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 displayed inner product on the first-order Sobolev space induces its usual Sobolev norm. A continuous reaction-diffusion weak formulation with unit diffusion and unit reaction has this exact bilinear form, so its conforming Galerkin method gives a best approximation with constant one in the first-order Sobolev norm.