Integer-order Sobolev spaces and their norms

• Juha Kinnunen, Sobolev Spaces, Chapter 1 §1.2, Definition 1.8 and the accompanying norm and local-space conventions, printed pp. 4–5. The source gives the finite-$p$ sum, the $p=\mathrm{\infty }$ sum and its equivalent maximum, and the convention $D^{0}u=u$. Its printed description of $U⋐\mathrm{\Omega }$ says that $U$ is compactly contained; this item uses the standard explicit form that $U$ is open and $\bar{U}$ is compact in $\mathrm{\Omega }$.
• John K. Hunter, Notes on Partial Differential Equations, Chapter 3, §§3.1–3.5, for the Sobolev-space conventions recorded in PDE-11.

Assume Countable Choice. Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be open with $n\ge 1$, $k\in {\mathbb{N}}_0$, $1\le p\le \mathrm{\infty }$, and $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$. Write ${\mathcal{A}}_k=\{\alpha \in {\mathbb{N}}_0^n:\mathrm{\mid }\alpha \mathrm{\mid }\le k\}\mathrm{.}$ This is a finite nonempty set: every coordinate of such an $\alpha$ lies in $\{0,\dots ,k\}$, and it contains the zero multi-index.

The real and complex $L^p$ classes use the conventions in The space $L^p(\mu )$ as the quotient by null functions and Complex Lp classes and Euclidean test-function conventions. Their quotient norm values are supplied by The $L^p$ norm descends to the quotient and makes $L^p$ a normed space for $1\le p\le \mathrm{\infty }$ for real scalars and Complex Holder, Minkowski, and the quotient norm for complex scalars.

Define $W^{k,p}(\mathrm{\Omega };\mathbb{K})$ to be the set of classes $u\in L^p(\mathrm{\Omega };\mathbb{K})$ such that, for every $\alpha \in {\mathcal{A}}_k$, there is an $L^p(\mathrm{\Omega };\mathbb{K})$ class with a locally integrable representative $v_{\alpha }$ satisfying ${\int }_{\mathrm{\Omega }}u{D}^{\alpha }\phi dx=(-1{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}{\int }_{\mathrm{\Omega }}{v}_{\alpha }\phi dx\text{for every }\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })\mathrm{.}$ Here the weak derivative is the one in Weak derivative of a locally integrable function. Under The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), local integrability of $L^p$ representatives and invariance under null-set changes follow from Weak differentiation ignores null-set changes, while Uniqueness of a weak derivative as an almost-everywhere class gives uniqueness of each locally integrable derivative class. Thus this condition is a condition on the $L^p$ class $u$, and each resulting derivative determines one $L^p$ class, denoted $D^{\alpha }u$. The test pairing is bilinear, without conjugation.

For the zero multi-index, set $D^{0}u=u$. Since ${\mathcal{A}}_0=\{0\}$, $W^{0,p}(\mathrm{\Omega };\mathbb{K})=L^p(\mathrm{\Omega };\mathbb{K})\mathrm{.}$ The Sobolev norm expression is $\mathrm{\parallel }u{\mathrm{\parallel }}_{W^{k,p}(\mathrm{\Omega })}=\{\begin{array}{ll}{\left({\displaystyle \sum _{\alpha \in {\mathcal{A}}_k}\mathrm{\parallel }{D}^{\alpha }u{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}^p}\right)}^{1\mathrm{/}p},& 1\le p<\mathrm{\infty },\\ {\displaystyle \underset{\alpha \in {\mathcal{A}}_k}{\max }\mathrm{\parallel }{D}^{\alpha }u{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\mathrm{\Omega })},}& p=\mathrm{\infty }\mathrm{.}\end{array}$ The formula is finite because each derivative class belongs to $L^p$ and ${\mathcal{A}}_k$ is finite; at $k=0$ it reduces to the $L^p$ norm expression. The finite maximum is defined because ${\mathcal{A}}_k$ is nonempty. The norm axioms for this expression are a separate assertion from this definition.

For open $U\subseteq \mathrm{\Omega }$ with $\bar{U}$ compact and contained in $\mathrm{\Omega }$, the notation $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{k,p}(\mathrm{\Omega };\mathbb{K})$ means that the restriction of the class belongs to $W^{k,p}(U;\mathbb{K})$ for every such $U$.

If $\mathrm{\Omega }=\mathrm{\varnothing }$, the $L^p$ space and every $W^{k,p}$ space contain only the zero class, and the displayed norm expression is zero.

The space $L^p(\mu )$ as the quotient by null functions

Complex Lp classes and Euclidean test-function conventions

The $L^p$ norm descends to the quotient and makes $L^p$ a normed space for $1\le p\le \mathrm{\infty }$

Complex Holder, Minkowski, and the quotient norm

Weak derivative of a locally integrable function

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Weak differentiation ignores null-set changes

Uniqueness of a weak derivative as an almost-everywhere class