H1 norm (source code)
= H1 norm
{title2=$\|\cdot\|_{H^1}$}
The inhomogeneous $H^1$ norm on a domain $D\subseteq\mathbb R^n$ is
$$
\|u\|_{H^1(D)}=\left(\int_D|u|^2+\sum_{j=1}^n|\partial_ju|^2\,dx\right)^{1/2},
$$
where the derivatives are <weak derivatives>. It measures both the size of the function and its first-order variation. The underlying <Sobolev space> consists of the $L^2$ functions whose first weak derivatives also lie in $L^2$.
For $u(x)=x$ on $(0,1)$, the squared norm is $1/3+1=4/3$, so the norm is $2/\sqrt3$. A nonzero constant has zero derivative but positive $H^1$ norm. Thus the gradient alone defines a seminorm on the full space, not the displayed norm.
On a suitable zero-boundary space over a bounded domain, a <Poincaré inequality> controls the $L^2$ term by the gradient and makes the gradient norm equivalent to the full norm. The boundary or normalization condition is essential for this replacement.
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