Rellich-Kondrachov 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.

On ${\mathbb{R}}^n$, choose a nonzero smooth compactly supported function $\varphi$ and centers $a_j$ far enough apart that the translated supports are disjoint. The translations of a function $u_j(x)=\varphi (x-a_j)$ have equal $W^{1,p}$ norms, but for $1\le p<\infty$,Consequently this bounded sequence has no strongly $L^p$ convergent subsequence, and the corresponding inclusion from the Sobolev space is not compact.

The same construction works in a domain containing disjoint translated copies of one fixed ball. Mere unboundedness of an arbitrary domain does not guarantee room for those copies. This example demonstrates the failure of a general unbounded-domain compactness claim; it does not assert noncompactness for every possible unbounded geometry.

This is the zero-boundary special case of the Rellich-Kondrachov theorem. 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.

Suppose $u_j\to u$ strongly in $L^2(B_R)$ for every $R>0$, and assume uniform tail controlThen $u\in L^2({\mathbb{R}}^n)$ and $u_j\to u$ strongly in L2 space on the whole space. Local convergence and the tail bound first bound the global L2 norms; Fatou's lemma gives the same tail bound for $u$. Splitting the error givesChoose $R$ to make the last two terms uniformly small, then increase $j$. Local Rellich-Kondrachov compactness theorem arguments and a diagonal subsequence supply the first hypothesis in many Sobolev space problems; the tail condition rules out loss of compactness at infinity.

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.