Compactness of $W_0^{1,p}(\mathrm{\Omega })\hookrightarrow L^p(\mathrm{\Omega })$ on bounded open sets

Given: the Axiom of Choice, $n\ge 1$, a bounded open $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$, $1\le p<\mathrm{\infty }$, and a sequence $(u_j)$ with $M:={\sup }_j\mathrm{\parallel }{u}_j{\mathrm{\parallel }}_{W_0^{1,p}(\mathrm{\Omega })}<\mathrm{\infty }$.

Zero extension. For each $j$ the zero extension ${\tilde{u}}_j$ of $u_j$ lies in $W^{1,p}({\mathbb{R}}^n)$ with $D_i{\tilde{u}}_j$ the zero extension of $D_i{u}_j$ and $\mathrm{\parallel }{\tilde{u}}_j{\mathrm{\parallel }}_{W^{1,p}({\mathbb{R}}^n)}=\mathrm{\parallel }{u}_j{\mathrm{\parallel }}_{W^{1,p}(\mathrm{\Omega })}$; the extension vanishes outside $\bar{\mathrm{\Omega }}$. (Zero extension of W_0^{1,p} has no boundary derivative, Zero-boundary Sobolev space as a norm closure, Integer-order Sobolev spaces and their norms)

Translation estimate. $\mathrm{\parallel }{\tau }_h{\tilde{u}}_j-{\tilde{u}}_j{\mathrm{\parallel }}_{L^p({\mathbb{R}}^n)}\le \mathrm{\mid }h\mathrm{\mid }\mathrm{\parallel }D{\tilde{u}}_j{\mathrm{\parallel }}_{L^p({\mathbb{R}}^n)}$, under the Axiom of Choice. (The translation estimate for $W^{1,p}$ functions on ${\mathbb{R}}^n$)

Automatic tails. A family in $L^p({\mathbb{R}}^n)$ whose members all vanish almost everywhere outside the fixed bounded set $\bar{\mathrm{\Omega }}$ satisfies the tightness condition of the Fr'echet--Kolmogorov criterion. (Uniformly supported families have vanishing tails, Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space)

The Fr'echet--Kolmogorov criterion. A bounded family in $L^p({\mathbb{R}}^n)$ with vanishing tails and uniform translation control is totally bounded, its closure is compact, and every sequence in it has an $L^p({\mathbb{R}}^n)$-convergent subsequence. (The Fr'echet--Kolmogorov compactness criterion in $L^p({\mathbb{R}}^n)$, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain)

Restriction is contractive. $\mathrm{\parallel }u{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}\le \mathrm{\parallel }u{\mathrm{\parallel }}_{W_0^{1,p}(\mathrm{\Omega })}$, and the $L^p(\mathrm{\Omega })$ norm of the restriction never exceeds the $L^p({\mathbb{R}}^n)$ norm of an extension. (Integer-order Sobolev spaces and their norms, The space $L^p(\mu )$ as the quotient by null functions)