Sobolev norm ($\Vert u{\Vert }_{W^{k,p}(D)}$)

For $1\le p<\infty$, an inhomogeneous Sobolev space norm is $\Vert u{\Vert }_{W^{k,p}(D)}=({\sum }_{|\alpha |\le k}\Vert D^{\alpha }u{\Vert }_{L^p(D)}^p{)}^{1/p}$, using weak derivatives. In particular, $\Vert u{\Vert }_{H^1(D)}^2={\int }_D(|u{|}^2+|\nabla u{|}^2)$. The gradient seminorm alone defines a different completion, the Dirichlet energy space, unless a suitable Poincaré inequality makes the two norms equivalent on the chosen zero-boundary space.

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.

For real-valued functions, the inner producton the H1 space induces its usual Sobolev norm. For complex-valued functions, use $u\overline{v}+\nabla u\cdot \overline{\nabla v}$ under the convention that the inner product is linear in its first argument. Omitting conjugation in the complex setting would fail positivity, for example on a purely imaginary constant function.

A real reaction-diffusion weak formulation with unit diffusion and unit reaction has this exact bilinear form. Its conforming Galerkin method is therefore an orthogonal projection and gives a best approximation with constant one in the first-order Sobolev norm.