Rellich–Kondrachov theorem

Remarks

The inclusion is bounded by the Sobolev inequality. For compactness, one mollifies: the mollified functions $u^\varepsilon$ form an equicontinuous and bounded family for each fixed $\varepsilon$, so the Arzelà–Ascoli theorem applies to them. The error $\norm{u^\varepsilon - u}_{L^1}$ is at most $\varepsilon \norm{\nabla u}_{L^1}$, uniformly on bounded sets, and interpolation with the bound in $L^{p^*}$ gives the same uniform control in $L^q$ for $q < p^*$. So the inclusion is a compact operator.

The hypotheses are needed. The inclusion into $L^{p^*}(U)$ itself is not compact: the rescaled functions $\lambda^{(n-p)/p} \varphi(\lambda x)$ for a fixed bump $\varphi$ are bounded in $W^{1,p}$ but concentrate at a point as $\lambda \to \infty$. On an unbounded set, such as $U = \R^n$, translates $\varphi(x - k e_1)$ of a bump are bounded in $W^{1,p}$ but have no convergent subsequence in $L^p$.

Going deeper

For a student

Compactness is how one passes to the limit in approximations of nonlinear equations: a weak limit in $W^{1,p}$ becomes a strong limit in $L^q$, so nonlinear terms in $u$ converge. It also implies that the inverse of an elliptic operator such as $-\Delta$ with Dirichlet conditions is compact on $L^2(U)$, which gives a discrete spectrum of eigenvalues. The Poincaré inequality for functions with zero mean is proved from it by contradiction.