Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 105 / 2 / f / Solution

The Rellich-Kondrachov compactness theorem says that if $U\subset {\mathbb{R}}^n$ is a bounded Lipschitz domain, thenfor $1\le q<p^{\ast }=np/(n-p)$ when $p<n$. When $p=n$, the embedding is compact into every finite $L^q$, and when $p>n$ it is compact into $C^0(\overline{U})$, hence into every $L^q$.

The boundedness of the domain is essential. Choose a nonzero $\varphi \in C_c^{\infty }(0,1)$ and setTranslation invariance gives $\Vert u_j{\Vert }_{W^{1,1}({\mathbb{R}}_{+})}=\Vert \varphi {\Vert }_{W^{1,1}}$, so after a fixed rescaling these functions lie in the unit ball. Their supports are pairwise disjoint andNo subsequence is Cauchy in $L^1$, so the unit ball is not compact. This is the standard failure of Rellich compactness on an unbounded domain.