x,\frac{x}{\epsilon}\right)\nabla u^{\epsilon}(x)\right)=f\ \mbox{in } \Omega,$$

the homogenized matrix $A_0$ is given in terms of solutions to the cell problem

$$-div\left(A\left(x,y\right)\nabla w_{\lambda}(x,y)\right)=-div\left(A(x,y)\lambda\right),$$ $$w_{\lambda}\mbox{ is periodic in } y,\mbox{ and } \lambda\in\mathbb{R}^d$$ where $x$ appears as a parameter.

Generally, $A$ is assumed to be bounded.

What condition do we require on $A$ to guarantee some regularity of $w_\lambda$, in order to carry out the homogenization? For example, if I were to use Tartar's method of oscillating test functions, I would require $w_{\lambda}$ to be at least $H^1$ in $x$, in order to carry out some calculations.

Is there a general theory of how solutions of partial differential equations depend on parameters or how they depend on the coefficients, in analogy to how there is a result for dependence of zeros of polynomials on coefficients?

**Edits**:

Allaire's paper deals with $A$ involving two scales. For the corrector result, Theorem 2.6 of his paper, $u_1$ is written as a linear combination of the solutions to the cell problem, $w_i$. However, in the cell problem, as above, $x$ appears as a parameter, therefore, the dependence of $w_i$, and therefore, of $u_1$ on $x$ is not known. Can this be concluded from regularity of $A$ with respect to $x$?

If we were to try extending Tartar's method of oscillating test functions (adapted for periodic homogenization) to the case where $A$ involves two scales, we will require information on regularity of $w_{\lambda}$ with respect to $x$, which only appears as a parameter in the cell problem.

Is there some general theory of how solutions to elliptic linear partial differential equations depend on parameters?

In general, how do solutions to linear partial differential differential equations depend on coefficients; is there a general theory compared to how zeros of polynomials depend continuously on coefficients?