Rellich-Kondrachov compactness theorem

On a bounded Lipschitz domain $\Omega \subset {\mathbb{R}}^d$, the Sobolev embedding $W^{1,p}(\Omega )\hookrightarrow L^q(\Omega )$ is compact whenever $q$ lies strictly below the critical Sobolev exponent.

Translations of one nonzero compactly supported smooth function have equal $W^{1,p}$ norms. On an unbounded domain they can have pairwise disjoint supports, so their mutual $L^p$ distances stay fixed and no subsequence converges in $L^p$.

For every bounded open $\Omega \subset {\mathbb{R}}^d$, the inclusion $H_0^1(\Omega )\hookrightarrow L^2(\Omega )$ is compact.

By the definition of $H_0^1(\Omega )$ as the closure of compactly supported smooth functions, extension by zero maps it continuously into $H^1({\mathbb{R}}^d)$ without creating a boundary distribution.

For $u\in H^1({\mathbb{R}}^d)$, Plancherel gives

Weak convergence on a bounded domain gives pointwise convergence of Fourier transforms and dominated convergence on bounded frequency balls. A uniform derivative bound controls the complementary high-frequency tails.

On bounded $\Omega$ with $V\in L^{\infty }(\Omega )$, weak convergence in $H_0^1(\Omega )$ makes the Dirichlet term lower semicontinuous and, by Rellich compactness, makes the potential term $\int V{u}^2$ continuous. Thusis weakly lower semicontinuous.

The infimum of $E(u)$ over $\Vert u{\Vert }_2=1$ is attained. A minimizing sequence is bounded in $H_0^1$ because $V$ is bounded below; weak compactness, Rellich strong $L^2$ convergence, and weak lower semicontinuity complete the direct-method argument.

Strong convergence on every bounded region combines with a tail estimate uniform in the sequence to give global strong convergence.

If local compactness makes every subsequential local limit agree with the global weak limit and the $L^2$ mass is uniformly tight, weak convergence upgrades to strong convergence.