 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.

The Rellich–Kondrachov theorem is a significant result in functional analysis and the theory of differential equations, particularly in the context of Sobolev spaces. It essentially states conditions under which the embedding of Sobolev spaces into Lp spaces is compact.