Interior and Boundary Sobolev Elliptic Regularity

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.

Assume Countable Choice. Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be open with $n\ge 1$, let $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$, let $u,v\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Omega };\mathbb{K})$, and fix $h\mathrm{\ne }0$ and $i\in \{1,\dots ,n\}$. Write ${\mathrm{\Omega }}_{i,h}=\{x\in \mathrm{\Omega }:x+h{e}_i\in \mathrm{\Omega }\}$ and ${\mathrm{\Omega }}_{i,-h}=\{x\in \mathrm{\Omega }:x-h{e}_i\in \mathrm{\Omega }\}$ (Difference quotients on a shrunken domain), so that ${\mathrm{\Omega }}_{i,-h}={\mathrm{\Omega }}_{i,h}+h{e}_i$. Then:

(i) Integration by parts, two-domain form. Whenever the displayed integrals converge absolutely, ${\int }_{{\mathrm{\Omega }}_{i,h}}{\delta }_h^{i}uvdx=-{\int }_{{\mathrm{\Omega }}_{i,-h}}u{\delta }_{-h}^{i}vdx+\frac{1}{h}({\int }_{{\mathrm{\Omega }}_{i,-h}}uvdx-{\int }_{{\mathrm{\Omega }}_{i,h}}uvdx)\mathrm{.}$ Consequently, if ${\int }_{{\mathrm{\Omega }}_{i,h}}uvdx={\int }_{{\mathrm{\Omega }}_{i,-h}}uvdx,$ in particular if $uv$ vanishes almost everywhere outside ${\mathrm{\Omega }}_{i,h}\cap {\mathrm{\Omega }}_{i,-h}$ — for instance when $uv$ has compact support in $\mathrm{\Omega }$ and $\mathrm{\mid }h\mathrm{\mid }<\mathrm{dist}(\mathrm{supp}(uv),\mathrm{\partial }\mathrm{\Omega })$ — or if ${\mathrm{\Omega }}_{i,h}={\mathrm{\Omega }}_{i,-h}$, then ${\int }_{{\mathrm{\Omega }}_{i,h}}{\delta }_h^{i}uvdx=-{\int }_{{\mathrm{\Omega }}_{i,-h}}u{\delta }_{-h}^{i}vdx$, with the integrals taken over their respective shrunken sets. For $\mathrm{\Omega }={\mathbb{R}}^n$ and $u\in L^p(\mathrm{\Omega })$, $v\in L^{p^{\mathrm{\prime }}}(\mathrm{\Omega })$ with $1\le p\le \mathrm{\infty }$ and $p^{\mathrm{\prime }}$ the Hölder conjugate exponent of Conjugate exponents, including the endpoint conventions ($1\mathrm{/}p+1\mathrm{/}{p}^{\mathrm{\prime }}=1$), both sides are absolutely convergent and ${\int }_{{\mathbb{R}}^n}{\delta }_h^{i}uvdx=-{\int }_{{\mathbb{R}}^n}u{\delta }_{-h}^{i}vdx\mathrm{.}$

(ii) Product rule. If $uv\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Omega })$, then a.e. on ${\mathrm{\Omega }}_{i,h}$, ${\delta }_h^i(uv)=\left({\tau }_{-h{e}_i}u\right){\delta }_h^{i}v+\left({\delta }_h^{i}u\right)v=u{\delta }_h^{i}v+\left({\delta }_h^{i}u\right)\left({\tau }_{-h{e}_i}v\right),$ with the translation ${\tau }_{-h{e}_i}$ of Translation of a function on ${\mathbb{R}}^n$. The identity is a pointwise a.e. algebraic identity; no local integrability of its shifted cross-products is asserted beyond the hypothesis $uv\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$, which makes the left side well defined.

(iii) Commutation with weak derivatives. If $u$ and all its weak derivatives $D^{\gamma }u$ with $\mathrm{\mid }\gamma \mathrm{\mid }\le \mathrm{\mid }\alpha \mathrm{\mid }$ have locally integrable representatives on $\mathrm{\Omega }$ (Weak derivative of a locally integrable function), then $D^{\alpha }({\delta }_h^{i}u)={\delta }_h^i(D^{\alpha }u)\text{weakly on }{\mathrm{\Omega }}_{i,h}\mathrm{.}$

All identities are identities of almost-everywhere classes (The space $L^p(\mu )$ as the quotient by null functions), and the proof uses no choice beyond the Sobolev interfaces.

Hunter's Proposition 4.52 (printed pp. 124-125) states the three properties on ${\mathbb{R}}^n$ (his parts (1)-(3)) with the forward-value notation $u_i^h(x)=u(x+h{e}_i)$; the two-domain correction term in (i) is the additional bookkeeping needed to read the identity on an arbitrary open set, and the published change-of-variables corollary supplies the substitution. Laugesen's identity (5.6) and the surrounding remarks (printed pp. 108-110) record the same calculus in the localized form used in the interior estimate. The scaffold's second product rule equality "${\delta }_h^i(uv)=({\delta }_h^{i}u)v+u({\delta }_h^{i}v)$" was repaired to the two correct shifted forms above; the counterpart repair for the cutoff commutator is carried out in The cutoff difference-quotient commutator estimate.

Assume Countable Choice. Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be open, $1\le p\le \mathrm{\infty }$, $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$, $u\in W^{1,p}(\mathrm{\Omega };\mathbb{K})$ and $\eta \in C_c^{\mathrm{\infty }}({\mathbb{R}}^n;\mathbb{R})$ with $M_{\eta }:=\mathrm{\parallel }D\eta {\mathrm{\parallel }}_{L^{\mathrm{\infty }}({\mathbb{R}}^n)}$ and compact support; the case $\eta \in C_c^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{R})$ is read by extending $\eta$ by zero. Then for every $i\in \{1,\dots ,n\}$ and every $h\mathrm{\ne }0$, on the shrunken set ${\mathrm{\Omega }}_{i,h}=\{x\in \mathrm{\Omega }:x+h{e}_i\in \mathrm{\Omega }\}$ of Difference quotients on a shrunken domain one has the exact identity ${\delta }_h^i(\eta u)-\eta {\delta }_h^{i}u=({\delta }_h^i\eta )({\tau }_{-h{e}_i}u)\text{a.e. on }{\mathrm{\Omega }}_{i,h},$ where $\tau$ is the published translation ${\tau }_{h}u(x)=u(x-h)$ of Translation of a function on ${\mathbb{R}}^n$, so that $({\tau }_{-h{e}_i}u)(x)=u(x+h{e}_i)$. Consequently $\mathrm{\parallel }{\delta }_h^i(\eta u)-\eta {\delta }_h^{i}u{\mathrm{\parallel }}_{L^p({\mathrm{\Omega }}_{i,h})}\le M_{\eta }\mathrm{\parallel }{\tau }_{-h{e}_i}u{\mathrm{\parallel }}_{L^p({\mathrm{\Omega }}_{i,h})}\le M_{\eta }\mathrm{\parallel }u{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })},$ and more generally, on the domain where both sides are defined, ${\delta }_h^i\left({\eta }^2{\delta }_h^{i}u\right)={\eta }^2{\delta }_h^i{\delta }_h^{i}u+({\delta }_h^i{\eta }^2)({\tau }_{-h{e}_i}{\delta }_h^{i}u)\mathrm{.}$ The statement is quantitative in $\mathrm{\parallel }D\eta {\mathrm{\parallel }}_{L^{\mathrm{\infty }}}$ and does not assume any regularity of $u$ beyond $W^{1,p}$.

Sobolev multiplier and support facts used below. For every integer $m\ge 0$, $q\in W^{m,\mathrm{\infty }}(U)$ and $z\in H^m(U)$ on an open set $U$, $qz\in H^m(U)$ and $D^{\alpha }(qz)={\sum }_{\beta \le \alpha }\left(\genfrac{}{}{0px}{}{\alpha }{\beta }\right)(D^{\beta }q)D^{\alpha -\beta }z,\mathrm{\mid }\alpha \mathrm{\mid }\le m,$ with $\mathrm{\parallel }qz{\mathrm{\parallel }}_{H^m(U)}\le C(n,m)\mathrm{\parallel }q{\mathrm{\parallel }}_{W^{m,\mathrm{\infty }}(U)}\mathrm{\parallel }z{\mathrm{\parallel }}_{H^m(U)}$. Also, every compactly supported $z\in H^m(U)$ lies in $H_0^m(U)$.

Hunter's proof of Theorem 4.27 (printed pp. 112-113) isolates exactly these commutator terms in the localisation step; Simon's Lecture 6 (printed pp. 60-62) records the product rule and the elementary properties of the difference operators in the same form. The scaffold wrote the identity with the opposite sign of the shift, $({\delta }_h^i\eta )({\tau }_{h{e}_i}u)$; with the published translation convention ${\tau }_{h}u(x)=u(x-h)$ the correct factor is $({\tau }_{-h{e}_i}u)(x)=u(x+h{e}_i)$, as stated and proved above.

Assume Countable Choice. Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be open, $U\subseteq \mathrm{\Omega }$ open, $1<p<\mathrm{\infty }$, $u\in L^p(\mathrm{\Omega };\mathbb{K})$, $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$, and fix a coordinate $i$. Suppose $h_0>0$ and $U\subseteq {\mathrm{\Omega }}_{i,h}$ for every $0<\mathrm{\mid }h\mathrm{\mid }<h_0$, and $\mathrm{\parallel }{\delta }_h^{i}u{\mathrm{\parallel }}_{L^p(U)}\le C(0<\mathrm{\mid }h\mathrm{\mid }<h_0)\mathrm{.}$ Then $D_{i}u\in L^p(U)$ with norm at most $C$, and ${\delta }_h^{i}u\rightharpoonup D_{i}u$ in $L^p(U)$ as $h\to 0$. No compact containment of $U$ and no bounds in other directions are required. In particular this applies to tangential quotients on boundary half-balls. For $U⋐\mathrm{\Omega }$ the shift condition holds whenever $h_0<\mathrm{dist}(U,\mathrm{\partial }\mathrm{\Omega })$; if bounds hold in every coordinate, $u\in W^{1,p}(U)$. Only Countable Choice is used.