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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?

Cross-posted from math.se

Edits:

  1. 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 $x$?

  2. 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.

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

  4. 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?

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1 Answer 1

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Let us be precise. Murat and Tartar's H-convergence result does not require anything on $A$ (except boundedness and ellipticity) to give the existence of an homogenized limit. So to obtain the existence of an A_0, you do not need any additional assumptions. It applies to sequences of matrices $A_\epsilon(x)$ satisfying $A_\epsilon(x)\xi\cdot\xi \geq \alpha \xi\cdot\xi$ and $A^{-1}_\epsilon(x)\xi\cdot\xi \geq \beta \xi\cdot\xi$, where $\alpha$ and $\beta$ are independent of $\epsilon$, no periodicity, or symmetry, is required. But it does not give a formula for the homogenised limit.

If your question is whether, when $A$ has two scales as described, it is always given by the periodic corrector formula, in 'Homogenization and Two-Scale Convergence', by G. Allaire, SIMA 23(6) 1992, the condition stated to obtain a corrector formula alike the one you want is $$ \lim_{\epsilon\to0}\int_\Omega A_{i,j}^2\left(x,\frac{x}{\epsilon}\right) dx = \int_{\Omega}\int_{Y} A_{i,j}^2(x,y) dx dy $$ for each $i$ and $j$ in $\{1,\ldots,d\}$, where $Y$ is the unit periodicity cell. The optimality of this requirement is discussed at length in the last section of the paper.

One issue for example is that if you just assume for example that $A$ is symmetric and $$\alpha \xi\cdot \xi\leq A(x,y)\xi\cdot\xi\leq \beta \xi\cdot \xi \mbox{ a.e in }\Omega\times Y, $$ for any $x\in\mathbb{R}^d$ and with $\alpha,\beta>0$, then $ x\to A\left(x,\frac{x}{\epsilon}\right) $ need not be measurable.

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  • $\begingroup$ So, Allaire's result is an improvement in that it can deal with two scales in $A$? It just seems that Tartar's result could be generalized to the situation with two scales in $A$, only some regularity in $x$ would be required. Besides, in Allaire's paper, the corrector result Theorem 2.6 pp 1496 requires regularity for the solutions of the cell problem. $\endgroup$
    – user47808
    Commented Nov 11, 2014 at 15:06
  • $\begingroup$ The corrector result is something else. This is for strong convergence, not convergence to an homogenised limit, where the homogenised tensor is given by the formula you wrote. The word corrector means both $w_\lambda$ and the first order ansatz. I assumed you asked about $w_\lambda$. Are you asking about convergence estimates? Then you should clarify you question. $\endgroup$
    – username
    Commented Nov 11, 2014 at 15:23

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