Difference quotients on a shrunken domain

Assume Countable Choice for the Sobolev interfaces used below. Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be open with $n\ge 1$, let $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$, and let $u\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Omega };\mathbb{K})$ (Locally integrable functions as regular distributions). For $h\in \mathbb{R}\setminus \{0\}$ and $i\in \{1,\dots ,n\}$ let $e_i$ be the $i$-th standard unit vector and put ${\mathrm{\Omega }}_{i,h}:=\{x\in \mathrm{\Omega }:x+h{e}_i\in \mathrm{\Omega }\}=\mathrm{\Omega }\cap (\mathrm{\Omega }-h{e}_i),$ an open subset of $\mathrm{\Omega }$, since it is the intersection of the open set $\mathrm{\Omega }$ with the preimage of $\mathrm{\Omega }$ under the homeomorphism $x\mapsto x+h{e}_i$. The $i$-th difference quotient of $u$ of size $h$ is ${\delta }_h^{i}u(x):=\frac{u(x+h{e}_i)-u(x)}{h},x\in {\mathrm{\Omega }}_{i,h},$ read on the almost-everywhere classes (The space $L^p(\mu )$ as the quotient by null functions); the difference-quotient vector is ${\delta }_{h}u:=({\delta }_h^{1}u,\dots ,{\delta }_h^{n}u),$ defined on the open intersection ${\mathrm{\Omega }}_h:={\bigcap }_{i=1}^n{\mathrm{\Omega }}_{i,h}$, while each component is defined on its own shrunken set ${\mathrm{\Omega }}_{i,h}$.

Throughout this page the translation notation is the published one of Translation of a function on ${\mathbb{R}}^n$, namely ${\tau }_{h}u(x)=u(x-h)$; consequently the forward shift that occurs in the product rule is the translate by $-h{e}_i$: $u(x+h{e}_i)=({\tau }_{-h{e}_i}u)(x),{\delta }_h^{i}u=\frac{{\tau }_{-h{e}_i}u-u}{h}\text{on }{\mathrm{\Omega }}_{i,h}\mathrm{.}$

For the open upper half-space $H=\{x_n>0\}$ and a tangential index $j<n$, the translation $x\mapsto x+h{e}_j$ maps $H$ onto $H$; hence ${\delta }_h^{j}u$ is defined on all of $H$. For the normal index $n$ and $h>0$ one has $H_{n,h}=H$ (since $x_n>0$ implies $x_n+h>0$), while for $h<0$ one has $H_{n,h}=\{x\in H:x_n>\mathrm{\mid }h\mathrm{\mid }\}$, a proper subset of $H$.

Well-definedness. Both values $u(x)$ and $u(x+h{e}_i)$ in the numerator are defined for every $x\in {\mathrm{\Omega }}_{i,h}$, and $\mathrm{\mid }{\delta }_h^{i}u(x)\mathrm{\mid }\le \frac{\mathrm{\mid }u(x+h{e}_i)\mathrm{\mid }+\mathrm{\mid }u(x)\mathrm{\mid }}{\mathrm{\mid }h\mathrm{\mid }},$ so ${\delta }_h^{i}u\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({\mathrm{\Omega }}_{i,h};\mathbb{K})$: on a compact $K⋐{\mathrm{\Omega }}_{i,h}$ the first term is controlled by ${\int }_K\mathrm{\mid }u(x+h{e}_i)\mathrm{\mid }dx={\int }_{K+h{e}_i}\mathrm{\mid }u(y)\mathrm{\mid }dy<\mathrm{\infty }$, because $K+h{e}_i$ is a compact subset of $\mathrm{\Omega }$, and the second term is controlled on $K⋐\mathrm{\Omega }$. Further, the definition depends only on the class of $u$: if $u=\tilde{u}$ almost everywhere on $\mathrm{\Omega }$ and $E\subseteq \mathrm{\Omega }$ is a null set with $u=\tilde{u}$ on $\mathrm{\Omega }\setminus E$, then ${\delta }_h^{i}u\mathrm{\ne }{\delta }_h^i\tilde{u}$ only at points of $({\mathrm{\Omega }}_{i,h}\cap E)\cup ({\mathrm{\Omega }}_{i,h}\cap (E-h{e}_i))$, a null set because $E$ is null and $E-h{e}_i$ is null by the choice-free translation invariance of Lebesgue measure (Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation).

Two warnings, part of the definition. First, a difference quotient is defined on the shrunken set ${\mathrm{\Omega }}_{i,h}$ inside $\mathrm{\Omega }$, which may equal $\mathrm{\Omega }$ when the shift preserves it; in particular it may be undefined on a strip of width $\mathrm{\mid }h\mathrm{\mid }$ along the boundary, so no estimate below differentiates a Sobolev class across the boundary. Second, extending a class $u\in H_0^1(\mathrm{\Omega })$ by zero does not enlarge the domain of its difference quotient: the quotient of the extension agrees with ${\delta }_h^{i}u$ on ${\mathrm{\Omega }}_{i,h}$, while values outside ${\mathrm{\Omega }}_{i,h}$ use points not both in $\mathrm{\Omega }$ and are not values of the original quotient. In the upper half-space, ${\mathrm{\Omega }}_{n,h}=H$ for $h>0$, whereas ${\mathrm{\Omega }}_{n,h}=\{x\in H:x_n>\mathrm{\mid }h\mathrm{\mid }\}$ for $h<0$. This is why every boundary argument on this page uses only tangential quotients of compactly supported localisations, for which the shrunken domain is the whole half-space and no extension across $\mathrm{\partial }\mathrm{\Omega }$ is invoked.