Uniformly elliptic periodic scalar diffusion data

Fix an integer $d\ge 1$ and the unit cell $Y=(0,1{)}^d$ with Lebesgue measure $\mathrm{\mid }Y\mathrm{\mid }=1$. Assume Countable Choice for the $L^2$ conventions of The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). A uniformly elliptic periodic scalar diffusion coefficient is a measurable real function $a:{\mathbb{R}}^d\to \mathbb{R}$ such that $a(y+k)=a(y)$ for almost every $y$ and every $k\in {\mathbb{Z}}^d$, and there are constants $0<\lambda \le \mathrm{\Lambda }<\mathrm{\infty }$ with $\lambda \le a(y)\le \mathrm{\Lambda }$ almost everywhere on $Y$. Coefficients are identified when they agree almost everywhere.

For $ϵ>0$, the physical scalar coefficient is $a_{ϵ}(x)=a(x\mathrm{/}ϵ)$ and the associated matrix is $a_{ϵ}(x)I_d$. The bounded-domain branch used on this page is $\mathrm{\Omega }=(0,1{)}^d$ with homogeneous zero boundary values. Its weak equation for datum $f\in L^2(\mathrm{\Omega };\mathbb{R})$ is ${\int }_{\mathrm{\Omega }}a(x\mathrm{/}ϵ)\mathrm{\nabla }{u}_{ϵ}\cdot \mathrm{\nabla }vdx={\int }_{\mathrm{\Omega }}fvdx(v\in H_0^1(\mathrm{\Omega };\mathbb{R}))\mathrm{.}$ No coefficient regularity beyond measurability and the displayed bounds is part of this definition.

For every $ϵ>0$, the map $x\mapsto x\mathrm{/}ϵ$ is a $C^1$ diffeomorphism of ${\mathbb{R}}^d$. By A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions, its pullback preserves Lebesgue measurability and takes null-set changes to null-set changes (apply the theorem also to the inverse map). Thus $a_{ϵ}$ is a well-defined almost-everywhere class independent of the representative of $a$. The same result applies to every affine translation and positive dilation used in periodic sampling below, so almost-everywhere coefficient bounds are preserved under those pullbacks.

The ellipticity bounds hold almost everywhere on all of ${\mathbb{R}}^d$, not just on $Y$: periodicity transfers the exceptional null set from $Y$ to each lattice translate, translations preserve null sets by the same change-of-variables theorem, and the countably many translates cover ${\mathbb{R}}^d$ up to their null boundaries.

Countable Choice is inherited through the real and complex $L^2$ and Sobolev class conventions. No Axiom of Choice or Dependent Choice is asserted here.