H1 norm ($\Vert \cdot {\Vert }_{H^1}$)

The inhomogeneous $H^1$ norm on a domain $D\subseteq {\mathbb{R}}^n$ iswhere 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 $2/\sqrt{3}$. 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.