The real H(div) space

Assume Countable Choice for the published Sobolev interfaces cited below. Let $D\subseteq {\mathbb{R}}^d$ be open and use real scalars. $L^2(D;{\mathbb{R}}^d)$ means the $d$-tuple of real $L^2$ equivalence classes, with norm $\mathrm{\parallel }v{\mathrm{\parallel }}_2^2={\sum }_{j=1}^d\mathrm{\parallel }{v}_j{\mathrm{\parallel }}_2^2$. The test space $C_c^{\mathrm{\infty }}(D;\mathbb{R})$ consists of real smooth functions compactly supported in $D$.

For $v\in L^2(D;{\mathbb{R}}^d)$, say that $g\in L^2(D;\mathbb{R})$ is its distributional divergence when

$${\int }_{D}v\cdot \mathrm{\nabla }\varphi dx=-{\int }_{D}g\varphi dx\text{for every }\varphi \in C_c^{\mathrm{\infty }}(D;\mathbb{R})\mathrm{.}$$

The $L^2$ class $g$ is unique when it exists. Indeed, if $g_1,g_2$ both satisfy the identity, put $h=g_1-g_2\in L^2(D)$. Every compact $C\subset D$ has finite Lebesgue measure by Lebesgue measure is sigma-finite, and every metrically bounded subset of ${\mathbb{R}}^n$ has finite outer measure, and Cauchy-Schwarz inequality for $L^2$ gives ${\int }_C\mathrm{\mid }h\mathrm{\mid }\le \mathrm{\mid }C{\mathrm{\mid }}^{1\mathrm{/}2}\mathrm{\parallel }h{\mathrm{\parallel }}_{L^2(D)}<\mathrm{\infty }$. Thus $h\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(D)$. Subtracting the two divergence identities gives ${\int }_{D}h\varphi =0$ for every real test $\varphi$; splitting a complex test into real and imaginary parts gives the same identity for complex tests. Injectivity of the regular-distribution map in Locally integrable functions embed in distributions therefore yields $h=0$ almost everywhere, hence $g_1=g_2$ as $L^2$ classes. Define

$$H(\mathrm{div};D):=\{v\in L^2(D;{\mathbb{R}}^d):\text{there is }g\in L^2(D;\mathbb{R})\text{ with }g=\mathrm{div}v\text{ distributionally}\},$$

and equip it with the graph norm

$$\mathrm{\parallel }v{\mathrm{\parallel }}_{H(\mathrm{div};D)}:=(\mathrm{\parallel }v{\mathrm{\parallel }}_2^2+\mathrm{\parallel }\mathrm{div}v{\mathrm{\parallel }}_2^2{)}^{1\mathrm{/}2}\mathrm{.}$$

The Sobolev notation $H^1(D)=W^{1,2}(D)$ and $H_0^1(D)=W_0^{1,2}(D)$ uses the norm and zero-boundary closure conventions of Integer-order Sobolev spaces and their norms and Zero-boundary Sobolev space as a norm closure. No pointwise boundary value is part of this definition.