Noncompactness of a Sobolev embedding by translation

Translations of one compactly supported bump have equal Sobolev norms. Widely separated translates have disjoint supports and remain a fixed positive distance apart in $L^2$, obstructing compactness on an unbounded domain.

A bounded sequence can fail to have a strongly convergent subsequence because its mass escapes to infinity even when its local regularity is uniformly controlled.

For $\varphi \in H^1(\mathbb{R})$ with $\Vert \varphi {\Vert }_2=1$, the dilationpreserves the $L^2$ norm and satisfies $\Vert u_R^{\prime }{\Vert }_2^2=R^{-2}\Vert {\varphi }^{\prime }{\Vert }_2^2$. Hence normalized functions on the line have Dirichlet-energy infimum zero, which no nonzero $L^2$ function attains.