Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 105 / 1 / d / Solution

The Rellich-Kondrachov compactness theorem states in particular that for a bounded open set $U\subset {\mathbb{R}}^n$ with smooth boundary, the embeddingis compact. More generally it is compact into $L^q(U)$ for $1\le q<2n/(n-2)$ when $n\ge 3$, for every finite $q$ when $n=2$, and into $C^0(\overline{U})$ in dimension one.

Boundedness is essential. Choose a nonzero $\phi \in C_c^{\infty }({\mathbb{R}}^n)$ and set $u_k(x)=\phi (x-k{e}_1)$ with pairwise disjoint supports. Their $H^1({\mathbb{R}}^n)$ norms are equal, whilefor $j\ne k$. Thus no subsequence converges in $L^2$.